Numbers and number systems

Counting is older than writing. Tally marks on bones, such as the Ishango bone from Central Africa, may date back more than 20,000 years. The first written numerals appeared in Mesopotamia and Egypt around 5,000 years ago, mainly for recording trade, taxes and harvests.

Early number systems varied widely. Egyptians used separate symbols for 1, 10, 100 and so on, repeating them as needed. Roman numerals worked in a similar way and are still used on clocks and in book chapters, though they are awkward for arithmetic.

The breakthrough was place value: the position of a digit determines its value. In 345, the 3 means three hundreds. The Babylonians used a place-value system based on 60, which survives in our 60 minutes in an hour and 360 degrees in a circle.

Our everyday system is base 10, probably because humans have ten fingers. The Maya used base 20. Computers use base 2, or binary, with only 0 and 1, while programmers also use base 16, hexadecimal, as a compact way to write binary.

A place-value system needs a way to mark an empty position. Babylonians eventually used a placeholder symbol, and the Maya had a zero symbol in their calendar system. But treating zero as a number in its own right, with rules for arithmetic, was developed in India.

The Indian mathematician Brahmagupta set out rules for zero and negative numbers in 628 CE. The Indian decimal system with zero spread through the Islamic world, where al-Khwarizmi wrote about it, and reached Europe in the Middle Ages, promoted by Leonardo of Pisa, known as Fibonacci, in his book Liber Abaci of 1202.

• Natural numbers: the counting numbers 1, 2, 3 and so on, sometimes including 0.
• Integers: whole numbers including negatives, such as minus 3, 0 and 7.
• Rational numbers: numbers that can be written as a fraction of two integers, such as one half.
• Irrational numbers: numbers that cannot be written as fractions, such as pi and the square root of 2, whose decimals never end or repeat.
• Real numbers: all rational and irrational numbers together, covering every point on the number line.
• Complex numbers: numbers that include the imaginary unit i, the square root of minus 1.

According to tradition, followers of Pythagoras were disturbed to discover that the diagonal of a square with sides of length 1 cannot be expressed as a ratio of whole numbers. The square root of 2 was the first known irrational number. Pi, the ratio of a circle's circumference to its diameter, was proved irrational by Johann Lambert in 1761.

Imaginary numbers were introduced by Renaissance mathematicians solving cubic equations, and the name was originally dismissive. Yet complex numbers turned out to be indispensable in physics and engineering, describing alternating current, waves and quantum mechanics.

There is no largest number, since you can always add one. But in the late nineteenth century Georg Cantor showed that infinities come in different sizes. The set of natural numbers and the set of fractions are the same size, since they can be paired off one to one.

The set of real numbers, however, is strictly larger: Cantor's diagonal argument proved that no list could ever include them all. His ideas were controversial at the time but became foundational to modern mathematics.