What is a number? Natural, rational, real and imaginary

A number is an abstract idea used to count, order and measure: "three" is what three apples, three days and three notes of music have in common, with the apples and days taken away. Mathematics has built a series of number systems, each containing the last: counting numbers, then zero and negatives, then fractions, then the irrational numbers that fill the line, then imaginary numbers. Each was added to answer a question the older numbers couldn't.

In short

• A number is the idea, not the symbol: 7, VII and 111 in binary are three numerals for one number.
• The main systems nest inside each other, natural ⊂ integer ⊂ rational ⊂ real ⊂ complex, and each extension solves an equation the previous one couldn't.
• Some numbers, such as √2, can't be written as any fraction, and a short proof from about 2,500 years ago shows why.

What does a number actually stand for?

Numbers do two everyday jobs. Cardinal numbers say how many: five chairs. Ordinal numbers say which position: the fifth chair. They do a third job when they measure: a board 1.8 meters long, a temperature of −4 °C. A measurement is not a count of anything; it is a comparison with a chosen unit, and it can land between whole numbers.

A number is also different from the way it is written. The marks "7", "VII" and the binary "111" are numerals, symbols that name the same number. That is why a computer can store any number as 0s and 1s, as the note on how binary works explains, without changing what the number is. The distinction sounds fussy, but it is what lets the same arithmetic work in Roman numerals, Chinese rods, decimal digits and a laptop.

What a number is philosophically, whether something that exists independently or an idea people made up, is still argued. Mathematics doesn't need to settle that to use them. Since the late nineteenth century, mathematicians have defined each system precisely from the one before, starting from a short list of rules for the counting numbers, so that every property can be proved rather than assumed.

How did the number systems grow?

One way to see why each kind of number exists is to look at the equation it was needed for. Each row of the table can do everything the rows above it can, plus one thing more.

System	Symbol	Example	The equation it can solve that the previous system couldn't
Natural numbers	ℕ	1, 2, 3, 47	(Counting: how many?)
Integers	ℤ	−3, 0, 12	x + 5 = 2, so x = −3
Rational numbers	ℚ	½, −7/4, 0.333…	3x = 2, so x = 2/3
Real numbers	ℝ	√2, π, 1.5	x² = 2, so x = √2
Complex numbers	ℂ	3 + 2i, i	x² = −1, so x = i

The letters are standard: ℤ from the German Zahlen, "numbers", and ℚ for quotient, the result of a division. Whether zero belongs among the natural numbers depends on the textbook; set theorists usually include it and number theorists often don't.

Zero and the negatives

Zero has two lives: as a placeholder in written numbers, like the 0 in 105, and as a number in its own right that you can add and multiply. The second idea took centuries and is the subject of its own note, who invented zero. Negative numbers came alongside it. According to Wikipedia's history of numbers, Chinese mathematicians used red counting rods for positive and black for negative quantities around two thousand years ago, and in India Brahmagupta set out rules for them in 628. In Europe they were widely resisted until the seventeenth century; many mathematicians threw away negative answers to equations as meaningless. Debts made the idea concrete: owing 3 is −3.

Fractions and the rationals

A rational number is any number that can be written as one integer divided by another, p/q, with q not zero. Every rational number has a decimal that either stops (¾ = 0.75) or repeats forever (⅓ = 0.333…, 1/7 = 0.142857142857…). That gives a simple test: a decimal that goes on forever without ever falling into a repeating block is not rational.

Why can't √2 be written as a fraction?

The square root of 2 is the length of the diagonal of a square with sides of 1, by the Pythagorean theorem. It is about 1.41421356. A Babylonian clay tablet known as YBC 7289, from about 1800 to 1600 BCE, already gives it correct to about six decimal places. But no fraction gives it exactly, and the proof, known to Greek mathematicians by the fifth century BCE, uses nothing beyond even and odd numbers.

1. Suppose it can. Say √2 = p/q, where p and q are whole numbers and the fraction is in lowest terms, meaning p and q share no common factor. Any fraction can be reduced this way.
2. Square both sides. 2 = p²/q², so p² = 2q². That means p² is even.
3. So p is even. The square of an odd number is odd, so if p² is even, p must be even. Write p = 2k for some whole number k.
4. Substitute. (2k)² = 2q² gives 4k² = 2q², so q² = 2k². That means q² is even, so q is even too.
5. Contradiction. Now p and q are both even, so both divide by 2. But the fraction was assumed to be in lowest terms. The assumption in step 1 must be false.

So √2 is irrational: no ratio of whole numbers equals it, which is what the word means, "not a ratio". The style of argument, assuming the opposite and deriving something impossible, is called proof by contradiction, and it is one of the main tools of a mathematical proof. Legend says the discovery was so unsettling to the followers of Pythagoras that one of them, Hippasus, was drowned for revealing it, but there is little historical evidence for the story.

Irrational numbers are not rare oddities. √3, √5, the golden ratio and pi are all irrational, and in a precise sense, which the note on infinity sets out, there are far more irrational numbers than rational ones. Together, the rationals and irrationals make up the real numbers: every point on a continuous number line, with no gaps.

What are imaginary numbers, and are they real?

No real number squared gives a negative result: 3² and (−3)² are both 9. So x² = −1 has no real solution. Imaginary numbers fill that gap by defining a new number, i, with i² = −1. A complex number combines both kinds, a + bi, such as 3 + 2i.

They arrived not to solve x² = −1, which mathematicians were happy to call impossible, but because Italian mathematicians of the 1500s, among them Gerolamo Cardano, found formulas for cubic equations that sometimes passed through square roots of negative numbers on the way to perfectly ordinary real answers. The name "imaginary" was meant as an insult; René Descartes is often credited with it, in 1637. They were fully accepted only after Caspar Wessel in 1799, and Carl Friedrich Gauss, who popularized the idea a few decades later, showed that complex numbers are points on a plane: a sideways number line at right angles to the real one.

That picture explains why they are useful. Multiplying by i rotates a point a quarter turn around zero, so complex numbers are a natural language for anything that rotates or oscillates. Electrical engineers use them to calculate alternating currents, and quantum mechanics is written in them. They are no less real than negative numbers, which also once seemed absurd.

Each step in the table also cost something. Going from integers to rationals loses the idea of a "next" number: there is no fraction right after ½. Going to complex numbers loses order: it makes no sense to ask whether 3 + 2i is bigger than 2 + 3i. Every extension is a trade.

What do people get wrong about numbers?

The most common mistake is to think of irrational numbers as approximations. √2 is an exact number, the precise length of that diagonal; 1.41421356 is the approximation. The second is to treat "imaginary" as meaning made-up, when it is only a name kept from an old argument. A third is to treat infinity as a very large number. It isn't one of the numbers in the table at all, and arithmetic with it follows different rules.

From here, numbers lead in two directions. Inward, to the structure of the whole numbers themselves, where prime numbers are the building blocks. Outward, to the question of how many numbers there are, and whether some infinite collections are bigger than others.

Check yourself

• Without looking back, give the equation each new number system was needed for, from integers to complex numbers.
• Walk through the proof that √2 is irrational, and say where the contradiction comes from.
• How can you tell from its decimal whether a number is rational?

Sources

• Number — Wikipedia
• Square root of 2 — Wikipedia
• Irrational number — Eric Weisstein, Wolfram MathWorld
• Complex number — Eric Weisstein, Wolfram MathWorld
• Girolamo Cardano — MacTutor History of Mathematics, University of St Andrews