Place value chart

A place-value chart makes visible the idea that the decimal system rests on: a digit's worth depends on where it stands. The 3 in 307 means three hundreds, the 3 in 37 three tens, and the zero is there to hold an empty place so the 3 stays put.

Positional writing is older than our digits. Babylonian scribes wrote numbers by position in base sixty from around 2000 BCE, but for most of that history without a zero, so an empty place had to be guessed from context. The base-ten system used across the world today comes from India. The Bakhshali manuscript, a birch-bark arithmetic text found in 1881 near Peshawar, uses a dot for zero, and radiocarbon dating published by Oxford's Bodleian Library in 2017 put some of its leaves in the third or fourth century CE. In 628 Brahmagupta wrote down rules for calculating with zero as a number in its own right.

The system travelled west through Baghdad, where al-Khwarizmi wrote on Indian reckoning around 825; Latin translations of that work gave Europe the word algorithm. Fibonacci's Liber Abaci of 1202 argued for the new numerals over Roman ones, which have no places at all.

Places to the right of the ones came later. Al-Uqlidisi used decimal fractions in Damascus in 952–953, and in 1585 the Flemish engineer Simon Stevin published De Thiende, a short pamphlet that set out to teach everyone to calculate with tenths, hundredths and thousandths. The decimal point itself was popularised by John Napier, who used it in 1617. Charts in Indian schools still group the places their own way, in thousands, lakhs and crores.

Key properties

• Each place is worth ten times the place to its right: a thousand is ten hundreds, a hundredth is a tenth of a tenth.
• The value of a digit is the digit times the value of its place, so the 5 in 52.4 is worth 5 × 10 = 50.
• Expanded form writes a number as the sum of those values: 4,307 = 4,000 + 300 + 7.
• A zero inside a number holds a place: remove the 0 from 4,307 and the 4 and 3 slide right to make 437.
• Multiplying by 10 moves every digit one place to the left, and dividing by 10 moves every digit one place to the right.
• Nine places reach 999,999,999, one less than a billion in the short scale used in English today.
• The places after the point, tenths, hundredths and thousandths, are the fractions 1/10, 1/100 and 1/1000.

Where they turn up

• The abacus and the Japanese soroban, where each rod is one place and a bead on it counts ones, tens or hundreds of that place.
• Odometers and mechanical counters, where each wheel is a place and rolls the next one over at ten.
• Cheques and bank forms that ask for an amount in figures and in words, which is the expanded form spelled out.
• Indian classrooms, whose charts name the places ones, tens, hundreds, thousands, ten thousands, lakhs, ten lakhs, crores, and group digits as 1,00,00,000.
• Money: in 52.38 the tenths and hundredths are dimes and cents, or tens and ones of pence.