Numbers

What are numbers?

The earliest numbers were used to count things.^[1] The natural numbers are the counting numbers 1, 2, 3, 4 and so on.^{[1, 8]} New kinds of number were later added to fill gaps in the old ones,^[7] and the idea of what a number is has changed greatly over the history of mathematics.^[13]

A note on convention: this page follows the OpenStax textbooks, which start the natural numbers at 1 and call the natural numbers plus zero the whole numbers.^{[1, 2]}

How it works: the number sets

The integers, rationals, reals and complex numbers each fill a gap left by the set before.^[7]

• Integers are the counting numbers, their opposites and zero.^{[3, 8]} A negative number is less than 0, and zero itself is neither positive nor negative.^{[3, 1]}
• Rational numbers can be written as p/q, where p and q are integers and q is not 0.^{[4, 1]} Every integer is rational, since it can be written over 1.^{[1, 4]}
• Irrational numbers cannot be written as a ratio of two integers, and their decimals never stop and never repeat.^[4] The circle number π is one.^[4]
• Real numbers are the rationals and irrationals together; a number cannot be both.^{[1, 4]} Each real number matches exactly one point on the number line, and each point exactly one real number.^[1]
• Complex numbers have the form a + bi, where i, the imaginary unit, is defined as the square root of −1.^[7]

The sets nest inside each other: every counting number is a whole number, every whole number is an integer, and every integer is rational.^[6] A real number is a complex number with b = 0.^[7]

Some natural numbers have only two divisors, 1 and themselves: these are the primes, such as 7 and 19.^[8] Numbers with more than two factors are composite.^[5]

The math (optional): worked examples

Place value. In our system a digit’s value depends on its position, so 537 and 735 are different numbers.^[2] The place values are powers of 10, which is why it is called base 10.^[9] For example, 138 is 100 + 30 + 8.^[2]

Decimals. A rational number’s decimal either stops, as in 15/8 = 1.875, or repeats, as in 4/11 = 0.363636….^[1]

Prime factors. The Fundamental Theorem of Arithmetic says every natural number except 1 is a product of primes in exactly one way, apart from their order.^{[8, 5]} For example, 80 = 2 × 2 × 2 × 2 × 5, written 2⁴ × 5.^[8]

Imaginary numbers. Squaring an imaginary number gives a negative real number, so i² = −1.^[7] Complex numbers do not fit on the number line, but they can be drawn on a plane: −2 + 3i is the point (−2, 3).^[7]

History

According to OpenStax, numbers were first used for counting in the Middle East 100 centuries ago.^[1]

Zero. MacTutor separates two uses of zero: an empty-place marker, and a number in its own right.^[11] The Babylonians used place value for over 1,000 years without a placeholder, and only around 400 BC put two wedge symbols in the empty place.^[11] The Maya used a base-20 place-value system with a zero symbol by 665.^[11] Sources differ on dates for India: OpenStax says zero was added to the number system about the fifth century CE, and credits Aryabhata with place-value notation in that century and Brahmagupta with the zero symbol roughly a century later.^{[1, 9]} MacTutor says zero was certainly in use as a number by around 650, and describes a Gwalior inscription dated 876 that writes 270 and 50 almost as we do.^[11]

Negatives. In 628 Brahmagupta wrote the Brahmasphutasiddhanta, defining zero as the result of subtracting a number from itself and giving rules in terms of fortunes and debts.^[12] OpenStax likewise notes that negative numbers were used in seventh-century India for equations and debts.^[1]

Spreading the digits. The numerals 0 to 9 began in India and passed through Persia to the Middle East, then to Europe, eventually replacing Roman numerals.^[9] Fibonacci’s Liber abaci of 1202 introduced them to Europe.^[18] In 1585 Stevin introduced decimal fractions and argued that irrational and negative numbers should all be treated as numbers.^[13]

Irrationals and complex numbers. Plato reports that Theodorus showed the square roots of 3, 5 and so on up to 17 are incommensurable with the unit, meaning no common measure fits both.^[13] Cubic formulas later forced square roots of negative numbers.^{[19, 16]} Bombelli, whose Algebra first appeared in 1572, was the first to write down the rules for adding, subtracting and multiplying complex numbers.^[16] Caspar Wessel was the first to publish a geometric picture of complex numbers; the approach attributed to Argand treats i as a rotation through 90°.^[17]

Rigour. In the nineteenth century Weierstrass, Heine, Méray, Cantor and Dedekind gave rigorous definitions of the real numbers.^[14] Cantor proved in 1873 that the rationals can be matched one-to-one with the natural numbers, and published in 1874 that the reals cannot.^[15]

Common misconceptions

“n ÷ 0 = ∞.” Bhaskara proposed this, but MacTutor explains that it would make every number equal.^[11] Brahmagupta’s claim that 0 ÷ 0 = 0 was also wrong.^[11]

“1 is a prime number.” The number 1 is neither prime nor composite.^[8]

“Imaginary numbers sit somewhere on the number line.” They cannot be plotted on it; they need a plane.^[7]

Going further

On this map, numbers lead on to Algebra, where letters stand for them, and to Calculus.