Multiplication table

The oldest decimal multiplication table yet found is Chinese. Among the Tsinghua bamboo strips, a hoard of Warring States manuscripts given to Tsinghua University in 2008 and dated to around 305 BCE, are twenty-one strips that together form a grid: read along a row and down a column and the product sits where the two meet. It works for whole numbers up to 99 and for halves, and it predates every other known decimal table by centuries.

Babylonian scribes kept multiplication tables much earlier, on clay tablets from roughly 2000–1600 BCE, but in base sixty, and each tablet usually listed the multiples of a single number. The square chart most of Europe calls the table of Pythagoras — tavola pitagorica, table de Pythagore — has no known link to Pythagoras. The attribution comes through later writers; a ten-by-ten grid of this kind appears in the Introduction to Arithmetic of Nicomachus of Gerasa, written around 100 CE.

Learning tables by heart is an old practice too. The Chinese rhyme was called jiujiu, "nine nines", because it was recited from 9 × 9 downwards; Japan took it up as kuku, and wooden tablets from the eighth century carry fragments of it. In 1617 John Napier cut the table into numbered rods, Napier's bones, so that long multiplication could be done by reading off and adding.

Why twelve? British and Indian charts stop at 12 × 12 largely for reasons of money and measure: there were twelve pence in a shilling and twelve inches in a foot. Much of Europe stops at 10 × 10. England made times tables a national test in 2022, when Year 4 pupils first sat the statutory multiplication tables check, which runs to 12 × 12.

Key properties

• Multiplication is commutative, so the chart is symmetric about its diagonal: a 12 × 12 chart has 144 cells but only 78 different pairs of factors.
• The diagonal of a multiplication chart holds the square numbers; on an addition chart it holds the doubles.
• Every row and every column is an arithmetic sequence: row k counts up in steps of k.
• The entries of an n × n chart add up to (n(n + 1)/2)², the square of a triangular number — 3,025 for 10 × 10 and 6,084 for 12 × 12.
• Many products repeat: a 12 × 12 chart contains only 59 different numbers, and a 10 × 10 chart only 42.
• The share of different products keeps shrinking as the chart grows. Paul Erdős showed in 1955 that it tends to zero, and Kevin Ford worked out the exact rate in 2008.
• In the nine times table up to 10 × 9, the digits of each product add up to 9, because 9 is one less than the base.

Where they turn up

• School exercise books in France, India and many other countries print the times tables on the back cover.
• England's statutory multiplication tables check asks Year 4 pupils 25 questions up to 12 × 12, each against a six-second timer.
• Napier's bones (1617) and the later Genaille–Lucas rulers turned the table into a hand calculator for long multiplication.
• Classroom walls the world over carry a times-table poster, and a blank chart is one of the most printed maths worksheets.
• The finger trick for the nine times table, folding down the k-th finger to read 9 × k, works because the product is always 10(k − 1) + (10 − k).