Saturday, March 14, 2009

Happy Pi day.

Today is March 14. In the ridiculous vernacular of the United States dates (MM/DD/YY), today is 3/14. Pi is approximated as 3.14159265. The US House of Representatives just passed a resolution designating March 14 as National Pi day.

Pi is an awesome number. Everybody knows that Pi is the ratio of the circumference of a circle to its diameter. That's for any circle in Euclidean geometry (what most of you are used to). But, this is just the beginning.

Pi is an irrational number. An irrational number is one that cannot be expressed as a ratio of two integers (as a fraction). This also means that the numbers after the decimal never stop and never repeat. You may find a pattern or two in Pi's numbers, but they won't survive long.

Pi is also a transcendental number, which means that there is no polynomial with rational coefficients of which Pi is the root. That is, there is no combination of addition, subtraction, multiplication, division, powers, roots, etc. and integers that will give you Pi. (You can do so with an imaginary number, though...more on that in a later post, perhaps.)

Pi's value to 50 decimal digits is:

3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510
You do not need more than 11 decimal points to calculate the circumference of a circle the size of the Earth to the precision of a millimeter. The Earth is not a circle, but a circle with a radius of 6378 km is a close approximation to the Earth's equatorial radius.

Back to Pi.

Pi can be approximated by:

Pi = 4/1 - 4/3 + 4/5 - 4/7 + 4/9 - 4/11 + 4/13 - 4/15 + ...

The more terms you include, the closer to Pi you get.
There are literally dozens of other ways to approximate Pi.

Pi is used a lot in physics and other physical sciences (geophysics, electromagnetism, hydrology, etcetera, etcetera, etcetera) because of its relation to circles, cycles, etc.

Pi is used in pretty much all mathematics.

Finally, here is my favorite mathematical relation:

e^(i*Pi) +1 = 0.

This combines the basic operators (addition, multiplication, and power) with some of the most fundamental numbers in mathematics:

square root of negative 1 = i
Pi
e, the base of the natural log, another transcendental, irrational number
1 is unity in multiplication
0 is unity in addition

Happy Pi Day (even though I despise the way of referring to dates as MM/DD/YYYY)!

1 comment:

Grumpator said...

With regards to the National Pi Day resolution, at least they can agree on something.