tag:blogger.com,1999:blog-51220590948595862672024-03-05T16:09:03.096-08:00FASCINATING FACTS OF MATHEMATICS"LIKE THE CREST OF A PEACOCK , SO IS MATHEMATICS AT THE HEAD OF ALL KNOWLEDGE"Malini Lingarajaprabhuhttp://www.blogger.com/profile/02400048832214425554noreply@blogger.comBlogger59125tag:blogger.com,1999:blog-5122059094859586267.post-30764357098308070732021-06-17T05:09:00.003-07:002021-06-17T05:09:44.648-07:00FACTS OF INTEGRATIONhttps://www.blogger.com/blog/post/edit/1704152347685259977/8341992096732285541Malini Lingarajaprabhuhttp://www.blogger.com/profile/02400048832214425554noreply@blogger.com0tag:blogger.com,1999:blog-5122059094859586267.post-46438470576791515982011-11-04T11:31:00.000-07:002011-11-04T11:42:07.137-07:00SIMPLE PUZZLES<div dir="ltr" style="text-align: left;" trbidi="on"><div><div align="center"><span style="color: #990000; font-family: courier new; font-size: 180%;"><b>PUZZLE 15</b></span></div><br />
<div style="color: blue; font-family: Arial,Helvetica,sans-serif;"><span style="font-size: large;">A two-digit number, read from left to right, is 4.5 times as large as the same number read from right to left. What is the number?</span></div><br />
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<div align="center"><span style="color: #990000; font-family: courier new; font-size: 180%;"><b>PUZZLE 16</b></span></div><br />
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<div style="color: blue; font-family: Arial,Helvetica,sans-serif;"><span style="font-size: large;">Some ducks are marching across a path. There's a duck in front of two ducks, there's a duck behind two ducks, and there's a duck in the middle of two ducks.</span></div><div style="color: blue; font-family: Arial,Helvetica,sans-serif;"><br />
</div><span style="color: blue; font-size: large;"><span style="font-family: Arial,Helvetica,sans-serif;"> </span></span><br />
<div style="color: blue; font-family: Arial,Helvetica,sans-serif;"><span style="font-size: large;">What's the least number of ducks that there could have been?</span></div><br />
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<div align="center"><b><span style="color: black;"><span style="font-family: verdana;"><span style="font-size: 180%;">SOLU</span><span style="font-size: 180%;">TIONS</span></span></span></b></div><br />
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<b><span style="color: #000099;"><span style="color: #003300;"><span style="font-size: 130%;"><span style="font-family: arial;"><span style="color: #000099;">PUZZLE 15 :</span> <span style="color: #990000;">18 and 81</span></span></span></span></span></b> <br />
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<span style="font-size: 130%;"><span style="color: #990000; font-family: arial;"><b><span style="color: #000099;">PUZZLE 16 :</span> 3 Ducks.</b></span></span><br />
<a href="http://www.blogger.com/goog_710321125"></a><br />
<div style="color: blue;"><b><a href="http://www.blogger.com/goog_710321125"><br />
</a></b></div><a href="http://malini-math.blogspot.com/2010/11/chocolate-puzzle.html" target="_blank"><span style="color: #0c343d; font-family: Arial,Helvetica,sans-serif; font-size: x-large;"><b>FOR MORE PUZZLES CLICK HERE</b></span></a></div></div>Malini Lingarajaprabhuhttp://www.blogger.com/profile/02400048832214425554noreply@blogger.com0tag:blogger.com,1999:blog-5122059094859586267.post-10054300547156064852011-11-02T06:33:00.000-07:002011-11-02T06:33:10.998-07:00LAPLACE TRANSFORM<div dir="ltr" style="text-align: left;" trbidi="on"><div style="font-family: Times,"Times New Roman",serif;"></div><div align="left" class="MsoNormal" style="font-family: Times,"Times New Roman",serif; text-align: left;"><span style="font-size: x-large;"><b><span style="letter-spacing: -0.75pt;"><span style="color: purple;">Introduction: </span></span></b></span></div><ul style="font-family: Times,"Times New Roman",serif; text-align: left;"><li><span style="font-size: large;"><b><span style="letter-spacing: -0.75pt;"> </span></b></span><span style="font-size: large;">The Laplace transform is named in honor of <span style="color: windowtext; text-decoration: none;">mathematician</span> and <span style="color: windowtext; text-decoration: none;">astronomer </span><span style="color: windowtext; text-decoration: none;">Pierre-Simon</span> <span style="color: windowtext; text-decoration: none;">Laplace</span>, who used the transform in his work on<span style="color: windowtext; text-decoration: none;"> probability theory. </span></span></li>
</ul><ul style="font-family: Times,"Times New Roman",serif; text-align: left;"><li><span style="font-size: large;">Like the Fourier transform, the Laplace transform is used for solving differential and integral equations.</span></li>
</ul><ul style="font-family: Times,"Times New Roman",serif; text-align: left;"><li><span style="font-size: large;"><span style="letter-spacing: -0.75pt;">Laplace transform </span>is a widely used integral transform. </span></li>
</ul><div style="font-family: Times,"Times New Roman",serif;"><ul style="text-align: left;"><li><span style="font-size: large;"><b><span style="font-weight: normal;">Laplace transform is just a shortcut for complex calculations</span></b>.</span></li>
</ul><span style="font-size: large;"><b> </b></span> <br />
<div align="left" class="MsoNormal" style="color: purple; text-align: left;"><span style="font-size: x-large;"><b> Real Life Applications:</b></span></div><ul style="text-align: left;"><li><!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w:TrackMoves/> <w:TrackFormatting/> <w:PunctuationKerning/> <w:ValidateAgainstSchemas/> <w:SaveIfXMLInvalid>false</w:SaveIfXMLInvalid> <w:IgnoreMixedContent>false</w:IgnoreMixedContent> <w:AlwaysShowPlaceholderText>false</w:AlwaysShowPlaceholderText> <w:DoNotPromoteQF/> <w:LidThemeOther>EN-US</w:LidThemeOther> <w:LidThemeAsian>X-NONE</w:LidThemeAsian> <w:LidThemeComplexScript>X-NONE</w:LidThemeComplexScript> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> <w:DontGrowAutofit/> <w:SplitPgBreakAndParaMark/> <w:DontVertAlignCellWithSp/> <w:DontBreakConstrainedForcedTables/> <w:DontVertAlignInTxbx/> <w:Word11KerningPairs/> <w:CachedColBalance/> <w:UseFELayout/> </w:Compatibility> <w:BrowserLevel>MicrosoftInternetExplorer4</w:BrowserLevel> <m:mathPr> <m:mathFont m:val="Cambria Math"/> <m:brkBin m:val="before"/> <m:brkBinSub m:val="--"/> <m:smallFrac m:val="off"/> <m:dispDef/> <m:lMargin m:val="0"/> <m:rMargin m:val="0"/> <m:defJc m:val="centerGroup"/> <m:wrapIndent m:val="1440"/> <m:intLim m:val="subSup"/> <m:naryLim m:val="undOvr"/> </m:mathPr></w:WordDocument> </xml><![endif]--><!--[if gte mso 9]><xml> <w:LatentStyles DefLockedState="false" DefUnhideWhenUsed="true"
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</style> <![endif]--> <div class="MsoListParagraph" style="text-align: left; text-indent: -0.25in;"><span style="font-size: large;"><span><span><span style="-moz-font-feature-settings: normal; -moz-font-language-override: normal; font-size-adjust: none; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal;"> </span></span></span></span><span style="font-size: large;">The Laplace transform turns a complicated nth order differential equation to a corresponding nth degree polynomial.</span></div><div class="MsoListParagraph" style="text-align: left; text-indent: -0.25in;"><br />
</div></li>
<li><span style="font-size: large;">In physics and engineering<span style="background: none repeat scroll 0% 0% rgb(255, 255, 255);">, it is used for analysis of linear time-invariant systems such as electrical circuits</span><span style="background-color: white;">, harmonic oscillators, optical devices, and mechanical </span>systems.</span></li>
</ul><ul style="text-align: left;"><li><span style="font-size: large;">The Laplace transform is one of the most important equations in digital signal processing and electronics. </span></li>
</ul><ul style="text-align: left;"><li><span style="font-size: large;"><span style="-moz-font-feature-settings: normal; -moz-font-language-override: normal; font-size-adjust: none; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal;"> </span>In <span style="color: windowtext; text-decoration: none;">Nuclear physics</span>, Laplace transform is used to get the correct form for radioactive decay.</span></li>
</ul><ul style="text-align: left;"><li><span style="font-size: large;">It has also been applied to the economic and managerial problems, and most recently, to <a href="http://en.wikipedia.org/wiki/Material_requirements_planning" target="_blank">Materials Requirement Planning</a> (MRP)<span style="line-height: 115%;"> </span></span></li>
</ul><span style="font-size: large;"><span style="line-height: 115%;"> </span></span></div><div style="font-family: Times,"Times New Roman",serif;"><ul style="text-align: left;"><li><span style="font-size: large;"><span style="line-height: 115%;">The Laplace transform reduces a linear differential equation to an algebraic equation, which can then be solved by the formal rules of algebra.</span></span><span style="font-size: x-large;"><a href="http://malini-math.blogspot.com/2009/07/laplace-transform-is-simple-way-of.html" target="_blank"><b>FOR MORE APPLICATIONS CLICK HERE</b></a></span></li>
</ul></div></div>Malini Lingarajaprabhuhttp://www.blogger.com/profile/02400048832214425554noreply@blogger.com3tag:blogger.com,1999:blog-5122059094859586267.post-28564115589503583902011-10-14T07:07:00.000-07:002011-10-14T07:07:50.719-07:00Conformal Mapping<div dir="ltr" style="text-align: left;" trbidi="on"><br />
<strong><span style="color: blue; font-size: large;">Geometric interpretation of a complex function. </span></strong><br />
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</span><br />
<span style="font-size: large;">If D is the domain of real-valued functions and u(x,y) and v(x,y) then the system of equations <strong><span style="color: purple;">u = u(x,y)</span></strong> and <span style="color: purple;"><strong>v = v(x,y)</strong></span> describes a <span style="color: #4c1130;"><strong>transformation (or mapping)</strong></span> from the x y - plane into the u v -plane, also called the w-plane.</span><br />
<br />
<span style="font-size: large;">Therefore, we consider the function <strong><span style="color: purple;">w= f(z) = u(x,y) + i v (x,y) </span></strong></span><br />
<span style="font-size: large;">to be a transformation (or mapping) from the set D in the z-plane onto the range R in the w-plane.</span><br />
<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhwIqte7i0dSSTGkXGmSse1HGZLtB4bKexUFqbP5-m9gek_Mnq7v4DaoZ3DNm9SXA3nFCftbxDDmli7cpn6-VkQ_m9LWddI1sjE-cflI-_y4HZPCfl4rW3A1PhoPLzF2GSddak95__nDwk/s1600/Untitled.png" imageanchor="1" style="clear: left; cssfloat: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="299" oda="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhwIqte7i0dSSTGkXGmSse1HGZLtB4bKexUFqbP5-m9gek_Mnq7v4DaoZ3DNm9SXA3nFCftbxDDmli7cpn6-VkQ_m9LWddI1sjE-cflI-_y4HZPCfl4rW3A1PhoPLzF2GSddak95__nDwk/s640/Untitled.png" width="640" /></a></div><span style="color: blue; font-size: large;"><strong>Conformal Mapping:</strong></span><br />
<span style="font-size: large;"><br />
</span><span style="font-size: large;">A function f: C → C is conformal at a point z₀ if and only if it is holomorphic and its derivative is everywhere non-zero on C.</span><br />
<span style="font-size: large;"><br />
</span><br />
<span style="font-size: large;">i.e., if f is analytic at z₀ and f’(z₀) ≠ 0</span><br />
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</span><br />
<span style="color: blue; font-size: large;"><strong>Isogonal Mapping:</strong></span><br />
<span style="font-size: large;"><br />
</span><br />
<span style="font-size: large;">An isogonal mapping is a transformation w = f (z) that preserves the magnitudes of local angles, but not their orientation.</span><br />
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</span><br />
<span style="color: blue; font-size: large;"><strong>Standard Transformations:</strong></span><br />
