Rational thinking

The first way we can expand our number system is by introducing all the negative numbers, forming the set of integers GLOSSARY integersThe set of all whole numbers, both positive and negative. The number 0 is definitely a part of this set. The natural numbers are a part of the integers. Examples: -2,-1,0,1,2, … , ℤ. However, it soon becomes clear that this won’t help us reach a larger cardinality—we’ve effectively doubled the set ℕ by adding in a negative number for each positive number, and we know from our experience with combining the odd and even numbers that doubling an infinite set is not going to change its cardinality.

The next thing we can try is to include all the numbers that can be written as fractions, creating the set of rational numbers GLOSSARY rational numbersThe set of all numbers, both positive and negative, that can be written as a fraction. Equivalently, the set of all numbers whose decimal expansions either terminate or eventually repeat. The natural numbers and the integers are a part of the rational numbers. Examples: -3, 1/3, 0.5, … , ℚ. We seem to have an awful lot of numbers in ℚ—for starters, there’s infinitely-many fractions between any two whole numbers! Clearly, it’s going to be difficult to pair off the set of natural numbers:

$$\mathbb{N}\text{ = \{0, 1, 2, 3, 4, 5, 6, … \}}$$

with the set of rational numbers:

$$\mathbb{Q}\text{ = \{ 0,} \frac{1}{2},\frac{1}{3}, \frac{2}{3}, \frac{1}{4}, \frac{3}{4}, … , 1, \frac{11}{2}, \frac{11}{3} … \}$$

Difficult … but not impossible. Cantor showed how this could be done by thinking about things a little differently. The key is to realise that in order to pair off the rational numbers with the natural numbers, we just need to find a way to list the rational numbers in some sort of order. We can then pair the first rational number in our list with the natural number 1, the second rational in the list with 2, and so on.

On the surface, this seems a tall order—how can we can list the rational numbers, which include fractions, in an order that captures every single one? There’s no such thing as the smallest fraction bigger than 0, for starters! But by doing a few mental zigzags, it turns out that it is possible to make such a list.

To start, we notice that every fraction is made from a pair of natural numbers—one on the top of the fraction (the numerator) and one on the bottom (the denominator). There are two caveats. First, we cannot have 0 on the denominator of the fraction. Second, we can write the same fraction in infinitely-many different ways using different pairs of numbers (for example, ¹⁄₂ is the same as ²⁄₄, and ³⁄₆, and so on).

• Why you can’t divide by zero

  Aside from calculators throwing up errors, and jokes about workbooks catching fire and black holes spontaneously forming, there’s a very valid (and very simple) reason why we cannot divide any number by zero.

  We learn early in school that there are four basic operations in mathematics: addition, subtraction, multiplication and division. To mathematicians, however, there are only two basic operations: addition and multiplication. We get subtraction and division for free, as the opposites of addition and multiplication.

  More specifically, every subtraction question can be thought of as an addition question, and every division question can be thought of as a multiplication question. For instance, a typical multiplication question might look like:

  $$\text{2 × 3 = ?}$$

  Whereas a typical division problem might look like:

  $$\text{6 ÷ 2 = ?}$$

  But what does this mean? When we ask, ‘What is six divided by two?’, we are really asking, ‘What do we multiply two by, in order to get six?’. In other words,

  $$\text{6 ÷ 2 = ?}$$

  is the same problem as:

  $$\text{2 × ? = 6}$$

  So a division problem is really a multiplication problem with the unknown number on the other side of the equation. With that in mind, let’s think about dividing by zero.

  $$\text{1 ÷ 0 = ?}$$

  is the same problem as asking, ‘What do we multiply zero by, to get one?’.

  $$\text{0 × ? = 1}$$

  But there is no number that works! No matter what we multiply zero by, we’re always going to get zero. That’s why we can’t divide one, or two or any other number by zero—there’s no sensible answer, because there’s no number that we can multiply by zero in order to get one, or two, or anything else—aside from zero.

  Which leads to one final point: what about dividing zero by zero?

  $$\text{0 ÷ 0 = ?}$$

  is the same problem as asking, ‘What do we multiply zero by, in order to get zero?’.

  $$\text{0 × ? = 0}$$

  This time the problem is not that there’s no number that works—it’s that every number works! Since multiplying any number by zero gives you zero, it follows that zero divided by zero could be literally any number at all. And, for that reason, it also doesn’t have a sensible answer.

  If we tried to be accommodating and let 0 ÷ 0 be any number you choose, then we’d quickly run into the following problem:

  Since 0 ÷ 0 = 1 and 0 ÷ 0 = 2, that means 1 = 0 ÷ 0 = 2.

  As 1 and 2 are most definitely not the same number, we can’t sensibly define 0 ÷ 0.

