C homework

[danbaron]

"e", is Euler's number. It can be defined so that the area under the curve, y = 1/x and above the x axis, and between the points x = 1 and x = e, equals 1. It is irrational - it's decimal expansion goes forever without repeating. It starts,

2.71828..

"pi", can be defined as the area of a circle with radius 1. It is irrational - it's decimal expansion goes forever without repeating. It starts,

3.14592..

"i", is defined as the square root of -1.

Here is maybe the most famous equation in mathematics, called, Euler's Identity.

e ^ (i * pi) + 1 = 0

(It is easy to prove, but it seems that no one understands what it means - why these three absolutely pivotal, but seemingly unrelated mathematical values, should be related in such a simple way. What does e have to do with pi?, - remember that both contain an infinity of digits, and neither has a repeating pattern in its infinity of digits.)

http://en.wikipedia.org/wiki/Euler%27s_identity



Note.

A rational number can be written as the ratio of two integers, a/b, where b is unequal to 0 (that's the definition).

At some point, the decimal expansion of a rational number always begins to repeat, and then does so infinitely.

For instance, 2 is rational. Its decimal expansion is,

2.000000000000.. i.e, 0 repeats forever.

Another rational number is,

453678123.56471234634524558326489706849256792567925679256792567..
(I am unable to remove the space character between the 2 and 5 (above), I don't know why.)

If you notice, after awhile, the sequence 92567 begins to repeat. And, it does so forever.

In the decimal expansion of every rational number, sooner or later, there is a pattern of one or more digits which repeats over and over infinitely. As soon as you get to the first instance of repetition, you basically know the entire number.

On the other hand, an irrational number cannot be written as the ratio of two integers. And, in its decimal expansion, there is never a repeating pattern.

So, in a sense, no one knows what the value of e is. For instance, even if you calculate it to a trillion decimal places, you still won't know what the trillion and first digit is. The same is true for pi.

[kryton9]

Very cool, thanks Dan. Your explanation of Euler's Identity is better than on wikipedia.
I think most of the math references there are for mathematicians and not good for the general public.

[danbaron]

I know you like math, Kent. In a certain way it is similar to space and the universe, big, mysterious, and surprising. And, by the way, while the universe is big, it is still finite. However, in 1931, Kurt Godel proved that math is infinite. There will never be a time when someone can (truthfully) say, "All of mathematics is now known.".

Einstein went with Godel, when Godel became a U.S. citizen. Einstein had to help shut him up, when he started spouting (what many people would consider to be) cuckoo talk. You can read about it here.

http://en.wikipedia.org/wiki/Kurt_Godel