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</span><br />
<span style="font-size: large;"><span style="color: purple;">• Translation</span> </span><br />
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</span><br />
<div><blockquote><span style="font-size: large;">- Maps of the form z → z + k, where k є C</span></blockquote></div><span style="font-size: large;"><br />
</span><br />
<span style="color: purple; font-size: large;">• Magnification and rotation</span><br />
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</span><br />
<div><blockquote><span style="font-size: large;">- Maps of the form z → k z , where k є C</span></blockquote></div><span style="color: purple; font-size: large;">• Inversion</span><br />
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</span><br />
<div><blockquote><span style="font-size: large;">- Maps of the form z → 1 / z </span></blockquote></div><span style="font-size: large;"><br />
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<div align="left"><span style="font-size: large;"></span></div></div>Malini Lingarajaprabhuhttp://www.blogger.com/profile/02400048832214425554noreply@blogger.com0tag:blogger.com,1999:blog-5122059094859586267.post-89711868694119001802011-10-14T06:14:00.000-07:002011-10-14T06:37:29.118-07:00Transformation w = a z<div dir="ltr" style="text-align: left;" trbidi="on"><span style="font-size: large;">Let w = a z, where a ≠ 0</span><br />
<span style="font-size: large;"><br />
</span><br />
<span style="font-size: large;">If a = │a│ e^(i α) and, z = │z │ e^(i θ), then </span><br />
<span style="font-size: large;"><br />
</span><br />
<strong><span style="font-size: large;">w = │a│ │z│ e^i(θ + α ) </span></strong><br />
<span style="font-size: large;"><br />
</span><br />
<span style="font-size: large;">The image of z is obtained by rotating the vector z through the angle α and magnifying or contracting the length of z by the factor │a│.</span><br />
<span style="font-size: large;"><br />
</span><br />
<span style="font-size: large;">Thus the transformation w = a z is referred to as a<strong> <span style="color: #4c1130;">rotation</span></strong> or <span style="color: #4c1130;"><strong>magnification</strong>.</span></span><br />
<span style="color: #4c1130;"><br />
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<div class="separator" style="clear: both; text-align: center;"><span style="font-size: large;"></span></div><a href="http://www.blogger.com/goog_1072551389"></a> <br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-DqqmGzzKYl0/Tpg3tFAfeTI/AAAAAAAAC1Q/uw8wVb7bdig/s1600/w%253D+cz.png" imageanchor="1" style="clear: left; cssfloat: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="302" oda="true" src="http://4.bp.blogspot.com/-DqqmGzzKYl0/Tpg3tFAfeTI/AAAAAAAAC1Q/uw8wVb7bdig/s640/w%253D+cz.png" width="640" /></a></div><span style="color: blue; font-size: large;"><strong>Example 1:</strong></span><br />
<span style="font-size: large;"><br />
</span><br />
<span style="font-size: large;">Find the image of the region y > 1 under the map w = ( 1 – i ) z</span><br />
<span style="font-size: large;"><br />
</span><br />
<span style="color: blue; font-size: large;"><strong>Solution:</strong></span><br />
<span style="font-size: large;"><br />
</span><br />
<span style="font-size: large;">Let w = u + i v ; z = x + i y </span><br />
<span style="font-size: large;"><br />
</span><br />
<span style="font-size: large;">Given w = ( 1 – i ) z</span><br />
<br />
<span style="font-size: large;">i.e., z = 1/2 ( 1 + i) w [ since ( 1 – i ) ( 1 + i) = 2]</span><br />
<span style="font-size: large;"><br />
</span><br />
<span style="font-size: large;">i.e., x + i y = 1/2 ( 1 + i) (u + i v )</span><br />
<br />
<span style="font-size: large;">i.e., x = (u- v )/2 ; y = (u+v)/2</span><br />
<span style="font-size: large;"><br />
</span><br />
<span style="font-size: large;">Hence the region y >1 is mapped on the region u + v > 2 in w –plane.</span><br />
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<div class="separator" style="clear: both; text-align: left;"><a href="http://3.bp.blogspot.com/-niTNhUpORHc/Tpg6BVMRahI/AAAAAAAAC1c/5b8N7jUIfiM/s1600/w+%253D+cz+1.png" imageanchor="1" style="clear: left; cssfloat: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="291" oda="true" src="http://3.bp.blogspot.com/-niTNhUpORHc/Tpg6BVMRahI/AAAAAAAAC1c/5b8N7jUIfiM/s640/w+%253D+cz+1.png" width="640" /></a><span style="color: blue; font-size: large;"><strong>Example 2 :</strong></span></div><span style="font-size: large;"><br />
</span><span style="font-size: large;">Determine the region R of the w plane into which the triangular region D enclosed by the lines</span><br />
<span style="font-size: large;">x = 0, y = 0, x + y = 3 is transformed under the transformation w = 2z.</span><br />
<span style="font-size: large;"><br />
</span><br />
<strong><span style="color: blue; font-size: large;">Solution:</span></strong><br />
<span style="font-size: large;">Let w = u +i v; z = x + i y </span><br />
<br />
<span style="font-size: large;">Given, w =2 z</span><br />
<br />
<span style="font-size: large;">i.e., u +i v = 2 (x + i y)</span><br />
<br />
<span style="font-size: large;">i.e., u = 2 x ; v = 2 y</span><br />
<br />
<br />
<span style="font-size: large;">When x = 0, u = 0 </span><br />
<br />
<span style="color: #4c1130;"><span style="font-size: large;"><strong>The line x = 0 is transformed into the line u = 0 in the w – plane</strong>.</span></span><br />
<span style="color: #4c1130;"><br />
<span style="font-size: large;"></span></span><br />
<span style="font-size: large;">When y = 0, v = 0 </span><br />
<br />
<strong><span style="color: #4c1130; font-size: large;">The line y = 0 is transformed into the line v = 0 in the w – plane.</span></strong><br />
<span style="font-size: large;"><br />
</span><br />
<span style="font-size: large;">When x + y = 3 , we get</span><br />
<span style="font-size: large;"><br />
</span><br />
<span style="font-size: large;">u/2 + v/2 = 3</span><br />
<span style="font-size: large;"><br />
</span><br />
<span style="font-size: large;">i.e., u + v = 6</span><br />
<br />
<span style="color: #4c1130; font-size: large;"><strong>The line x + y = 3 is transformed into the line u + v = 6 in the w – plane.</strong></span><br />
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<div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-5mYFt6JnJjY/Tpg6yYoimbI/AAAAAAAAC1o/52WFunLGBuI/s1600/cz+2.png" imageanchor="1" style="clear: left; cssfloat: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="260" oda="true" src="http://2.bp.blogspot.com/-5mYFt6JnJjY/Tpg6yYoimbI/AAAAAAAAC1o/52WFunLGBuI/s640/cz+2.png" width="640" /></a></div><div class="separator" style="clear: both; text-align: center;"><br />
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</span></div>Malini Lingarajaprabhuhttp://www.blogger.com/profile/02400048832214425554noreply@blogger.com0tag:blogger.com,1999:blog-5122059094859586267.post-50164697439728813262011-08-10T08:34:00.000-07:002011-08-10T08:34:02.598-07:00CIRCLE AND ITS RELATED TERMS<div dir="ltr" style="text-align: left;" trbidi="on"><div style="font-family: Verdana,sans-serif;"><span style="font-size: large;"><b><strong>Circle</strong> is the set of all points in a plane that are equidistant from a given point in the plane known as the center of the circle.</b></span></div><div class="separator" style="clear: both; font-family: Verdana,sans-serif; text-align: center;"><span style="font-size: large;"><b><a href="http://4.bp.blogspot.com/-W0jK4FqatwE/TkKeTSXEQKI/AAAAAAAACZo/93HZiTMGr9w/s1600/tp2ch1_image3.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><br />
</a></b></span></div><div class="separator" style="clear: both; font-family: Verdana,sans-serif; text-align: center;"><span style="font-size: large;"><b><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgZ6chAAqP03QuZcaTaysT4Nwze1ycMhR2Faq6uoQfFdyM1epSZrAK0bwZbtYSuGgURxY2kanjToOc9jdhbeT6eDUhHhjclzgqAY2YoyXx06Ps8iu90M_6SMk779Za3IY8mjkNEe0gl_yA/s1600/220px-CIRCLE_LINES.svg.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgZ6chAAqP03QuZcaTaysT4Nwze1ycMhR2Faq6uoQfFdyM1epSZrAK0bwZbtYSuGgURxY2kanjToOc9jdhbeT6eDUhHhjclzgqAY2YoyXx06Ps8iu90M_6SMk779Za3IY8mjkNEe0gl_yA/s400/220px-CIRCLE_LINES.svg.png" width="396" /></a></b></span></div><div style="color: blue; font-family: Verdana,sans-serif;"><span style="font-size: large;"><b><br />
<strong></strong></b> <b><br />
<strong>Chord -</strong></b> <b> is a segment whose endpoints lie on a circle</b></span></div><div style="color: blue; font-family: Verdana,sans-serif;"><br />
</div><div style="color: blue; font-family: Verdana,sans-serif;"><span style="font-size: large;"><b><strong><br />
<span style="color: lime;">Secant -</span></strong><span style="color: lime;"> is a line that intersects the circle at two points</span></b></span></div><div style="color: blue; font-family: Verdana,sans-serif;"><br />
</div><div style="color: blue; font-family: Verdana,sans-serif;"><span style="font-size: large;"><b><span style="color: lime;"><span style="color: magenta;">Diameter is the longest secant in a circle.</span> </span></b></span></div><div style="color: blue; font-family: Verdana,sans-serif;"><br />
</div><div style="color: red; font-family: Verdana,sans-serif;"><span style="font-size: large;"><b><strong>Tangent -</strong> is a line in the plane of the circle that intersects the circle at exactly one point </b></span></div><div class="separator" style="clear: both; font-family: Verdana,sans-serif; text-align: center;"><span style="font-size: large;"><b><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjFLCGansLfaB6UGFjKxYsxI6iL7Lzk2YFaX2oqye312ih3TjwOCWHlkBZSz84viTQGXR6atHKtFIq2D8Wz82-lLco_L8GhYN-Ln-EvdTLwB_nRhNNEbS_lJwAYHaSmdfrGx7O_qz3WA84/s1600/figure_28.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjFLCGansLfaB6UGFjKxYsxI6iL7Lzk2YFaX2oqye312ih3TjwOCWHlkBZSz84viTQGXR6atHKtFIq2D8Wz82-lLco_L8GhYN-Ln-EvdTLwB_nRhNNEbS_lJwAYHaSmdfrGx7O_qz3WA84/s1600/figure_28.gif" /></a></b></span></div><div style="color: orange; font-family: Verdana,sans-serif;"><span style="font-size: large;"><b>Sector -The part of a circle enclosed by two radii of a circle and an arc.</b></span></div><div class="separator" style="clear: both; color: #20124d; font-family: Verdana,sans-serif; text-align: center;"><span style="font-size: large;"><b><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgPbRWkr3zbNMYVGNDKfMKY0cdetHhyphenhyphenZy0WXTHAaajMIlbMaXISWhD6sQjK30XfwM-V8VDgXyy4vQKepuw3sESP9d-dSL0vkxBO_Nu8StmKHvdkrAzif_ZAVnWn-md0A_eTTYai2oFllpM/s1600/PIZZA.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgPbRWkr3zbNMYVGNDKfMKY0cdetHhyphenhyphenZy0WXTHAaajMIlbMaXISWhD6sQjK30XfwM-V8VDgXyy4vQKepuw3sESP9d-dSL0vkxBO_Nu8StmKHvdkrAzif_ZAVnWn-md0A_eTTYai2oFllpM/s320/PIZZA.jpg" style="cursor: move;" width="320" /></a></b></span></div><div style="font-family: Verdana,sans-serif;"><span style="font-size: large;"><b style="color: #4c1130;"><span style="background-color: #fce5cd;">The cut piece of pizza is minor sector and the remaining is the major sector.</span></b></span></div><div style="font-family: Verdana,sans-serif;"><span style="font-size: large;"><b><br style="color: purple;" /></b></span></div><div style="font-family: Verdana,sans-serif;"><span style="font-size: large;"><b><br style="color: purple;" /></b></span></div><div class="separator" style="clear: both; font-family: Verdana,sans-serif; text-align: center;"><span style="font-size: large;"><b><a href="http://4.bp.blogspot.com/-06QsgyaS8GQ/TkKiPjKowSI/AAAAAAAACaA/bDerFwSjDb4/s1600/glossary_segment.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-06QsgyaS8GQ/TkKiPjKowSI/AAAAAAAACaA/bDerFwSjDb4/s1600/glossary_segment.gif" /></a></b></span></div><div style="font-family: Verdana,sans-serif;"><span style="font-size: large;"><b><br style="color: purple;" /></b></span></div><div style="font-family: Verdana,sans-serif;"><span style="font-size: large;"><b><br />
