Thursday, March 14, 2019
There are a number of fascinating constants, ... well, fascinating in you like mathematics.
If you have a circle on a perfectly flat plain, and it has a radius of one, then the area of that circle is some number, and that number is given a Greek letter, the letter Pi.
At one point in my youth, for no good reason at all I memorized the first 100 digits of Pi. The first part of that mantra is still stuck in my head today:
I've long forgotten the other 85 digits that I once new. But it doesn't matter much. To put it in perspective, those 15 digits, if they were the accuracy of the distance from the Sun to the earth, in inches, it would be to an accuracy of about half an inch. Knowing that to more accuracy than that is pretty useless too.
Why is this number Pi "irrational"? Because it is not a ratio of two integers. There is no repeating pattern in those digits of Pi. If there were, like say this, four digits over and over:
X = 3.141614161416....
then multiply by just enough to slide those 4 digits to the left by four spots:
1000 X = 31416.14161416...
If you subtract the X from 1000 X on the left you get 999X, if you subtract the numbers on the right the repeating pattern has slid exactly enough to cancel out on the right:
1000X - X = 31416.14161416 ... - 3.14161416 ...
999X = 31413
X = 31413/999
A rational number. So trust me there is no repeating pattern ever in Pi.
I found this somewhere, but here is another expression of Pi:
Pi/4 = 1/1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 ...
Happy area of a circle of radius one day.