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 nonrepeating, nonrational 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/