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Mathematicians like all kinds of numbers these days, but it has apparently not always been so. In fact, they used to be positively hostile to accepting new types of number. You can see this process in the names that different kinds of number still have.
Natural numbers, or positive whole numbers starting from 1, have been there from the beginning. Even the Romans knew how to write them down. I will not be talking much about the Romans when I write about the history of maths, as they were by and large not very good at it, as you can tell from the ridiculous numbering system they invented.
Negative whole numbers seem almost as innocent, and indeed the Chinese were relaxed about them as long ago as 200 BC. The Greeks, despite being much better at maths than the Romans, were not quite as relaxed. Diophantus, a third-century Greek mathematician known as the "father of algebra", wrote that the equation "4 x + 20 = 4" is absurd, because then x would have to be -4. But the Greeks accepted negative numbers too in due course.
Then there was zero. Zero was a late arrival on the scene, and has a fascinating history for a mere nothing, which I will write about another time.
Next we have rational numbers, otherwise known as ratios or fractions, in all their flavours, proper, mixed and improper. "Ratio" is the Latin word for "reason", and the Greeks (the more thoughtful ones, anyway, if not the worshippers of Dionysius) loved rationality. Pythagoras especially loved it, and he thought all numbers could be expressed as ratios, so they were all rational. He was horrified, therefore, when one of his students proved that the square root of two cannot be expressed as a ratio. It was irrational! Help!
It's not hard to prove that's the case. Suppose √2 can be expressed as a fraction, p/q, where p and q are integers (whole numbers) with no common factors; you can express any fraction in this way. Then p²/q² = 2, so p² = 2 q². That means p is even, because only even numbers have even squares. So p = 2r, where r is another whole number. But then 2 q² = p² = 4 r², so q² = 2 r², so q is even as well. In other words, both p and q turn out to have 2 as a factor. But we assumed they didn't have any common factors, didn't we? So something is very wrong, and it must be our original supposition, because everything else follows from that. Therefore, the square root of two can't be expressed as a fraction. This kind of trick is called proof by contradiction, and we'll be seeing it again.
It turns out, though, that some irrational numbers are more irrational than others. The square root of two isn't rational, but at least it's a solution to a simple sort of equation, consisting only of powers of a variable x multiplied by integers. In this case, the equation is x² = 2. But it turns out that some numbers are not just irrational, they're transcendental: there's no equation of that simple sort in which they can be substituted for the "x" to make it true. π is like that. So is e, which we'll meet later. This was proved relatively recently (well, relative to the ancient Greeks, anyway): 1873 in the case of π, and 1882 in the case of e.
And if √2 is weird, √-1 is even weirder. In fact it's so weird that it's unreal; try multiplying any ordinary number by itself, and the result will never be negative. So mathematicians call numbers like √-1 imaginary, and in the process admit, grudgingly, that all the other numbers we've seen so far, even the highly irrational ones, are real by comparison. But that takes us into an area that is rather...complex, so we won't go there, at least for now. And if you want to go further in a slightly different direction, there are superreal, hyperreal and surreal numbers to explore. But that might be a bit too much reality to consume in one sitting.
I'm sure the mention of transcendental numbers has made you think of people sitting cross-legged and chanting, so I'll finish with a joke that probably dates back to the 1960s:
Q. Why doesn't the Maharishi need anaesthetic when he has his teeth filled?
A. Because he's learned to transcend dental medication.
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