Since every rational number can be thought of as a pair of natural numbers, we can draw up the following infinite table to capture every possible rational number:

We only want each fraction to appear once in the table, and at the moment this table captures each fraction over and over again, infinitely-many times. So we take care to eliminate all the duplicates by crossing out each entry in the table where the fraction can be simplified (all the entries where the numerator and denominator share a common factor are removed).

Having done that, we can now zigzag our way through the table to catch every single fraction, like so:

Then all we have to do then is list every fraction in the order we encountered it during the zigzag:

And, just like that, we’ve made our bijection between the rational numbers ℚ and the natural numbers ℕ! We can check that every natural number has a partner (the list of fractions never ends) and that every rational number has a partner (every rational appears exactly once in the table, after we carefully remove the duplicates). Although we didn’t capture all the negative rational numbers in our mapping, the same idea can be easily modified to take them into account.

So it turns out that the set of rational numbers is countable too—it has cardinality ℵ₀. It may seem like we’ve run out of options; that there really is only one type of infinity. But Cantor had one final trick up his sleeve, which he published in his famous 1874 paper. There is one more source of numbers we can draw on.

The real surprise

Apart from natural numbers and rational numbers, there is one more set of numbers that most people are familiar with—the set of real numbers GLOSSARY real numbersThe set of all numbers that live on the number line. The natural numbers, integers and rational numbers are a part of the real numbers—but the real numbers also include numbers that cannot be written as a fraction, like π and the square root of pi, whose decimal expansions never terminate or repeat. Examples: -2, 0, 1/4, √3, log 2, … , ℝ. The set of real numbers includes all the whole numbers and all the fractions, as well as lots of other numbers that can’t be written as a fraction, like π (pi) and the square root of 2.

Cantor was able to prove that it was impossible to create a bijection between the set of real numbers ℝ and the set of natural numbers ℕ. No matter how hard you try, you’ll always end up using up all of the natural numbers and still having real numbers left over, unpaired. There are definitely more real numbers, in a mathematically precise way, than there are natural numbers.

To prove this, Cantor used an elegant mathematical technique known as ‘proof by contradiction’. Cantor began by assuming that you could create the bijection, listing all real numbers in some order, just like we did for the rational numbers, in order to match them with the natural numbers. He then showed that no matter how you had created the list, he’d be able to show you a real number that didn’t appear on your list, thus contradicting your result and proving it couldn’t be done.

Even more incredible, Cantor’s ‘diagonalisation argument’ doesn’t even need all the real numbers to be considered. It turns out that any piece of the number line, no matter how tiny, contains ‘more’ numbers in it than all of the natural numbers (and all of the rationals too)! To illustrate Cantor’s argument, we’ll just take the set of real numbers between 0 and 1 and show that you can’t possibly match them up with the set of all natural numbers.

Suppose that we’ve created our bijection, producing a list containing every single real number between 0 and 1 in some order. Cantor then hands us another real number between 0 and 1 that can’t possibly be on our list. To create this number, he takes the first decimal place of our first number on our list and adds 1 to it, and uses the result as the first decimal place of his new number. (If the first decimal place of our first number was 9, he changes it to a 0). He then takes the second decimal place of our second number and adds 1 to it, using the result as the second decimal place of his new number. This process continues, resulting in a real number that Cantor claims won’t be on our list.

Is he right? Well, let’s try and find Cantor’s number in our list. It can’t be the first number, because the first decimal place (at least) doesn’t match up. It can’t be the second number on our list, because the second decimal place doesn’t match. It can’t be the millionth number on our list, because the millionth decimal place doesn’t match, and so on. No matter how we go about making the list, we can’t even capture every real number between 0 and 1!

Cantor’s argument works for the real numbers because the decimal expansion of a real number doesn’t have to have a pattern to it. This is in contrast to the decimal expansions of rational numbers, which must eventually repeat themselves.

So at last we have finally found a larger infinity than ℵ₀! Perhaps not surprisingly, this new infinity—the cardinality of the set of real numbers ℝ—is called ℵ₁. It’s the second transfinite cardinal number, and our first example of a bigger infinity than the ℵ₀ infinity we know and love.

Conclusion

So although Buzz Lightyear’s goals of travelling ‘to infinity and beyond’ doesn’t make sense in the literal sense—infinity is not a place, nor a destination, nor a final point—you could argue that mathematics allows us to explore infinity and beyond. Different infinite sets can have different cardinalities, and some are larger than others. Beyond the infinity known as ℵ₀ (the cardinality of the natural numbers) there is ℵ₁ (which is larger) … ℵ₂ (which is larger still) … and, in fact, an infinite variety of different infinities.

It may seem esoteric, but the understanding of infinity—and set theory—is vital to understanding the very foundations of mathematics. It’s knowledge that was hard won over millennia, and that knowledge provides us with tools for understanding very real complexities in the world around us.