<strong></strong></b> <b><span style="color: #e06666;">Segment -part of a circle bounded by a </span><a class="glossaryLink" href="" style="color: #e06666;" title="glossary">chord</a><span style="color: #e06666;"> and an </span><a class="glossaryLink" href="" style="color: #e06666;" title="glossary">arc</a></b></span></div><div class="separator" style="clear: both; font-family: Verdana,sans-serif; text-align: center;"><span style="font-size: large;"><b><a href="http://4.bp.blogspot.com/-kNP1Om4mFf4/TkKjje2UqcI/AAAAAAAACaI/P1wEd7VnrLg/s1600/semicircle.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-kNP1Om4mFf4/TkKjje2UqcI/AAAAAAAACaI/P1wEd7VnrLg/s1600/semicircle.jpg" /></a></b></span></div><div style="font-family: Verdana,sans-serif;"><span style="font-size: large;"><b><br style="background-color: #6aa84f;" /><strong style="background-color: #6aa84f; color: #e06666;"><span><span style="background-color: #d9ead3;"></span></span><span><span style="background-color: #d9ead3;"></span></span><span style="background-color: #d9ead3;">Semi-circle -</span></strong></b> <b><span style="background-color: #d9ead3; color: #e06666;"> is an arc whose endpoints are endpoints of a diameter</span><br />
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</strong></b> </span></div></div>Malini Lingarajaprabhuhttp://www.blogger.com/profile/02400048832214425554noreply@blogger.com12tag:blogger.com,1999:blog-5122059094859586267.post-26728774662367546802011-08-09T08:32:00.000-07:002011-08-09T08:32:39.894-07:00Applications of Trigonometry in Real life<div dir="ltr" style="text-align: left;" trbidi="on"><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-vNLyiL1eZUw/TkFMFREj0FI/AAAAAAAACZA/dMRTagZ410Q/s1600/distancetoisland.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="167" src="http://2.bp.blogspot.com/-vNLyiL1eZUw/TkFMFREj0FI/AAAAAAAACZA/dMRTagZ410Q/s320/distancetoisland.jpg" width="320" /></a></div><ul style="font-family: Verdana,sans-serif; text-align: left;"><li><b>Trigonometry is commonly used in finding the height of towers and mountains.</b></li>
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<div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-868OTsjlAik/TkFPMVn_9nI/AAAAAAAACZY/zxLzNnxQ6K4/s1600/trig__6bafcd9e.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="229" src="http://2.bp.blogspot.com/-868OTsjlAik/TkFPMVn_9nI/AAAAAAAACZY/zxLzNnxQ6K4/s320/trig__6bafcd9e.jpg" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjTLGM755hWUm6T13GUpY09zMYTW8QzA2c21QejNnj3nlLmmqnv43kEBtLQ9peZ2rw4IE-9nJGbL4gIUbkCEGtQ4mQmXu9VREP7-WDLxf9qWo2DLu9i1zx07Gpn7DcJHSty7OGCHHM2_oM/s1600/matterhorn-problem.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="231" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjTLGM755hWUm6T13GUpY09zMYTW8QzA2c21QejNnj3nlLmmqnv43kEBtLQ9peZ2rw4IE-9nJGbL4gIUbkCEGtQ4mQmXu9VREP7-WDLxf9qWo2DLu9i1zx07Gpn7DcJHSty7OGCHHM2_oM/s320/matterhorn-problem.jpg" width="320" /></a></div><br />
<ul style="font-family: Verdana,sans-serif; text-align: left;"><li><b>It is used in navigation to find the distance of the shore from a point in the sea.</b></li>
</ul><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEil07-LODuBk0VK_s5xfAOQ6L90o_A2E54PMfFd1nWZgT_ndXg7HokcQV3Ff5qB_09xMtGlEE4m6YZp8rEKmWa2__VhoCY6MK99eElIpJM3yIY221E6MuZM0QTVAm3_TPvHhrACfu_UIlA/s1600/Sextant-PSF.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="244" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEil07-LODuBk0VK_s5xfAOQ6L90o_A2E54PMfFd1nWZgT_ndXg7HokcQV3Ff5qB_09xMtGlEE4m6YZp8rEKmWa2__VhoCY6MK99eElIpJM3yIY221E6MuZM0QTVAm3_TPvHhrACfu_UIlA/s320/Sextant-PSF.png" width="320" /></a></div><br />
<ul style="font-family: Verdana,sans-serif; text-align: left;"><li><b> It is used in oceanography in calculating the height of tides in oceans</b></li>
</ul><br />
<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjUM8yzMxrrzdzUphEVNJZvrTf4KlnVDqkxOTYsbxNd_mFa5AyCbCrJnxREra5uJirpDvI2nqzKfXdqEIw3xNFTD8oFyIc6n3T_ipHJnZUCYay3s5wEPjawHSU_bjjXSZZ4y1cZP6wifjc/s1600/tp2ch1_image3.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="224" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjUM8yzMxrrzdzUphEVNJZvrTf4KlnVDqkxOTYsbxNd_mFa5AyCbCrJnxREra5uJirpDvI2nqzKfXdqEIw3xNFTD8oFyIc6n3T_ipHJnZUCYay3s5wEPjawHSU_bjjXSZZ4y1cZP6wifjc/s320/tp2ch1_image3.jpg" width="320" /></a></div><br />
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<ul style="font-family: Verdana,sans-serif; text-align: left;"><li><b>It is used in finding the distance between celestial bodies</b></li>
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgz3oWmTuueO_yOkceP06NWQ4Y5-zF-jg57fF1ZF5PHRvqDnDfg3Sym1uZAnyYrS6MEjlgfa6g_6sEAS0SBSedaeSOUfqQ8Go-D14ZSiNZMHMwBV_cSQal48oFl2901xS_ha_y4Dh2RlEM/s1600/triang1.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgz3oWmTuueO_yOkceP06NWQ4Y5-zF-jg57fF1ZF5PHRvqDnDfg3Sym1uZAnyYrS6MEjlgfa6g_6sEAS0SBSedaeSOUfqQ8Go-D14ZSiNZMHMwBV_cSQal48oFl2901xS_ha_y4Dh2RlEM/s320/triang1.gif" width="312" /></a></div><br />
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<ul style="text-align: left;"><li><b style="font-family: Verdana,sans-serif;">The sine and cosine functions are fundamental to the theory of periodic functions such as those that describe sound and light waves</b>.</li>
</ul><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjfwZ5BF3jFnDZnj0ubXAulyafW8JVLyUb5GFBBeRj8dSC0r6MSiufJPIRI-L2iN7C-_1nyt-hxatJkVrka_aUWIK-Eq1EDJi_sl-9_OQ1hK-sw3rPCw1QHpTDnqrzAT0gOCZefEtb0Y-4/s1600/soundwaves515x321.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="199" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjfwZ5BF3jFnDZnj0ubXAulyafW8JVLyUb5GFBBeRj8dSC0r6MSiufJPIRI-L2iN7C-_1nyt-hxatJkVrka_aUWIK-Eq1EDJi_sl-9_OQ1hK-sw3rPCw1QHpTDnqrzAT0gOCZefEtb0Y-4/s320/soundwaves515x321.png" width="320" /></a></div><br />
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<figure class="Thumbnail articlePhoto"> </figure> <ul style="font-family: Verdana,sans-serif; text-align: left;"><li><b>Architects use trigonometry to calculate structural load, roof slopes, ground surfaces and many other aspects, including sun shading and light angles</b></li>
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</div>Malini Lingarajaprabhuhttp://www.blogger.com/profile/02400048832214425554noreply@blogger.com145tag:blogger.com,1999:blog-5122059094859586267.post-69173440156021603822011-03-25T10:03:00.000-07:002011-03-29T08:23:26.665-07:00THE NUMBER 13<div dir="ltr" style="text-align: left;" trbidi="on"><span style="font-size: large;"><b> Two interesting facts about the number 13: </b></span><br />
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<div style="font-family: Georgia,"Times New Roman",serif;"><span style="font-size: small;"><b>(i) <span style="font-family: Arial,Helvetica,sans-serif;"> (1 / 13 ) = 0.</span><span style="color: blue; font-family: Arial,Helvetica,sans-serif;">076923</span><span style="color: #274e13; font-family: Arial,Helvetica,sans-serif;">076923</span><span style="color: purple; font-family: Arial,Helvetica,sans-serif;">076923</span><span style="color: red; font-family: Arial,Helvetica,sans-serif;">076923</span><span style="font-family: Arial,Helvetica,sans-serif;">...</span></b></span></div><div style="font-family: Georgia,"Times New Roman",serif;"><span style="font-size: large;"><b> </b></span><b>(rational periodic infinite decimal)</b></div><ul style="text-align: left;"></ul><br />
<blockquote style="color: blue;"><span style="font-size: large;"><b>On multiplying the <u>periodic number </u><u>76923</u> by successive multiples of 13, the following curious numerical pattern is obtained. </b></span></blockquote><span style="font-size: large;"><b>76923 * 013 = 0999999</b></span><br />
<span style="font-size: large;"><b>76923 * 026 = 1999998 </b></span><br />
<span style="font-size: large;"><b>76923 * 039 = 2999997 </b></span><br />
<span style="font-size: large;"><b>76923 * 052 = 3999996</b></span><br />
<span style="font-size: large;"><b>76923 * 065 = 4999995</b></span><br />
<div style="text-align: center;"><div style="text-align: justify;"><span style="font-size: large;"><b>76923 * 078 = 5999994</b></span></div><div style="text-align: justify;"><span style="font-size: large;"><b>76923 * 091 = 6999993</b></span></div><div style="text-align: justify;"><span style="font-size: large;"><b>76923 * 104 = 7999992</b></span></div><div style="text-align: justify;"><span style="font-size: large;"><b>76923 * 117 = 8999991</b></span></div><div style="text-align: justify;"><span style="font-size: large;"><b>76923 * 130 = 9999990</b></span><br />
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</div><div style="text-align: justify;"></div><div style="text-align: justify;"><span style="font-size: small;"><b> </b><b style="font-family: Georgia,"Times New Roman",serif;">(ii) The numerical pattern <span style="color: red;">abcabc</span> is divisible by 13.</b></span></div></div><ul style="text-align: left;"></ul></div>Malini Lingarajaprabhuhttp://www.blogger.com/profile/02400048832214425554noreply@blogger.com0tag:blogger.com,1999:blog-5122059094859586267.post-89308502784733519992011-02-04T06:57:00.000-08:002011-02-04T06:57:50.371-08:00An interesting fact about the powers of 10<div dir="ltr" style="text-align: left;" trbidi="on"><br />
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<span style="font-size: large;"><b>There seems to be <u>only 11 powers of ten</u> that are products of 2 integers <u>without any zero digits</u>.</b></span><br />
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<div style="font-family: Arial,Helvetica,sans-serif;"><b><span style="font-size: small;">10 ^ 0 = 1 * 1</span></b></div><div style="font-family: Arial,Helvetica,sans-serif;"><b><span style="font-size: small;">10 ^ 1 = 2 * 5</span></b></div><div style="font-family: Arial,Helvetica,sans-serif; text-align: left;"><b><span style="font-size: small;">10 ^ 2 = 4 * 25</span></b></div><div style="font-family: Arial,Helvetica,sans-serif; text-align: left;"><b><span style="font-size: small;">10 ^ 3 = 8 * 125</span></b></div><div style="font-family: Arial,Helvetica,sans-serif; text-align: left;"><b><span style="font-size: small;">10 ^ 4 = 6 * 625</span></b></div><div style="font-family: Arial,Helvetica,sans-serif; text-align: left;"><b><span style="font-size: small;">10 ^ 5 = 32 * 3125 </span></b></div><div style="font-family: Arial,Helvetica,sans-serif; text-align: left;"><b><span style="font-size: small;">10 ^ 6 = 64 * 15625</span></b></div><div style="font-family: Arial,Helvetica,sans-serif; text-align: left;"><b><span style="font-size: small;">10 ^ 7 = 128 * 78125</span></b></div><div style="font-family: Arial,Helvetica,sans-serif; text-align: left;"><b><span style="font-size: small;">10 ^ 9 = 512 * 1953125 </span></b></div><div style="font-family: Arial,Helvetica,sans-serif; text-align: left;"><b><span style="font-size: small;">10 ^ 18 = 262144 * 3814697265625 </span></b></div><div style="text-align: left;"><b style="font-family: Arial,Helvetica,sans-serif;"><span style="font-size: small;">10 ^ 33 = 8589934592 * 116415321826934814453125</span></b></div><br />
</div>Malini Lingarajaprabhuhttp://www.blogger.com/profile/02400048832214425554noreply@blogger.com6tag:blogger.com,1999:blog-5122059094859586267.post-52456189013016106142010-11-02T08:14:00.000-07:002011-11-04T11:35:11.585-07:00CHOCOLATE PUZZLE<div dir="ltr" style="text-align: left;" trbidi="on">A shop sells chocolates @ Re.1 each. You can exchange 3 wrappers for 1 chocolate.<br />
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If you have Rs.15/- how many chocolates can you get totally???? <br />
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<span style="color: red;"><b>Find the solution..</b></span> <br />
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<b><span style="color: red;">TO CHECK YOUR ANSWER ... SCROLL DOWN</span></b> <br />
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<a href="http://malini-math.blogspot.com/2009/08/simple-math-puzzles.html" target="_blank"><span style="font-size: x-large;"><b><span style="font-family: Arial,Helvetica,sans-serif;"> FOR MORE PUZZLES CLICK HERE</span></b></span></a><br />
