3/14 is Pi Day - Let's Celebrate

Every year, math teachers around the world celebrate Pi Day on March 14 (as in 3/14 or 3.14).We sometimes say that pi is 3.14, but we really mean that pi to two decimal places is 3.14. In fact pi doesn't have a decimal expansion that terminates (1/2 = .5) or repeats (1/3 = .33333 repeating forever). Pi is an example of an irrational number - it cannot be expressed as a ratio of two integers. There are some interesting irrational numbers:

i) pi - the ratio of the circumference of a circle to its diameter (about 3.14);
ii) e - the base for the natural logarithms and an important constant in population growth (about 2.718);
iii) the square root of 2 (about 1.414);

but there are a lot of irrational numbers that are just random collections of digits without any particular purpose. That would be OK if it didn't turn out that there are so many of them. In the late 1800's German mathematician George Cantor was turning mathematics upside down with his study of infinite sets. He stated that even though we can't count all the elements in an infinite set, we can assign them a "count" by putting them into a one - to - one correspondence with another set. From there he postulated that infinite sets come in different sizes. The smallest infinite set is the set of natural numbers -  1, 2, 3, 4, and so on. He used the Hebrew symbol aleph for counting infinite sets and assigned the size of the natural numbers to aleph null (here null is like zero).
That's where it started to get a little weird. Unlike finite sets, infinite sets don't behave well when you add elements. For example, suppose we looked at the set of integers: . . ., -3, -2, -1, 0, 1, 2, 3, . . . You might think this set is roughly twice as big as the natural numbers. But Cantor said that this set is the same size as the natural numbers. And he showed it by matching up the two sets

naturals       1     2      3     4      5      6      7      8      9     10     11      . . .

integers       0     1     -1     2     -2      3     -3     4     -4      5      -5      . . .

Neither set is going to run out before the other one does, so there is a one - to - one correspondence between the two sets. Therefore the sets are the same size. Cantor went on to show that the set of all rational numbers, i. e., fractions like 2/3, 17/5, -6/11, and so on, is the same size as the natural numbers as well. At that point, we might be forgiven if we think that, OK,  all infinite sets of numbers are the same size. It turns out the irrational numbers are a bigger set than aleph null. Here's how we show that. The key to matching up to the natural numbers is to put your set in some kind of order. If you look up a few lines, you can see that's what we did with the integers. We found a way to list them in an order that could then be matched up with 1, 2, 3, 4, and so on. Suppose we had some kind of order for all the irrational numbers.

first:        .328123986512970245 and so on
second:   .776823919374529476 and so on
third:      .554376920472539402 and so on
fourth:    .019283524377253628 and so on

It doesn't matter what the order is, just assume we have one and every irrational number is somewhere on the list. We can now make a new number that can't be on our list. We do that by making a decimal that is different from the first one in the first decimal place, different from the second one in the second decimal place, different from the third one in the third decimal place, and so on. Let's arbitrarily go one bigger each time with a 9 rolling over to a 0. Looking at our ordering, a new irrational number would have a 4 in the first decimal place (one bigger than the 3 in the first number), an 8 in the second decimal place (one bigger than the 7 in the second number), a 5 in the third decimal place (one bigger than the 4 in the third number), and a 3 in the fourth decimal place (one bigger than the 2 in the fourth number). Remember each irrational number keeps going to the right forever and our list keeps going down forever. So we keep up the process we outlined above. The new number we make is different from the first number (4 instead of 3), different from the second number (8 instead of 7), different from the third number (5 instead of 4), and so on forever. So it's a new number that wasn't on the list. So our ordering can't include every irrational number and the set of irrational numbers must be bigger than the set of natural numbers.Ta da!

Unfortunately, Cantor raised almost as many questions as he answered. The set of real numbers, which includes the rational and irrational numbers together, is commonly called the continuum, and the letter c is used to stand for the size of the set of real numbers. They couldn't use an aleph name because they didn't know if it should be aleph - one or aleph - two or whatever (remember aleph - zero was the natural numbers). So the continuum hypothesis is that c is really aleph - one, that is, the real numbers form the next size of infinity up from the naturals. Unfortunately nobody has been able to prove or disprove this in the 120 years since the question was first posed.

So, enjoy Pi day. I prefer apple or pecan or cherry pie myself, but it's the thought that counts. You may not be a pie person, but there are a lot of them out there. For example, as Nathan would tell you, if you hang around with World War 2 veterans for any length of time, you're going to eat a lot of pie.

HAPPY  PI  DAY !!!