The counting numbers that we learn as children are so familiar that using them is second nature. They bear the appropriate name natural numbers. From then on, names of numbers become less and less apposite.
The integers include all whole numbers, both negative and positive. We are heading into difficulty: the term negative is so often used in a pejorative sense, implying prohibition, refusal or lack of positive attributes.
The notion of a negative number seems to conflict with our primitive concept of counting. However, when all the integers are arranged in a row – a “number line” – the negative numbers simply reflect the positive ones, stretching to the left instead of to the right.
Rational numbers – or ratios of whole numbers – come next. All the fractions are rational; but the term fraction has the ring of breaking bones, or even fractiousness. The Pythagoreans discovered to their consternation that the square root of two is not a ratio of whole numbers. Somehow, they found such quantities absurd, and the term surd is a hangover of this. But there is nothing absurd about the diagonal of a unit square.
Numbers like √2, that are not whole-number ratios, are called irrational numbers. Again, we have a “negative” connotation, that they are unreasonable or even illogical. Irrational numbers occur “naturally” as the solutions of quadratic and higher-order algebraic equations. Indeed, we have the algebraic numbers, solutions of polynomial equations with integer coefficients.
However, not all numbers are algebraic. Cantor’s theory of the cardinality of sets showed that almost all numbers fall outside this category and are called transcendental numbers. Wow: this suggests that they are unrealizable: abstract, obscure, beyond the Pale.
The algebraic and transcendental numbers together give us the real numbers. But this solid, concrete name has a down-side: it implies that all other numbers are unreal. To add insult to injury, quantities like the square root of minus one, or √(-1), have the unfortunate name imaginary numbers. Cardano described these entities as being “as subtle as they are useless”.
We have Descartes to thank – or to blame – for the catastrophic appellation “imaginary”, which has caused untold confusion, trouble and misunderstanding. It is arguable that root-minus-one is neither more nor less real than root-plus-one.
Combining reals and imaginaries, we get the complex numbers. They are essentially two-dimensional, comprising two parts, and “complex” indicates a whole composed of several parts. But it also has a forbidding tone, hinting at complications and difficulties; complexity, in short. William Rowan Hamilton showed that complex numbers are simply ordered pairs of real numbers – tuples – with straightforward rules for their manipulation.
If we ever introduce EarthSpeak, a universal language, we should try to choose number-names more wisely. But as long as we are using “natural” languages, we are saddled with these unfortunate names for our beloved numbers.