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<span style="font-size: x-large;"><b><span style="color: red;">SOLUTION: </span></b></span><br />
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<span style="font-size: x-large;"><b><span style="color: blue;">22 CHOCOLATES</span></b></span><br />
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<b><span style="color: blue;"> </span></b>for 15 rupees you get <span style="font-size: large;"><b>15 chocolates</b>.</span><br />
on returning 15 wrappers you get <span style="font-size: large;"><b>5 chocolates.</b></span><br />
on returning 3 wrappers (keeping two wrappers in hand) you get <span style="font-size: large;"><b>1 chocolate.</b></span><br />
now this one chocolate. wrapper and the twp wrapers you had will add on three.<br />
atlast on returning 3 wrappers you get <span style="font-size: large;"><b>1 chocolate.</b></span><br />
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i.e., <span style="font-size: large;"><b>15+5+1+1 = 22</b></span><br />
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</div>Malini Lingarajaprabhuhttp://www.blogger.com/profile/02400048832214425554noreply@blogger.com3tag:blogger.com,1999:blog-5122059094859586267.post-58912967647715307322010-08-30T12:55:00.000-07:002010-08-30T12:55:36.389-07:00MATHS AND NATURE<div class="separator" style="clear: both; text-align: center;"></div><div style="text-align: left;"><span style="font-size: large;">"The laws of nature are but the mathematical thoughts of God"</span></div><div style="text-align: left;"><span style="font-size: large;"> - Euclid</span></div><div style="text-align: left;"><br />
</div><div style="text-align: left;">Mathematics is everywhere in this universe. We seldom note it. We enjoy nature and are not interested in going deep about what mathematical idea is in it. Here are a very few properties of mathematics that are depicted in nature.</div><div style="text-align: center;"><strong><span style="font-size: large;">SYMMETRY </span></strong></div><div style="text-align: center;"><br />
</div>Symmetry is everywhere you look in nature . <br />
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<em>Symmetry is when a figure has two sides that are mirror images of one another. It would then be possible to draw a line through a picture of the object and along either side the image would look exactly the same. This line would be called a line of symmetry. </em><br />
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<strong>There are two kinds of symmetry.</strong> <br />
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One is <strong>bilateral symmetry</strong> in which an object has two sides that are mirror images of each other. <br />
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<em>The human body would be an excellent example of a living being that has bilateral symmetry.</em><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi5Vt1mmBGKJzemzrhyphenhyphen3jCnDTwS53_rj-GdibtdcNSpymYIU1xNaXh4WRqYegNsfshWFEzRVtiZt60Ano0wOiPYvM2f6Wo5hmji54BCBc3moWDBKGgIxB3H0soLA8HLP7BpmJPFdW1T6Pw/s1600/bilateral_symmetry.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi5Vt1mmBGKJzemzrhyphenhyphen3jCnDTwS53_rj-GdibtdcNSpymYIU1xNaXh4WRqYegNsfshWFEzRVtiZt60Ano0wOiPYvM2f6Wo5hmji54BCBc3moWDBKGgIxB3H0soLA8HLP7BpmJPFdW1T6Pw/s320/bilateral_symmetry.jpg" width="210" /></a><br />
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<strong><em>Few more pictures in nature showing bilateral symmetry</em></strong>.<br />
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<div style="border-bottom: medium none; border-left: medium none; border-right: medium none; border-top: medium none;"><br />
<a href="http://3.bp.blogspot.com/_NYaJQllvyA0/THv0sQicKRI/AAAAAAAABv0/jBXUEed2FTY/s1600/butterfly.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/_NYaJQllvyA0/THv0sQicKRI/AAAAAAAABv0/jBXUEed2FTY/s320/butterfly.jpg" /></a> <br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh_D4yIycxpVT5wlyDvkCvj6wtNMMxVqjUvhHiBhfRs4s3-iQ518AXvWJwlnM3c-qNN4UBL1gEDodJho21Y1870XVFMT1w0knpf0Gvv-GWcMjZL3CdRUz6v6Q8xQ7ae3sBWfpwz1Uk0ooI/s1600/untitled.bmp" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh_D4yIycxpVT5wlyDvkCvj6wtNMMxVqjUvhHiBhfRs4s3-iQ518AXvWJwlnM3c-qNN4UBL1gEDodJho21Y1870XVFMT1w0knpf0Gvv-GWcMjZL3CdRUz6v6Q8xQ7ae3sBWfpwz1Uk0ooI/s320/untitled.bmp" /></a> <br />
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<a href="http://3.bp.blogspot.com/_NYaJQllvyA0/THv2PB6ZBKI/AAAAAAAABwA/2fFRfyX2tdc/s1600/220px-Leaf_1_web.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="http://3.bp.blogspot.com/_NYaJQllvyA0/THv2PB6ZBKI/AAAAAAAABwA/2fFRfyX2tdc/s320/220px-Leaf_1_web.jpg" /></a><br />
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</div>The other kind of symmetry is <strong>radial symmetry.</strong> This is where there is a center point and numerous lines of symmetry could be drawn. <br />
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</div><em>The most obvious geometric example would be a circle.</em><br />
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<div class="separator" style="clear: both; text-align: center;"></div><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh2eHPO6_81Me04C_aueMkZ7fiKe2wovjCATvTBZUa7K8QlqNhISy_E593E6NDOmYdv-pWG4PPupWVE4FGZ31pvFvqCsmrNvcy1cUdbN5BK3fnIBUHJDEuTnAXo6ng_aeAQ0BiRkrm2dDw/s1600/circle%20lines%20of%20symmetry.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh2eHPO6_81Me04C_aueMkZ7fiKe2wovjCATvTBZUa7K8QlqNhISy_E593E6NDOmYdv-pWG4PPupWVE4FGZ31pvFvqCsmrNvcy1cUdbN5BK3fnIBUHJDEuTnAXo6ng_aeAQ0BiRkrm2dDw/s320/circle%2520lines%2520of%2520symmetry.jpg" /></a><br />
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<strong><em>Few more pictures in nature showing radial symmetry.</em></strong><br />
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</div><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgxg5Nd1hbV4ijo3XuQMaYj-B2AsupdXhHhZBDXUgTSqiLZ2iu3rMvo2fMV-xqIJ1-wC4hxzGmKXMUr0JdKl-fRzvsipl57WrvrEMO1z9y5ugDletYpss9Phip4JTHd1EzAxx5y8yP2EEw/s1600/stock-photo-isolated-half-cut-orange-with-perfect-geometrical-shape-11920849.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgxg5Nd1hbV4ijo3XuQMaYj-B2AsupdXhHhZBDXUgTSqiLZ2iu3rMvo2fMV-xqIJ1-wC4hxzGmKXMUr0JdKl-fRzvsipl57WrvrEMO1z9y5ugDletYpss9Phip4JTHd1EzAxx5y8yP2EEw/s320/stock-photo-isolated-half-cut-orange-with-perfect-geometrical-shape-11920849.jpg" /></a> <br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiCmKG7DOgNuko-hVW9yBY_p6aI_HpKrgM-CMDsY1t9U7DOw8pN85Nz2DghZE24PaA9SCwVHmQiAa2bZoRU7GV_gzNTaxnSOthShFNpCtUu6Hs5BeG8FEFtFrgjr6pXH5pcdwOn65fwAl0/s1600/Starfish-1.jpg" imageanchor="1" style="clear: left; cssfloat: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiCmKG7DOgNuko-hVW9yBY_p6aI_HpKrgM-CMDsY1t9U7DOw8pN85Nz2DghZE24PaA9SCwVHmQiAa2bZoRU7GV_gzNTaxnSOthShFNpCtUu6Hs5BeG8FEFtFrgjr6pXH5pcdwOn65fwAl0/s320/Starfish-1.jpg" /></a><br />
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<div style="text-align: center;"><strong><span style="font-size: large;">SHAPES</span></strong></div><div style="text-align: center;"><br />
</div>Geometry is the branch of mathematics that describes shapes.<br />
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<strong>Sphere:</strong><br />
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<em>A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball.</em> <br />
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The shape of the Earth is very close to that of an oblate spheroid, a sphere flattened along the axis from pole to pole such that there is a bulge around the equator.<br />
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<strong>Hexagons:</strong><br />
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<em>Hexagons are six-sided polygons, closed, 2-dimensional, many-sided figures with straight edges.</em><br />
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For a beehive, close packing is important to maximise the use of space. Hexagons fit most closely together without any gaps; so hexagonal wax cells are what bees create to store their eggs and larvae. <br />
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<strong>Cones:</strong><br />
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<em>A cone is a three-dimensional geometric shape that tapers smoothly from a flat, usually circular base to a point called the apex or vertex.</em><br />
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Volcanoes form cones, the steepness and height of which depends on the runniness (viscosity) of the lava. Fast, runny lava forms flatter cones; thick, viscous lava forms steep-sided cones. <br />
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEja6UPBatutsqd9ZkCzqg5Ix86x4dAzYycd-aybhU0msh56waDtEjpJ2EMwqAIo5or_srQdta8SeCAewkjNlNVjQKRfTPANCs_yAh__ynAC32iBn4PZ9p0k_RPk1JQeAqy5s1dRM7t6bj0/s1600/Amazing-Volcanoes-Screensaver_hfn2.jpg" imageanchor="1" style="clear: left; cssfloat: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEja6UPBatutsqd9ZkCzqg5Ix86x4dAzYycd-aybhU0msh56waDtEjpJ2EMwqAIo5or_srQdta8SeCAewkjNlNVjQKRfTPANCs_yAh__ynAC32iBn4PZ9p0k_RPk1JQeAqy5s1dRM7t6bj0/s320/Amazing-Volcanoes-Screensaver_hfn2.jpg" /></a></div><br />
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<strong>Few more cones in nature:</strong><br />
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<strong>Parallel lines:</strong><br />
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<em>In mathematics, parallel lines stretch to infinity, neither converging nor diverging.</em> <br />
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These parallel dunes in the Australian desert aren't perfect - the physical world rarely is.<br />
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<strong>Fibonacci spiral:</strong><br />
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<em>If you construct a series of squares with lengths equal to the Fibonacci numbers (1,1,2,3,5, etc) and trace a line through the diagonals of each square, it forms a Fibonacci spiral</em>. <br />
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Many examples of the Fibonacci spiral can be seen in nature, including in the chambers of a nautilus shell.<br />
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjNLFvDGu0ikUe42N1tZ9TYp6aUZUL0JfHnx2coD3uZuWr5iJVv0oW8mJOyXed-_r9y1h8sl0eSFBzKRe4dnEC5aGhBsB_UN9GWJrvdyroWYyW80_jhOcguHif5vkobgSWmrji-9XdbA-w/s1600/12.jpg" imageanchor="1" style="clear: left; cssfloat: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjNLFvDGu0ikUe42N1tZ9TYp6aUZUL0JfHnx2coD3uZuWr5iJVv0oW8mJOyXed-_r9y1h8sl0eSFBzKRe4dnEC5aGhBsB_UN9GWJrvdyroWYyW80_jhOcguHif5vkobgSWmrji-9XdbA-w/s320/12.jpg" /></a></div><br />
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<div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/_NYaJQllvyA0/THwHdcz-HeI/AAAAAAAABx8/bHm1ttgOzAI/s1600/sunflower.jpg" imageanchor="1" style="clear: left; cssfloat: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" ox="true" src="http://2.bp.blogspot.com/_NYaJQllvyA0/THwHdcz-HeI/AAAAAAAABx8/bHm1ttgOzAI/s320/sunflower.jpg" /></a></div>Malini Lingarajaprabhuhttp://www.blogger.com/profile/02400048832214425554noreply@blogger.com79tag:blogger.com,1999:blog-5122059094859586267.post-71839315802826908452010-07-29T09:38:00.000-07:002010-07-29T09:38:37.123-07:00VAMPIRE NUMBERThe vampire numbers were introduced by Clifford A. Pickover in 1994.<br />
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A Vampire number z is a number which can be written as a product of two numbers x and y containing the same digits the same number of times as the vampire number. Here x and y are called FANGS.<br />
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FOR EXAMPLE: <br />
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1827 can be written as a product of two numbers 21 and 87. <br />
i.e., 1827 = 21 * 87<br />
Here the digits 1,8, 2,7 are repeated the same number of times but in different order. 21 and 87 are called fangs.<br />
