You probably have some idea of what infinity means. Something that is infinite goes on forever. But it might surprise you to know that there are different kinds of infinity, and some are bigger than others!
As an example, a circle has an infinite number of points. (Yet the circumference of the circle has finite length.) Compare that to a straight line with infinite length. Both have infinitely many points but does the finite length circle have fewer? [The answer is no.]
To understand all this, we have to first talk a bit about numbers.
We can start with the counting numbers. They have been with us the longest. From the beginning humans have found it useful to count things: sheep, coins, people; even ships and mountains.
Before numbers were really understood, humans used a one-to-one relationship between what they were counting and some way of representing the count. Hash marks in clay, pebbles in a bag, knots in a rope, these were all ways of representing how many of something else you had.
If you’ve ever counted using the “prisoner’s hash marks,” you’ve used this early counting method. Each mark stands for one of whatever you’re counting.
Technically, this is called base one notation. One mark (or knot or pebble) stands for one something. (And a rope with 30 knots is a lot easier to carry around than 30 sheep. Or 30 ships!)
Because the process of counting things is ancient, natural, organic, free-range, and sugar-free, the counting numbers are formally known as the natural numbers.
Mathematicians have a symbol that represents the set of all natural numbers: is N (or ℕ).
Many of the natural numbers have names: one, two, three,… fifteen, fifty, hundred, and so on. Others are named by tacking names together: fifty-seven, twelve-hundred-and-ninety-nine, and so on.
It might surprise you, but there is no agreement on whether zero is included in the natural numbers. Some include it; others don’t. It’s generally included in a formal definition of the natural numbers as the first (and only) number explicitly defined.
(Not including it is due to zero having come later in the mathematics of many cultures. For many, at first there was no symbol to stand for the unmarked clay, the unknotted rope, the empty pebble bag.)
The next step along the way was the need to sometimes represent a part of something.
If one day’s work was worth one loaf of bread; a half day’s work is worth a half loaf. How many loaves does a worker who worked 10 half days get?
Somehow you had to add up 10 half loaves.
That example is simple enough, but it’s just the start.
Once you develop a way to handle fractions of things, it gets harder. In school, you’ve probably had to add 2/3 to 1/5, and that’s not as intuitive as adding ten halves!
Numbers which are fractions, that can be represented as a/b, are called rational numbers; their symbol is Q (or ℚ), for quotient.
All natural numbers are also rational numbers. For instance, five is 5/1.
Rational numbers have a decimal expansion form. For example, 1/2 is 0.5, 1/5 is 0.2 and 1/3 is 0.333…
That last one illustrates an important point: the decimal form of rational numbers extends forever. It always extends forever, even if it’s just zeros. Technically speaking, 1/2 is 0.500…, and 1/5 is 0.2000… In both cases, and in all like them, the zeros go on forever.
Another important case comes from 1/11, which is 0.090909…
The decimal expansion of rational numbers either repeats the same digit (including zero) or it repeats a pattern of digits. That pattern can be as short as two digits (as with 1/11), or it can be indeterminately long.
For example. the more complex pattern that comes from 2/7, which is 0.285714285714285714… Can you see where it repeats? (If not, here’s a hint: it appears three times.) Once a pattern begins to repeat, it repeats forever.
If there are rational numbers, you might think there could be irrational numbers.
You’d be right! All numbers with a decimal point are called real numbers; they have the symbol R (or ℝ). Real numbers with repeating sequences are rational; real numbers that never repeat are irrational.
Irrational numbers cannot be represented with fractions (because all fractions result in rational numbers).
And it turns out that nearly all real numbers are irrational numbers! The most famous of which is π (pi). The digits of pi go on forever and never repeat. There is no fraction that equals pi. In fact, pi is transcendental (a special kind of real number), so there is no formula for the exact value of pi.
Now we’re finally ready to get back to the idea of infinity.
You know the natural numbers are infinite.
We can start counting them and keep counting them forever. No matter how big a number we consider, we can always add a one to it to make a bigger number.
This fact is actually the theoretical basis of the natural numbers. The idea is to start by defining a starting number, usually zero but sometimes one, and then just adding one to it to make any other number we want. We use zero to define one, one to define two, two to define three, and so on.
The process of adding a one to a number to get the next number in line is called a successor function.
We can count — or enumerate — the natural numbers because they have a successor function (the successor function called “add one”).
Which brings us to the first important concept of infinity: There is a type of infinity that can be counted.
The counting goes on forever, but we can always name the next number in line, no matter where we are in the count.
It turns out that even the rational numbers are countable.
We can design a successor function that enumerates the rational numbers, because rational numbers have the form a/b, and since ‘a‘ and ‘b‘ are natural numbers (which are countable), rational numbers are countable.
This brings us to the second important concept of infinity: All countable infinities are the same “size.”
Suppose we list the natural numbers: 1, 2, 3,…
And then we also list them by tens: 10, 20, 30,…
Even though the second list skips nine numbers for every number on the list, both lists are the same size. (The same logic applies if we take, for example, just the even numbers: 2, 4, 6…)
We prove this by showing that, for every number on the first list, there is a corresponding number on the second list (and vice-versa).
We call this mapping one set (list) onto another. If the map is one-to-one, then the size of the two sets is the same.
Since we can enumerate them, the set of natural numbers and the set of rational numbers are the same size.
You might think the set of fractions was bigger with an infinite number of fractions between each natural number, but the two sets can indeed be mapped onto each other, which means they must be the same size.
The same size of infinity. Countable infinity.
But now consider the irrational numbers.
What is the first number after pi?
We know the digits of pi go on forever with no pattern. A successor function adds something to the current number to generate a new number. What can we add to pi to generate the next number? We need something along the lines of an infinity of zeros followed by a one.
But infinities go on forever, so the zeros never end, which means we never get to the one!
Which means we can never name the next number after pi. Or the one right below it.
In fact, for any given real number (even if it’s also natural or rational), we can never name the next real number.
Which brings us to the third important concept about infinity: The irrational numbers cannot be counted, because there is no successor function.
What this ends up meaning is that the set of all irrational numbers is a larger infinity than the set of natural (or rational) numbers. There is no map between the two sets.
About 130 years ago, a mathematician by the name of Georg Cantor came up with an elegant proof demonstrating that the real numbers were uncountable.
His proof uses a trick, called the diagonal argument, that I’ll explain when we take this topic up again.
For now, remember these facts about infinity:
- The natural and rational numbers are infinite but countable.
- All countable infinities are the same size.
- The irrational numbers are infinite and uncountable.
- Uncountable infinities have a larger “size” than countable infinities.
Until next time, don’t count your chicken enchiladas until they’re served!
Update 11/10/2014: The diagonal argument is discussed in detail in the Sideband #54: Cantor’s Diagonal post.