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Tuesday, October 20, 2009


A circle has 360 degrees, but it also has 400 gradients and approximately 6.2831853 radians. It all depends on what *units* you measure your angles with.

You think 360 is a terrible number, and you think that you want a circle to have 100 "somethings" in it. Well, you divide up the circle into 100 equal angles, all coming out from the center, and then you call one of these angles a "deeg." Then you've just defined a new way to measure a circle. 100 deegs are in a circle.

This invented unit, the deeg, is much like the degree, except the degree is smaller (why?). They are both angles. Just as 1 inch = 2.54 centimeters, although the centimeter is smaller, the inch and centimeter are both units of length. So the ancient Babylonians (not the Greeks), decided that a circle should contain 360 degrees. In one degree there are 60 minutes (though they have the same name, one minute-angle is not the same as one minute-time). Furthermore, in one minute there are 60 seconds (again, one second-angle is not one second-time, though they are the same word).

The British military chose a different way to divide the circle, specifically, 400 gradients in one circle. So one gradient is a tad bit smaller than a degree. And what's a radian? It's what mathematicians use because there's a way to divide the circle into a number of parts that happen to make certain computations easy. The way they decided this was that they took a circle, say with radius 1 cm. Then they took a piece of string, and made marks on it, evenly spaced 1 cm apart. Then they took the string and wrapped it around the circle. They then asked how many little 1 cm pieces of string fit around the circle, and they got the answer of about 6.2831853 pieces. They decided that the angle that a 1 cm piece of string covers as it is wrapped about the edge of a circle of radius 1 cm should be called one radian. Weird but true.

Now, one might wonder why the Babylonians chose the number 360. The reason is that their number system was based on the number 60. To compare, we base our number system on 10. For us, 10 is a nice, round number and we find it very convenient to count in multiples of 10, like millimeter, centimeter, meter, kilometer, etc. But the Babylonians liked 60.

Why this was nice for them, nobody knows, but 60 is a nice number too, because 60 = 2 x 2 x 3 x 5 and 360 = 2 x 2 x 2 x 3 x 3 x 5. What's so neat about that, you ask? Well, you will find that 360 is divisible by 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, and 20. There are few other numbers as small as 360 that have so many different factors. This makes the degree a very nice unit to divide the circle into an equal number of parts. 120 degrees is 1/3 of a circle, 90 degrees is 1/4, and so on.


  • Pi is an irrational number. It means that it cannot be written as the ratio of two integer numbers. 22/7 is a popular one used for Pi but it is only an approximation, which equals to 3.142857143...
  • Another characteristic of pi as an irrational number is the fact that it takes an infinite number of digits to give its exact value, i.e. you can never get to the end of it.
  • One of the most accurate fractions for Pi is 104348 / 33215. it is accurate to 0.00000001056%
  • First 100 digits
    3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 ...
  • 1.2411 trillion digits (1,241,100,000,000) digits of pi have been dicovered
  • You can determine your hat size by measuring the circumference of your head, then divide by Pi and round off to the nearest one_eighth inch.
  • The height of an elephant (from foot to shoulder) = 2 * Pi * the diameter of its foot.
  • The Babylonians, in 2000 B.C. were the first people known to find a value for Pi.
  • Pi day is celebrated on March 14 at the Exploratorium in San Francisco (March 14 is 3/14) at 1:59 PST which is 3.14159.
  • Pi Approximation Day is on the 22 / 7 - that is, July 22. For the past few years, people at Chalmers University have celebrated it.

Real Life Applications of Imaginary Numbers

  • Complex numbers enter into studies of physical phenonomena in ways that most people can't imagine.
  • For example: A differential equation, with coefficients like the a, b, and c in the quadratic formula, that models how electrical circuits or forced spring/damper systems behave. The movement of the shock absorber of a car as it goes over a bump is an example of the latter. The behavior of the differential equations depends upon whether the roots of a certain quadratic are complex or real. If they are complex, then certain behaviors can be expected. These are often just the solutions that one wants.
  • Closely related to the electrical engineering example is the use of complex numbers in signal processing, which has applications to telecommunications (cellular phone), radar (which assists the navigation of airplanes), and even biology (in the analysis of firing events from neurons in the brain).

  • In modeling the flow of a fluid around various obstacles, like around a pipe, complex analysis is very valuable for transforming the problem into a much simpler problem. When everything from large structures of riveted beams to economic systems are analyzed for resilience, some very large matrices are used in the modeling.
  • In everyday use, industrial and university computers spend some fraction of their time solving polynomial equations. The roots of such equations are of interest, whether they are real or complex
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