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<strong><span style="color: blue;">A true Vampire number isa number which can be written with 2 fangs having the same number of digits not both ending in zero.</span></strong><br />
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FOR EXAMPLE :<br />
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1260 is a vampire number, with 21 and 60 as fangs, since 21 × 60 = 1260. However, 126000 – 210 × 600 – is not, as both 210 and 600 have trailing zeroes.<br />
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<strong><span style="color: blue;">All vampire numbers must clearly have an no. of digits.</span></strong><br />
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Some vampire numbers:<br />
117067 = 167*701<br />
124483 = 281 * 443<br />
536539 = 563 · 953<br />
23287176 = 2673 * 8712.<br />
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<span style="color: purple;"><strong>Vampire number having two distinct pairs of fangs :</strong></span><br />
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125460 = 204 *615 = 246 * 510<br />
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<strong><span style="color: purple;">Vampire number having three distinct pairs of fangs :</span></strong> <br />
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13078260 = 1620*8073 = 1863*7020 = 2070*6318Malini Lingarajaprabhuhttp://www.blogger.com/profile/02400048832214425554noreply@blogger.com2tag:blogger.com,1999:blog-5122059094859586267.post-86571883805719054042010-06-22T00:48:00.000-07:002011-11-04T11:34:02.246-07:00SIMPLE MATH PUZZLES<div dir="ltr" style="text-align: left;" trbidi="on"><div style="text-align: center;"><span style="color: #660000; font-size: large;"><b>PUZZLE 8</b></span></div><br />
<span style="color: blue; font-family: Arial,Helvetica,sans-serif; font-size: large;">What mathematical symbol can be put between 5 and 9, to get a number bigger than 5 and smaller than 9? </span><br />
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<div style="text-align: center;"><span style="color: #660000; font-size: large;"><b>PUZZLE 9</b></span></div><br />
<span style="color: blue; font-family: Arial,Helvetica,sans-serif; font-size: large;">The following multiplication example uses every digit from 0 to 9 once (not counting the intermediate steps). Fill in the missing numbers.</span><br />
<div style="text-align: center;"><span style="color: blue; font-family: Arial,Helvetica,sans-serif; font-size: large;"><b>7 _ _ * 4 _ = _ _ _ _ _</b></span></div><br />
<div style="text-align: center;"><span style="color: #660000; font-size: large;"><b>PUZZLE 10</b></span></div><br />
<span style="color: blue; font-family: Arial,Helvetica,sans-serif; font-size: large;">Here's a simple multiplication problem in which each letter represents a different digit. Can you solve it?</span><br />
<div style="text-align: center;"><span style="color: blue; font-family: Arial,Helvetica,sans-serif; font-size: large;"><b>IF * AT = FIAT</b></span></div><div align="center"><br />
</div><div style="text-align: center;"><span style="color: #660000;"><span style="font-size: large;"><b>PUZZLE 11</b></span>: </span></div><br />
<span style="color: blue; font-family: Arial,Helvetica,sans-serif; font-size: large;">Write down the next number in this series:</span><br />
<span style="color: blue; font-family: Arial,Helvetica,sans-serif; font-size: large;"></span><br />
<div style="text-align: center;"><span style="color: blue; font-family: Arial,Helvetica,sans-serif; font-size: large;"><b>18,46,94,63,52, ?</b></span></div><br />
<div style="text-align: center;"><span style="color: #660000;"><span style="font-size: large;"><b>PUZZLE12:</b></span> </span></div><br />
<span style="color: blue; font-family: Arial,Helvetica,sans-serif; font-size: large;">Find the next letter in the series:</span><br />
<span style="color: blue; font-family: Arial,Helvetica,sans-serif; font-size: large;"></span><br />
<div style="text-align: center;"><span style="color: blue; font-family: Arial,Helvetica,sans-serif; font-size: large;"><b>O,T,T,F,F,S,S,E,?</b></span></div><br />
<div style="text-align: center;"><span style="color: #660000; font-size: large;"><b>PUZZLE 13:</b></span></div><br />
<span style="color: blue; font-family: Verdana,sans-serif; font-size: large;">How many two-digit positive whole numbers are there?</span><br />
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<div style="text-align: center;"><span style="color: #660000;"><span style="font-size: large;"><b>PUZZLE 14:</b></span> </span></div><br />
<span style="color: blue; font-family: Arial,Helvetica,sans-serif; font-size: large;">Find four consecutive prime numbers that add up to 220.</span><br />
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<div style="text-align: center;"><span style="font-size: x-large;"><b>SOLUTIONS </b></span></div> <br />
<span style="font-family: Arial,Helvetica,sans-serif;"><span style="font-size: large;"><b><span style="color: blue;">PUZZLE 8 :</span></b> <span style="color: #660000;">A Decimal Point. <span style="color: black;"><b>5.9</b></span> </span></span></span><br />
<span style="font-family: Arial,Helvetica,sans-serif; font-size: large;"> </span><br />
<span style="font-family: Arial,Helvetica,sans-serif;"><span style="font-size: large;"><b><span style="color: blue;">PUZZLE 9</span> <span style="color: blue;">:</span></b> <span style="color: #660000;">715 * 46 = 32890 </span></span></span><br />
<span style="font-family: Arial,Helvetica,sans-serif; font-size: large;"> </span><br />
<span style="font-family: Arial,Helvetica,sans-serif;"><span style="font-size: large;"><span style="color: blue;"><b>PUZZLE 10</b></span> <b><span style="color: blue;">:</span></b> <span style="color: #660000;">IF = 41. AT = 35. FIAT = 1435<b>.</b></span></span></span><br />
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<span style="font-family: Arial,Helvetica,sans-serif;"><span style="font-size: large;"><span style="color: blue;"><b>PUZZLE 11</b></span> <b><span style="color: blue;">:</span> </b><span style="color: #660000;"><b>61.</b> Each is a perfect square read from back to front</span></span></span><br />
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<span style="font-family: Arial,Helvetica,sans-serif;"><span style="font-size: large;"><span style="color: blue;"><b>PUZZLE 12 :</b></span> <span style="color: #660000;"><b>N</b> – The series is the first letter of the numbers 1, 2,3,4,5,6,7,8, ? .so the next letter is N the first letter of Nine. 1[<span style="color: blue;"><b>O</b></span>ne],2[<span style="color: blue;"><b>T</b></span>wo], 3 [<span style="color: blue;"><b>T</b></span>hree], … 8[<span style="color: blue;"><b>E</b></span>ight], 9 [<span style="color: blue;"><b>N</b></span>ine].</span></span></span><br />
<span style="font-family: Arial,Helvetica,sans-serif;"><br />
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<span style="font-family: Arial,Helvetica,sans-serif;"><span style="font-size: large;"><b><span style="color: blue;">PUZZLE 13:</span></b> <span style="color: #660000;">90</span></span></span><br />
<span style="font-family: Arial,Helvetica,sans-serif;"><br />
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<span style="font-family: Arial,Helvetica,sans-serif;"><span style="font-size: large;"><span style="color: blue;"><b>PUZZLE 14:</b></span> <span style="color: #660000;">47,53,59,61.</span></span></span><br />
<a href="http://www.blogger.com/goog_1595546237"><span style="color: #660000; font-size: x-large;"><span style="font-family: Arial,Helvetica,sans-serif;">FOR</span></span></a><span style="font-family: Arial,Helvetica,sans-serif;"><span style="font-size: large;"><span style="color: #660000;"><a href="http://malini-math.blogspot.com/2010/11/chocolate-puzzle.html" target="_blank"><span style="font-size: x-large;"> MORE PUZZLES CLICK HERE</span></a> </span></span></span></div>Malini Lingarajaprabhuhttp://www.blogger.com/profile/02400048832214425554noreply@blogger.com4tag:blogger.com,1999:blog-5122059094859586267.post-26554085200002658502010-06-14T03:13:00.000-07:002010-06-14T03:13:00.925-07:00FASCINATING FACTS OF MATHEMATICS: PERMUTATIONS AND COMBINATIONS<a href="http://malini-math.blogspot.com/2010/06/permutations-and-combinations.html">FASCINATING FACTS OF MATHEMATICS: PERMUTATIONS AND COMBINATIONS</a>Malini Lingarajaprabhuhttp://www.blogger.com/profile/02400048832214425554noreply@blogger.com0tag:blogger.com,1999:blog-5122059094859586267.post-87971463498987450832010-06-14T03:02:00.000-07:002010-06-14T03:22:18.987-07:00PERMUTATIONS AND COMBINATIONSIn common, students who come across the topic <strong>“Permutation and Combination”</strong> feel it to be confusing. <br />
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<span style="color: #990000;"><strong><em>The basic idea of the topic is selection and arrangement of things. They are widely used in solving problems of probability, genetic engineering and life sciences</em></strong>.</span><br />
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<span style="color: #0b5394;"><strong>Combination to unlock a safe</strong>..</span> Consider a 3 digit number 653 which you use as a combination to the safe. Here the order of the number is important as 563 will not work to unlock the safe. This is permutation where the order of elements is very important.<br />
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<span style="color: #0b5394;"><strong>Combination of three colors</strong>..</span> Consider 4 colors red, blue, yellow and green. If I ask you to select 3 colors and you choosing them as Red, green and blue is same when you choose it as blue, red and green. This is combination where the order does not matter.<br />
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To make it simple, mathematically it can be stated as:<br />
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<strong><span style="color: blue;">“Permutation is an ordered Combination”.</span></strong><br />
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<div style="text-align: left;"></div>An easy way to memorize:<br />
<blockquote>Permutation - <strong>P</strong>osition important <br />
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Combination - <strong>C</strong>hosen important</blockquote><strong><span style="color: purple; font-size: large;">Permutation:</span></strong><br />
<blockquote><span style="font-family: Georgia, "Times New Roman", serif;"><span style="color: #b45f06;">Permutation is arrangement of values in all possible ways.The word arrangement is used to emphasize that the order of the things is important</span>.</span></blockquote>Consider the three letter word “PEN”. In how many ways can we rearrange it.<br />
<blockquote> <strong>PEN PNE EPN ENP NEP NPE</strong></blockquote>A three letter word can be arranged in 6 ways. You can easily find it out. How about a 10 letter word. Quite a long process isn’t it. Hence we use FACTORIAL to find the number of possible ways to arrange the elements.<br />
For a <strong>3 letter word</strong>, <br />
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<strong> <span style="font-size: x-small;">3! = 3*2*1 = 6 ways.</span></strong><br />
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For a <strong>10 letter word</strong>, <br />
<div><blockquote><strong><span style="font-size: x-small;">10! = 10*9*8*7*6*5*4*3*2*1= 3628800</span></strong></blockquote><strong><span style="color: #674ea7;">Permutation with Repetition:</span></strong><br />
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Number of permutations of n-things, taken ‘r’ at a time when each thing can be repeated r-times is given by = n ^ r.<br />
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A child has 3 pocket and 4 coins. In how many ways can he put the coins in his pocket.<br />
First coin can be put in 3 ways, similarly second, third and forth coins also can be put in 3 ways.<br />
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So<strong> total number of ways</strong> = 3 x 3 x 3 x 3 <br />
= 3 ^ 4 = 81<br />
