June 1 – Heads for the Hills

Today’s Factismal: While he was a prisoner of war, John Kerrich flipped a coin 10,000 times and got 5,017 heads and 4,983 tails.

You probably didn’t know that today is National Flip A Coin Day, which is  dedicated to improving our understanding of statistics. Though the subject may seem boring, statistics plays an essential part in our lives. From determining if a heat wave is unusual to discovering if the predictions being made by NOAA are worth heeding to simply telling if traffic will be heavier than usual, statistics drives many of our public policy decisions as well as serving as the way we judge the outcomes of experiments. And the amazing thing about statistics is that many of the most powerful statistical tools are based in, you guessed it, coin flips.

But if you’d rather go have a root canal from Orin Scrivello, DDS, than do statistics, then may I suggest an alternative? Panamath is an ongoing citizen science project that is trying to understand our “number sense”, that is, the way that we estimate the number of people in a group or the number of slices of pie left on the table. If you’d like to help, then head on over to
http://panamath.org/

March 3 – IDIC

Today’s factismal: There are more real numbers between 0 and 1 than there are integers between -∞ and +∞.

Today is Georg Cantor’s 169th birthday; we know this because we can count integers. But it took Cantor to show that there are some things that we couldn’t count and that there’s infinite and really infinite.

Since the days of the Greeks, philosophers have argued over what “infinity” means. Was it just a really big number or was it something even more interesting. In the 1920s, Hilbert described infinity best through paradox; he pointed out that a hotel with an infinite number of rooms would always have room for more guests even if every room was already occupied. Because the rooms are discrete units, you could even (if you had an infinite amount of time) count them all without skipping any; this is what mathematicians call “countably infinite”. And it turns out  that countably infinite is the smallest infinity possible.

The natural numbers (1, 2, 3, 4, … , +∞) are countably infinite. You start at 1 and keep moving forward in regular, discrete intervals until you hit +∞ after an infinite time. And it turns out that the integers are also countably infinite, because you can match every negative integer to a positive one. As a result, the integers, which includes both positive and negative numbers, is exactly as large as the whole numbers, which just has positive integers.

But then things get weird in mathland as the rational numbers, which are defined as of ratios of integers, turns out to be another countably infinite series. To see this, you need to look at Cantor’s clever diagram. He made a square with the numerator across the top (since that’s where numerators belong) and denominators down the side. Because every integer can be matched up with every numerator and every denominator, the total number of rational numbers is just countably infinite.

Numerators
1 2 3 4
Denominators 1 1/1 2/1 3/1 4/1 ∞/1
2 1/2 2/2 3/2 4/2 ∞/2
3 1/3 2/3 3/3 4/3 ∞/3
4… 1/4 2/4 3/4 4/4 ∞/3
1/∞ 2/∞ 3/∞ 4/∞ ∞/∞

So, by applying a little logic and ignoring his common sense, Cantor was able to show that the natural numbers, the whole numbers, the integers, and the rational numbers were all equally big. That would have been enough to satisfy just about anyone except a mathematician. What Cantor and his fellow mathematicians wanted to know was “Is there anything bigger than countably infinite?” And it turns out there there is.

Numerators
1 2 3 4
Denominators 1 0/1 2/1 3/1 4/1 ∞/1
2 1/2 1/2 3/2 4/2 ∞/2
3 1/3 2/3 2/3 4/3 ∞/3
4… 1/4 2/4 3/4 3/4 ∞/4
1/∞ 2/∞ 3/∞ 4/∞ 17/∞

Cantor proved that there were larger infinities by changing the diagram argument slightly. He decided to use the diagonal of the diagram to define a new number. And he defined each diagonal number by saying that it couldn’t be equal to the previous one. As a result, he created a non-repeating, non-rational number; what we call an irrational number. Even better, because the values didn’t match any of the other values, the new number couldn’t be found anywhere in the diagram and so couldn’t be counted. The new number was uncountably infinite. And there were more of these irrational numbers than there were rational numbers. Or, as another mathematician pointed out, there were more real numbers between 0 and 1 than there integers between 0 and ∞.

Though the result seems contradictory, the types of infinity turn out to have some very practical uses. Cantor’s sets have been used to help develop new and better data encryption techniques and to improve computer animation. And we’re still working on new ideas in math, including how our intuitions about numbers can be wrong. If you’d like to help scientists as they try to understand how people think about numbers then head over to Panamath and take their “how many?” test. I promise that the number won’t be infinite and that the test won’t take infinitely long!
http://panamath.org/

February 5 – Pi, schmi

Today’s Factismal: In 1879, the Indiana State House of Representatives unanimously passed a bill making pi equal to 3.2, 4.0, or 3.23 (you had your choice).

Perhaps the most maligned number in all of history is pi (also written as π). When the Babylonians first proposed that the ratio of the diameter of a circle to its circumference was a constant, they didn’t know the trouble that they would cause. The Greeks fought over whether it was a rational number that could be calculated by the ratio of two other numbers(e.g., 142/45) or an irrational number that could only be approximated (e.g., by throwing needles on the floor); during a long sea voyage, one sect even threw a member overboard when he proved that pi had to be irrational.

And the confusion continued to the modern day. Every year, there are people who try to demonstrate that pi is really a rational number just because it offends them that it might be irrational. And every once in a while, those people manage to get someone to listen.

That’s what happened in Indiana in 1879. A physician by the name of Ed Goodwin who had a love/hate relationship with pi made friends with T.I. Record who was state representative. Goodwin persuaded his friend to introduce a bill into the Indiana House of Representatives that would “revolutionize mathematics”, which was putting it mildly. Record managed to bring the bill to his committee, which happened to be the Committee on Swamp Lands. The other legislators decided that they weren’t the right group to debate it and sent the bill over to the Committee on Education which amazingly gave it a “do pass” recommendation. (Obviously, there was a need for more education in the Committee on Education.) It went to the full House, which passed it unanimously.

The next step in turning the bill into a law was getting it passed by the State Senate. The senators put the bill in the Committee on Temperance, which again gave it a “do pass” recommendation. The bill was then read out on the floor of the State Senate where a professor of mathematics from Purdue University just happened to be sitting in the gallery, listening to the arguments. The professor went to the office of his State Senator and explained what a bunch of idiots they would look like if they passed the bill. The senator then passed the word to his colleagues, and the bill was allowed to die quietly on the floor without ever being brought up for a vote.

While pi is irrational, you don’t have to be. You can take part in a mathematics experiment by growing sunflowers and counting the seeds in order to prove an idea by Alan Turing, one of the great mathematicians of the last century.
http://www.turingsunflowers.com/