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<span style="color: #674ea7;"><strong>Permutation without Repetition:</strong></span></div><br />
If we have to select 3 numbers from a set of 9 numbers say, from 1 to 9 , then what are the possibilities?<br />
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For the first number we have <strong>9 choices{</strong>1,2,3,4,5,6,7,8,9}. Let us choose 4.<br />
<br />
For the second number we have <strong>8 choices</strong>{1,2,3,5,6,7,8,9}. Let us choose 8.<br />
<br />
For the third number we have <strong>7 choices</strong>{1,2,3,5,6,7,9}. Let us choose 3.<br />
<br />
We had 9 choices at first, then 8 and then 7.Therefore the total no. of options would be <strong>9*8*7 = 504.</strong><br />
<br />
So, if you wanted to select all of the 9 numbers, the permutation would be:<br />
<br />
<blockquote><strong>9! =9*8*7*6*5*4*3*2*1 = 362880.</strong></blockquote><br />
But here you wanted to select just 3, then you have to stop the multiplying after 7. How do you do that? There is a simple trick ... you divide by 6! ...<br />
<br />
<span style="font-size: x-small;">9! ÷6! = (9*8*7*6*5*4*3*2*1) / (6*5*4*3*2*1) </span><br />
<span style="font-size: x-small;"> = 9*8*7 = 504.</span><br />
<div align="left"><br />
</div><strong>Mathematically,</strong> the number of permutations of ‘r ‘objects chosen from a set of ‘n’ objects is expressed as:<br />
<div style="text-align: center;"> <strong><span style="font-size: large;">nPr = n! / (n-r)!</span></strong></div><div style="text-align: left;"><br />
</div><div style="text-align: left;"><strong><span style="color: purple; font-size: large;">Combination:</span></strong></div><br />
<br />
<span style="font-family: Georgia, "Times New Roman", serif;"></span><br />
<blockquote><span style="color: #b45f06; font-family: Georgia, "Times New Roman", serif;">Combination means selection of things. The word selection is used, when the order of things has no importance.</span></blockquote><br />
If 9 players are selected to form a team from 20 players, the order in which the 9 players are selected doesn’t matter as they are all in the team. This is combination. There will be a change in combination iff a player in the team is changed.<br />
<br />
In permutation the order is important as 234 is different from 324. but in combination we are concerned only with the numbers 2, 3 and 4 that have been selected. The combination of 234 is same as 324.<br />
<br />
<strong>The combinations of abcd taken 3 at a time :</strong><br />
<br />
abc abd acd bcd.<br />
<br />
Each of these four combinations will give rise to 3! Permutations:<br />
<div style="text-align: center;"> <span style="color: blue;"><strong>abc</strong></span> <span style="color: purple;"><strong>abd</strong> </span> <strong><span style="color: #674ea7;"><span style="color: #e06666;">acd</span> </span> <span style="color: magenta;">bcd</span></strong></div><div style="text-align: center;"><br />
</div><div style="text-align: center;"> <span style="color: blue;">acb </span> <span style="color: purple;">adb </span> <span style="color: #ea9999;"><span style="color: #e06666;">adc</span> </span> <span style="color: magenta;">bdc</span></div><div style="text-align: center;"><br />
</div><div style="text-align: center;"> <span style="color: blue;">bac </span> <span style="color: purple;"> bad</span> <span style="color: #e06666;">cad </span> <span style="color: #0c343d;"> </span><span style="color: magenta;">cbd</span></div><div style="text-align: center;"><br />
</div><div style="text-align: center;"> <span style="color: blue;"> bca</span> <span style="color: purple;"> bda</span> <span style="color: #e06666;">cda </span> <span style="color: #0c343d;"> </span><span style="color: magenta;">cdb</span></div><div style="text-align: center;"><br />
</div><div style="text-align: center;"> <span style="color: blue;">cab </span> <span style="color: purple;">dab </span> <span style="color: #e06666;"> dac</span><span style="color: #20124d;"> </span> <span style="color: magenta;">dbc</span></div><div style="text-align: center;"><br />
</div><div style="text-align: center;"> <span style="color: blue;">cba </span><span style="color: purple;"> dba</span> <span style="color: #20124d;"> </span><span style="color: #e06666;">dca </span> <span style="color: magenta;">dcb</span></div><br />
Each column is the permutation of that combination. But they are all one combination as order has no importance.Since the order does not matter in combinations, there are clearly fewer combinations than permutations. Hence combinations can be stated as :<br />
<br />
<span style="color: blue;"><strong>"Combinations are subsets of permutation"</strong>.</span><br />
<br />
Thus if we want to figure out how many combinations we have of n objects then r at a time, we just create all the permutation and then divide by r! variant.<br />
<br />
<strong>Mathematically</strong>, the number of combinations of n things taken r at a time is:<br />
<div style="text-align: center;"><strong><span style="font-size: large;">nCr = nPr / r!</span></strong></div><span style="font-size: large;"><br />
</span><br />
<div style="text-align: center;"><strong><span style="font-size: large;"> = n! / [r! (n- r)!]</span></strong><br />
<br />
</div><strong><span style="color: #660000;"><span style="color: black;">nCr is known as</span> n choose r. <span style="color: black;">It is also known as</span> Binomial co-efficient</span>.</strong>Malini Lingarajaprabhuhttp://www.blogger.com/profile/02400048832214425554noreply@blogger.com14tag:blogger.com,1999:blog-5122059094859586267.post-36767289819260581462010-06-02T09:14:00.000-07:002010-06-02T09:14:25.306-07:00INCREDIBLE TRICK - SHE LL BLOW YOUR MIND<a href="http://www.regiftable.com/regiftingrobinpopup.html">http://www.regiftable.com/regiftingrobinpopup.html</a><br />
<br />
<br />
<span style="color: red; font-size: large;"><strong>If you are smart enough post the simple trick hidden behind this ...</strong></span>Malini Lingarajaprabhuhttp://www.blogger.com/profile/02400048832214425554noreply@blogger.com1tag:blogger.com,1999:blog-5122059094859586267.post-29931866395332626792010-04-05T06:13:00.000-07:002010-04-05T06:13:38.625-07:00STATISTICS IN DAILY LIFE• <b>Statistics a mathematical science form a basic tool in business and manufacturing as well.</b><br />
<br />
<b>• MEAN:</b><br />
<blockquote>If in a tour, the total money spent by 10 friends is Rs.50000, then the average money spent by each person is Rs.5000.Here<b> 5000</b>(50000 /10 )<b> is the mean.</b></blockquote><b>• MEDIAN:</b><br />
<blockquote>If you have 15 things lined up next to each other by their cost, the median cost will be the cost of the thing in the very middle. <b>Here the cost of middle thing is the median.</b></blockquote><b> • MODE:</b><br />
<blockquote>A shopkeeper, selling shirts, keeps more stock of that size of shirt which has more sale. Here the <b>size of that shirt is the mode</b> among other.</blockquote>Malini Lingarajaprabhuhttp://www.blogger.com/profile/02400048832214425554noreply@blogger.com1tag:blogger.com,1999:blog-5122059094859586267.post-58322109540823786392010-04-05T05:17:00.000-07:002010-04-05T05:17:41.454-07:00MATHS IN COOKINGCooking is an art. While preparing a delicious meal, measuring the ingredients and calculating the time it will take to cook involve maths. If a person adds too much of salt or sugar it will spoil the food. One must be aware in what proportion it should be added. Converting measurements when doubling and tripling a recipe needs basic math skills of multiplication. Knowing how to measure can make the difference between a great meal and a horrible. People who enjoy cooking understand that they need to have sharp math skills in order to prepare good food.Malini Lingarajaprabhuhttp://www.blogger.com/profile/02400048832214425554noreply@blogger.com0tag:blogger.com,1999:blog-5122059094859586267.post-47896258972747724402010-04-05T05:14:00.000-07:002010-04-05T05:14:42.428-07:00MATHS IN DAILY LIFE<span style="font-size: small;"></span><span style="font-size: small;"></span><span style="font-size: small;">The language of mathematics is numbers and it is the only language shared by all people regardless of culture , religion or gender.</span><blockquote><span style="font-size: small;">Roger Bacon, an English philosopher and scientist stated<b> :<i></i></b></span></blockquote><blockquote><span style="font-size: small;"><b><i>"Neglect of mathematics works injury to all knowledge, since he who is ignorant of it cannot know the other sciences or the things of the world." </i></b></span></blockquote><span style="font-size: small;">Adding up the cost of a basket full of groceries involves the same math process regardless of whether the total is expressed in dollars, rubles, or yen.With this universal language, all of us, no matter what our unit of exchange, are likely to arrive at math results the same way.</span><span style="font-size: small;">Math can help us to shop wisely, buy the right insurance, remodel a home within a budget, understand population growth, or even bet on the horse with the best chance of winning the race.</span><span style="font-size: small;"><br />
</span>Malini Lingarajaprabhuhttp://www.blogger.com/profile/02400048832214425554noreply@blogger.com0tag:blogger.com,1999:blog-5122059094859586267.post-66556362076869188062010-03-08T06:38:00.000-08:002010-03-08T06:38:47.078-08:00ZERO<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhpF7pby5ZzV1K7uz3wc0ReKWe_Nq1XwsLLpB4pEz0b7OzGrTPJF0ngeRbAenDjQ885AiIj9LA6Wd_btTR8qRxJFmmowryB_yilHFU9qEahQe6jv8a9UsNEDwDgnwC2o9Y8sDofrmq_i0s/s1600-h/zero.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhpF7pby5ZzV1K7uz3wc0ReKWe_Nq1XwsLLpB4pEz0b7OzGrTPJF0ngeRbAenDjQ885AiIj9LA6Wd_btTR8qRxJFmmowryB_yilHFU9qEahQe6jv8a9UsNEDwDgnwC2o9Y8sDofrmq_i0s/s320/zero.jpg" /></a></div><div style="text-align: center;"><span style="color: #660000; font-size: x-large;"><b>ZERO</b></span></div><div style="color: #3d85c6; text-align: left;"><span style="font-size: large;"><b>... a fine and wonderful refuge of the divine spirit - almost an amphibian between being and non being. --- Gottfried Leibniz</b></span></div><div style="text-align: left;"><br />
</div><span style="font-family: Georgia,"Times New Roman",serif;"><b><span style="color: purple;"><span style="font-family: Verdana,sans-serif;"> </span></span></b></span><br />
<ul><li><span style="font-family: Georgia,"Times New Roman",serif;"><b><span style="color: purple;"><span style="font-family: Verdana,sans-serif;">The first thing to say about zero is that the use of zero is extremely importa</span><span style="font-family: Verdana,sans-serif;">nt</span></span><span style="color: purple;"> .</span></b></span></li>
</ul><div></div><span style="color: blue; font-family: Georgia,"Times New Roman",serif;"></span><br />
<blockquote><span style="color: blue; font-family: Georgia,"Times New Roman",serif;"><span style="color: blue; font-family: Georgia,"Times New Roman",serif;"><b>It is used as an empty place indicator in our place-value number system. Hence in a number like <span style="font-family: Arial,Helvetica,sans-serif;">1502 , </span>the zero is used so that the positions of the<span style="font-family: Arial,Helvetica,sans-serif;"> 1</span> and <span style="font-family: Arial,Helvetica,sans-serif;">5 </span>are correct. Clearly<span style="font-family: Arial,Helvetica,sans-serif;"> 152</span> means something quite different</b></span></span></blockquote><span style="color: blue; font-family: Georgia,"Times New Roman",serif;"><br />
</span><br />
<ul><span style="color: blue; font-family: Georgia,"Times New Roman",serif;">
<li><b><span style="color: purple; font-family: Verdana,sans-serif;">The name "zero" derives ultimately from the Arabic sifr which also gives us the word "cipher".</span></b></li>
</span></ul><span style="color: blue; font-family: Georgia,"Times New Roman",serif;"><br />
</span><br />
<ul><span style="color: blue; font-family: Georgia,"Times New Roman",serif;">
<li><span style="color: purple;"><span style="font-family: Verdana,sans-serif;"> <b>The first use of the symbol which we recognise today as the notation for zero "0" is omicron, the first letter of the Greek word for nothing namely "ouden".</b></span></span></li>
</span></ul><span style="color: blue; font-family: Georgia,"Times New Roman",serif;"><br style="color: purple;" /></span><br />
<ul><li><span style="color: blue; font-family: Georgia,"Times New Roman",serif;"> <span style="color: purple; font-family: Verdana,sans-serif;"><b>THE FIRST RECORD OF INDIAN USE OF ZERO:</b></span></span></li>
</ul><blockquote><span style="color: blue; font-family: Georgia,"Times New Roman",serif;"><span style="color: blue; font-family: Georgia,"Times New Roman",serif;"><i><b>We have an inscription on an stone tablet which contains a date which translates to 876. T he inscription concerns the town of Gwalior , 400 km south of Delhi, where they planted a garden 187 by 270 hastas which would produce enough flowers to allow 50 garlands per day to be given to the local temple.</b></i></span><i> </i></span></blockquote><blockquote><span style="color: blue; font-family: Georgia,"Times New Roman",serif;"><span style="color: blue; font-family: Georgia,"Times New Roman",serif;"><i><b>Both of the numbers 270 and 50 are denoted almost as they appear today although 0 is smaller and slighty raised.</b></i></span></span></blockquote><div class="separator" style="clear: both; text-align: center;"></div><ul><li><span style="color: blue; font-family: Georgia,"Times New Roman",serif;"> <b><span style="color: purple; font-family: Verdana,sans-serif;"> INDIAN MATHEMATICIANS </span></b></span></li>
</ul><span style="color: blue; font-family: Georgia,"Times New Roman",serif;"> </span><br />
<blockquote><span style="color: blue; font-family: Georgia,"Times New Roman",serif;"><span style="font-family: Georgia,"Times New Roman",serif;"><span style="color: #660000;"><span style="font-family: Georgia,"Times New Roman",serif;"><span style="color: #660000;"><b><span style="color: #20124d;"><span style="font-size: large;">Brahmagupta</span> attempted to give the rules for arithmetic involving zero and negative numbers in the seventh century</span></b>.</span></span></span></span></span><br />
<span style="color: blue; font-family: Georgia,"Times New Roman",serif;"><span style="font-family: Georgia,"Times New Roman",serif;"><span style="color: #660000;"><br />
</span></span></span><br />
<span style="color: blue; font-family: Georgia,"Times New Roman",serif;"><span style="font-family: Georgia,"Times New Roman",serif;"><span style="color: #660000;"><b><span style="font-family: Georgia,"Times New Roman",serif;"> He gave the following rules for addition which involve zero:- </span></b></span></span></span><br />
<span style="color: blue; font-family: Georgia,"Times New Roman",serif;"><span style="font-family: Georgia,"Times New Roman",serif;"><span style="color: #660000;"><br />
</span></span></span><br />
<span style="color: blue; font-family: Georgia,"Times New Roman",serif;"><span style="font-family: Georgia,"Times New Roman",serif;"><span style="color: #660000;"><i><span style="color: #351c75;"><span style="font-family: Georgia,"Times New Roman",serif;"><span style="color: blue;">The sum of zero and a negative number is negative, the sum of a positive number and zero is positive, the sum of zero and zero is zero.</span> </span></span></i></span></span></span><br />
<span style="color: blue; font-family: Georgia,"Times New Roman",serif;"><span style="font-family: Georgia,"Times New Roman",serif;"><span style="color: #660000;"><span style="color: #351c75; font-family: Georgia,"Times New Roman",serif;"></span></span></span></span><br />
<div></div><span style="color: blue; font-family: Georgia,"Times New Roman",serif;"><span style="font-family: Georgia,"Times New Roman",serif;"><span style="color: #660000;"><span style="font-family: Georgia,"Times New Roman",serif;"><b>He gave the following rules for Subtraction which involve zero:-</b> </span></span></span></span><br />
<span style="color: blue; font-family: Georgia,"Times New Roman",serif;"><span style="font-family: Georgia,"Times New Roman",serif;"><span style="color: #660000;"><span style="font-family: Georgia;"></span> </span></span></span><br />
<div><span style="color: blue; font-family: Georgia,"Times New Roman",serif;"><span style="font-family: Georgia,"Times New Roman",serif;"><span style="color: #660000;"><i><span style="color: blue; font-family: Georgia,"Times New Roman",serif;">A negative number subtracted from zero is positive, a positive number subtracted from zero is negative, zero subtracted from a negative number is negative, zero subtracted from a positive number is positive, zero subtracted from zero is zero. </span></i></span></span></span></div><div></div><span style="color: blue; font-family: Georgia,"Times New Roman",serif;"><span style="font-family: Georgia,"Times New Roman",serif;"><span style="color: #660000;"><br />
</span></span></span><br />
<div><span style="color: blue; font-family: Georgia,"Times New Roman",serif;"><span style="font-family: Georgia,"Times New Roman",serif;"><span style="color: #660000;"><span style="font-family: Georgia,"Times New Roman",serif;"><b>Brahmagupta was not able to explain clearly about division.</b> </span></span></span></span></div></blockquote><span style="color: blue; font-family: Georgia,"Times New Roman",serif;"><span style="font-family: Georgia,"Times New Roman",serif;"><span style="color: #660000;"></span></span></span><br />
<div></div><div><blockquote style="font-family: Georgia,"Times New Roman",serif;"><span style="color: blue;"><b><span style="color: purple;"><span style="color: black; font-size: large;">Bhaskara</span> tried to solve the problem by writing n/0 = ∞.</span></b></span></blockquote><blockquote><span style="color: blue; font-family: Georgia,"Times New Roman",serif;"><span style="color: blue; font-family: Georgia,"Times New Roman",serif;"><i> If this were true then <span style="font-family: Arial,Helvetica,sans-serif;">0</span> times <b>∞ </b>must be equal to every number n, so all numbers are equal.</i></span></span></blockquote><blockquote style="font-family: Georgia,"Times New Roman",serif;"><div><span style="color: blue;"><span style="color: purple;"><b>Bhaskara did correctly state other properties of zero such as </b></span></span></div><div style="text-align: center;"><span style="color: blue;"><span style="color: purple;"><b> <span style="font-family: Verdana,sans-serif;">0 ^ 2 = 0 and √ 0 = 0</span>. </b></span></span></div></blockquote></div><div></div><div style="font-family: Times,"Times New Roman",serif;"><span style="font-size: large;"><b><span style="color: blue;">The Indian mathematicians could not bring themselves to the point of admitting that one could not divide by zero. </span></b></span></div><div></div><br />
<ul style="color: purple; font-family: Verdana,sans-serif;"><li><b>The number zero is neither positive nor negative, neither a prime number nor a composite number, nor it is a unit. It is a even number.</b></li>
</ul><br />
<span style="color: blue; font-family: Georgia,"Times New Roman",serif;"><span style="font-family: Verdana,sans-serif;"></span><br />
</span><br />
<div></div><span style="color: blue; font-family: Georgia,"Times New Roman",serif;"><br />
</span><br />
<div></div><span style="color: blue; font-family: Georgia,"Times New Roman",serif;"></span>Malini Lingarajaprabhuhttp://www.blogger.com/profile/02400048832214425554noreply@blogger.com7tag:blogger.com,1999:blog-5122059094859586267.post-73865579923868978182009-12-20T00:27:00.000-08:002009-12-20T01:43:39.804-08:00RAMANUJAM<div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/_NYaJQllvyA0/Sy3wtOEdIOI/AAAAAAAAAdE/XtmufCWdG5k/s1600-h/Ramanujan.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ps="true" src="http://1.bp.blogspot.com/_NYaJQllvyA0/Sy3wtOEdIOI/AAAAAAAAAdE/XtmufCWdG5k/s320/Ramanujan.jpg" /></a><br />
</div><div style="text-align: center;"><strong><span style="color: purple;">"An equation for me has no meaning unless it expresses a thought of God."-</span> RAMANUJAM</strong><br />
</div><div style="text-align: center;"><br />
</div>Srinivasa Ramanujan (22 December 1887 – 26 April 1920) was one of India's greatest mathematical geniuses. He made substantial contributions to the analytical theory of numbers and worked on elliptic functions, continued fractions, and infinite series. His most famous work was on the number p(n) of partitions of an integer n into summands. <br />
<br />
By age 11, he had exhausted the mathematical knowledge of two college students who were lodgers at his home. He was later lent a book on <strong>Advanced Trigonometry</strong> written by S. L. Loney. He completely mastered this book by the age of 13 and discovered sophisticated theorems on his own.<br />
<br />
When he was 16, Ramanujan came across the book "<strong>A Synopsis of Elementary Results in Pure and Applied Mathematics" </strong> by George S. Carr. This book was a collection of 5000 theorems, and it introduced Ramanujan to the world of mathematics. The next year, <span style="color: blue;"><em><span style="font-family: Times, "Times New Roman", serif;">he had independently developed and investigated the Bernoulli numbers and had calculated Euler's constant up to 15 decimal place</span></em>s</span>.<br />
<br />
Ramanujan, with the help of Ramaswami Iyer(founder member of the Indian Mathematical Society) , had his work published in the Journal of Indian Mathematical Society.<br />
<br />
In January 1913 Ramanujan wrote to G .H. Hardy having seen a copy of his book <strong>Orders of infinity. </strong>Hardy, together with Littlewood, studied the long list of unproved theorems which Ramanujan enclosed with his letter. <br />
<br />
Hardy wrote back to Ramanujan and in 1914, Hardy brought Ramanujan to Trinity College, Cambridge, to begin an extraordinary collaboration.<br />
<br />
<strong>On 6 December 1917, he was elected to the<span style="color: #351c75;"> London Mathematical Society</span>. </strong><br />
<br />
<strong>In 1918, he became a <span style="color: #990000;">Fellow of the Royal Society</span> , and <span style="color: purple;">he was the youngest Fellow in the entire history of the Royal Society</span></strong><span style="color: purple;">.</span><br />
<br />
<strong>On 13 October 1918, he became <span style="color: #38761d;">the first Indian</span> to be elected a <span style="color: #783f04;">Fellow of Trinity College, Cambridge.</span></strong><br />
<br />
<br />
<strong><span style="color: blue;">TAXICAB NUMBER:</span></strong><br />
<strong>The number derives its name from the following story:</strong><br />
G. H. Hardy told about Ramanujan. I remember once going to see him when he was ill . I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather dull one, and that I hoped it was not an unfavorable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."<br />
<strong>1729 is the second taxicab number (the first is 2= 1^3 + 1^3).</strong> The number was also found in one of Ramanujan's notebooks dated years before the incident.<br />
<br />
<strong><span style="color: blue;"><em>Every positive integer is one of Ramanujan's personal friends" -</em> John Littlewood, on hearing of the taxicab incident</span></strong><br />
<br />
<span style="color: black;">Ramanujan had problems</span> settling in London. He was an orthodox Brahmin and right from the beginning he had problems with his diet. Ramanujan sailed to India on 27 February 1919 arriving on 13 March. However his health was very poor and, despite medical treatment, he died on April 6, 1920.Malini Lingarajaprabhuhttp://www.blogger.com/profile/02400048832214425554noreply@blogger.com18tag:blogger.com,1999:blog-5122059094859586267.post-53805978773687015842009-12-17T06:46:00.000-08:002009-12-17T06:46:07.740-08:00SUM OF ADJACENT INTEGERS <span style="color: #38761d; font-family: Arial, Helvetica, sans-serif;"><strong>4 + 5 + 6 = 7 + 8</strong></span><br />
<span style="color: #38761d; font-family: Arial, Helvetica, sans-serif;"><strong></strong></span><br />
<span style="color: #38761d;"><br />
<strong><span style="font-family: Arial, Helvetica, sans-serif;"></span></strong></span><br />
<span style="color: #38761d; font-family: Arial, Helvetica, sans-serif;"><strong> 9 + 10 + 11 + 12 = 13 + 14 + 15</strong></span><br />
<span style="color: #38761d;"><br />
<strong><span style="font-family: Arial, Helvetica, sans-serif;"></span></strong></span><br />
<span style="color: #38761d; font-family: Arial, Helvetica, sans-serif;"><strong> 16 + 17 + 18 + 19 + 20 = 21 + 22 + 23 + 24</strong></span><br />
<span style="color: #38761d;"><br />
<strong><span style="font-family: Arial, Helvetica, sans-serif;"></span></strong></span><br />
<span style="color: #38761d; font-family: Arial, Helvetica, sans-serif;"><strong> 25 + 26 + 27 + 28 + 29 + 30 = 31 + 32 + 33 + 34 + 35</strong></span><br />
<span style="color: #38761d;"><br />
<strong><span style="font-family: Arial, Helvetica, sans-serif;"></span></strong></span><br />
<span style="color: #38761d; font-family: Arial, Helvetica, sans-serif;"><strong> 36 + 37 + 38 + 39 + 40 + 41 + 42 = 43 + 44 + 45 + 46 + 47 + 48</strong></span><br />
<span style="color: #38761d;"><br />
<strong><span style="font-family: Arial, Helvetica, sans-serif;"></span></strong></span><br />
<span style="color: #38761d;"><strong><span style="font-family: Arial, Helvetica, sans-serif;">49 + 50 + 51 + 52 + 53 + 54 + 55 + 56 = 57 + 58 + 59 + 60 + 61 + 62 +</span><span style="font-family: Arial, Helvetica, sans-serif;"> 63</span></strong></span>Malini Lingarajaprabhuhttp://www.blogger.com/profile/02400048832214425554noreply@blogger.com0tag:blogger.com,1999:blog-5122059094859586267.post-74520722192631989242009-12-17T06:36:00.000-08:002009-12-17T06:38:46.876-08:00NUMBER PATTERN USING 1, 2, 8 AND 9<div style="text-align: left;"> <span style="font-size: large;"> <span style="color: #351c75;"> </span><span style="color: purple;"> <strong><span style="font-family: Verdana, sans-serif;">9 x <span style="color: lime;">2</span> = <span style="background-color: white;"><span style="color: blue;">1</span></span><span style="background-color: white; color: red;">8</span></span></strong></span></span><br />
</div><div style="text-align: left;"><strong><span style="color: purple; font-family: Verdana, sans-serif;"><span style="font-size: large;"> 99 x <span style="color: lime;">2</span> = <span style="color: blue;">1</span>9<span style="color: red;">8</span></span></span></strong><br />
</div><div style="text-align: left;"><strong><span style="color: purple; font-family: Verdana, sans-serif;"><span style="font-size: large;"> 999 x <span style="color: lime;">2</span> = <span style="color: blue;">1</span>99<span style="color: red;">8</span></span></span></strong><br />
</div><div style="text-align: left;"><strong><span style="color: purple; font-family: Verdana, sans-serif;"><span style="font-size: large;"> 9999 x <span style="background-color: white; color: lime;">2</span> = <span style="color: blue;">1</span>999<span style="color: red;">8</span></span></span></strong><br />
</div><div style="text-align: left;"><strong><span style="color: purple; font-family: Verdana, sans-serif;"><span style="font-size: large;"> 99999 x <span style="color: lime;">2</span> = <span style="color: blue;">1</span>9999<span style="color: red;">8</span></span></span></strong><br />
</div><div style="text-align: left;"><strong><span style="color: purple; font-family: Verdana, sans-serif;"><span style="font-size: large;"> 999999 x <span style="color: lime;">2</span> = <span style="color: blue;">1</span>99999<span style="color: red;">8</span></span></span></strong><br />
</div><div style="text-align: left;"><strong><span style="color: purple; font-family: Verdana, sans-serif;"><span style="font-size: large;"> 9999999 x <span style="color: lime;">2</span> = <span style="color: blue;">1</span>999999<span style="color: red;">8</span></span></span></strong><br />
</div><div style="text-align: left;"><strong><span style="color: purple; font-family: Verdana, sans-serif;"><span style="font-size: large;"> 99999999 x <span style="color: lime;">2</span> = <span style="color: blue;">1</span>9999999<span style="color: red;">8</span></span></span></strong><br />
</div><div style="text-align: left;"><strong><span style="color: purple; font-family: Verdana, sans-serif;"><span style="font-size: large;">999999999 x <span style="color: lime;">2 </span>= <span style="color: blue;">1</span>99999999<span style="color: red;">8</span></span></span></strong><br />
</div>Malini Lingarajaprabhuhttp://www.blogger.com/profile/02400048832214425554noreply@blogger.com0tag:blogger.com,1999:blog-5122059094859586267.post-89254728610877241842009-12-17T06:20:00.000-08:002009-12-17T06:20:16.619-08:00PALINDROMIC NUMBER PATTERN WITH 1<strong><span style="color: blue; font-family: Verdana, sans-serif; font-size: large;">111111111 </span></strong><br />
<strong><span style="font-family: Verdana, sans-serif;"><span style="font-size: large;"><span style="color: blue;"> * 111111111</span> </span></span></strong><br />
<strong><span style="font-family: Verdana, sans-serif; font-size: large;">--------------------------</span></strong><br />
<strong><span style="font-family: Verdana, sans-serif;"><span style="font-size: large;"> <span style="color: purple;">12345678987654321</span></span></span></strong>Malini Lingarajaprabhuhttp://www.blogger.com/profile/02400048832214425554noreply@blogger.com1tag:blogger.com,1999:blog-5122059094859586267.post-86943750656435204692009-11-09T08:30:00.000-08:002011-10-20T08:32:32.147-07:00EUCLID<div dir="ltr" style="text-align: left;" trbidi="on"><div></div><div style="text-align: center;"><b><span style="font-size: x-large;"><br />
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</div><div></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjxHiDgca4O9Hh61e4HuXgE6limz1K6hEHasemU5HJAaHQqGRaITf8CuzT4dnJ5XQuMgZ_bAzThZUss-seinoW0W8L5oKxrH2E6519i9dDsBvBLwsUQIbFT6la5HoAjb6Hh1YLymS6lqJM/s1600-h/225px-Euklid-von-Alexandria_1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" sr="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjxHiDgca4O9Hh61e4HuXgE6limz1K6hEHasemU5HJAaHQqGRaITf8CuzT4dnJ5XQuMgZ_bAzThZUss-seinoW0W8L5oKxrH2E6519i9dDsBvBLwsUQIbFT6la5HoAjb6Hh1YLymS6lqJM/s320/225px-Euklid-von-Alexandria_1.jpg" /></a></div><div class="separator" style="clear: both; text-align: center;"><b><span style="font-size: large;">FATHER OF MATHEMATICS</span></b></div><div style="text-align: center;"><br />
</div><div style="text-align: left;"><span style="font-family: Georgia,"Times New Roman",serif;">The Greek mathematician Euclid’s referred to as “THE FATHER OF GEOMETRY” is well known for his most famous work “ The Elements” which is a collection of geometrical theorems and “Euclidean theorem”. </span></div><div style="text-align: left;"><br />
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</div><div style="text-align: left;"><span style="font-family: Georgia,"Times New Roman",serif; font-size: large;"><b>EUCLID’S FAMOUS QUOTES:</b></span></div><div style="text-align: center;"><span style="font-family: Georgia,"Times New Roman",serif;"></span><span style="font-family: Georgia,"Times New Roman",serif;"></span></div><blockquote style="text-align: center;"><div style="text-align: left;"><span style="font-family: Georgia,"Times New Roman",serif;"><b><i>“The laws of nature are but the mathematical thoughts of God”</i></b></span></div><div style="text-align: left;"><br />
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</div><div style="text-align: left;"><span style="font-family: Georgia,"Times New Roman",serif;"><b><i>“ There is no other Royal path which leads to geometry”.</i></b></span></div></blockquote><div style="text-align: left;"><b><span style="font-family: Georgia,"Times New Roman",serif; font-size: large;">THE ELEMENTS:</span></b></div><div style="text-align: center;"><br />
</div><div style="text-align: left;"><span style="font-family: Georgia,"Times New Roman",serif;">The Elements is divided into <span style="font-family: Arial,Helvetica,sans-serif;">13 </span>books. </span></div><div style="text-align: left;"><ul><li><span style="font-family: Georgia,"Times New Roman",serif;">The first <span style="font-family: Arial,Helvetica,sans-serif;">6 </span>books deals with plane geometry.</span></li>
<li><span style="font-family: Georgia,"Times New Roman",serif;">Books <span style="font-family: Arial,Helvetica,sans-serif;">7</span>to <span style="font-family: Arial,Helvetica,sans-serif;">9</span> deals with number theory.</span></li>
<li><span style="font-family: Georgia,"Times New Roman",serif;">Book<span style="font-family: Arial,Helvetica,sans-serif;"> 10</span> deals with the theory of irrational numbers .</span></li>
<li><span style="font-family: Georgia,"Times New Roman",serif;">Books <span style="font-family: Arial,Helvetica,sans-serif;">11 </span>to<span style="font-family: Arial,Helvetica,sans-serif;"> 13</span> deals with three-dimensional geometry .</span></li>
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<div style="text-align: left;"><span style="font-family: Georgia,"Times New Roman",serif;"><b>Euclid's Elements is remarkable for the clarity with which the theorems are stated and proved. </b></span></div><br />
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</div><div style="text-align: left;"><span style="font-family: Georgia,"Times New Roman",serif; font-size: large;"><b>EUCLID'S OTHER WORKS :</b></span><span style="font-family: Georgia,"Times New Roman",serif;"></span><span style="font-family: Georgia,"Times New Roman",serif;"> </span></div><div style="text-align: left;"><ul><li><span style="font-family: Georgia,"Times New Roman",serif;"><span style="font-family: Georgia,"Times New Roman",serif;"><b>ON DIVISION</b> deals with plane geometry.</span> </span></li>
<li><span style="font-family: Georgia,"Times New Roman",serif;">The book <b>DATA</b> discusses plane geometry and contains propositions.</span></li>
<li><span style="font-family: Georgia,"Times New Roman",serif;"><b>PHAENOMENA</b> is a work by what we call today as applied mathematics, concerning the geometry of spheres for use in astronomy. </span></li>
<li><span style="font-family: Georgia,"Times New Roman",serif;"><b>THE OPTICS</b>, corrects the belief held at the time that the sun and other heavenly bodies are actually the size they appear to be to the eye.</span></li>
<li><span style="font-family: Georgia,"Times New Roman",serif;"><b>CONICS</b> was a work on conic sections.</span></li>
</ul></div><b><span style="font-family: Georgia,"Times New Roman",serif;"> <span style="font-size: large;">EUCLID'S APPROACH:</span></span></b><br />
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</div><div style="text-align: left;"><span style="font-family: Georgia,"Times New Roman",serif;">Euclid used an approach called the "synthetic approach" to present his theorems. Using this method, one progresses in a series of logical steps from the known to the unknown. </span></div><div style="text-align: left;"><br />
</div><span style="font-family: Georgia,"Times New Roman",serif; font-size: large;"><b>EUCLID’S CLASSICAL PROOF ON PRIME NUMBERS:</b></span><br />
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</div><div style="text-align: left;"><span style="font-family: Georgia,"Times New Roman",serif;">Euclid proved that it is impossible to find the "largest prime number," because if you take the largest known prime number, add 1 to the product of all the primes up to and including it, you will get another prime number. Euclid's proof for this theorem is generally accepted as one of the "classic" proofs because of its conciseness and clarity. Millions of prime numbers are known to exist, and more are being added by mathematicians and computer scientists.</span></div></div>Malini Lingarajaprabhuhttp://www.blogger.com/profile/02400048832214425554noreply@blogger.com11