Wednesday, 8 January 2014

Oldest Multiplication Table Found?

"From an online article in Nature, Jan 2014:
From a few fragments out of a collection of 23-century-old bamboo strips, historians have pieced together what they say is the world's oldest example of a multiplication table in base 10.
(The Egyptian scrolls of 17th Century BC give a method equivalent to multiplication in base two ,although the scrolls are generally in base 60, see below)

Five years ago, Tsinghua University in Beijing received a donation of nearly 2,500 bamboo strips. Muddy, smelly and teeming with mould, the strips probably originated from the illegal excavation of a tomb, and the donor had purchased them at a Hong Kong market. Researchers at Tsinghua carbon-dated the materials to around 305 bc, during the Warring States period before the unification of China.

Each strip was about 7 to 12 millimetres wide and up to half a metre long, and had a vertical line of ancient Chinese calligraphy painted on it in black ink. Historians realized that the bamboo pieces constituted 65 ancient texts and recognized them to be among the most important artefacts from the period.

When the strips are arranged properly, says Feng, a matrix structure emerges. The top row and the rightmost column contain, arranged from right to left and from top to bottom respectively, the same 19 numbers: 0.5; the integers from 1 to 9; and multiples of 10 from 10 to 90.

The researchers suspect that officials used the multiplication table to calculate surface area of land, yields of crops and the amounts of taxes owed. “We can even use the matrix to do divisions and square roots,” says Feng. “But we can’t be sure that such complicated tasks were performed at the time.”

An older multiplication method (although not technically a multiplication table) is presented in ancient Egyptian scrolls)

In mathematics, ancient Egyptian multiplication (also known as Egyptian multiplication, Ethiopian multiplication, Russian multiplication, or peasant multiplication), one of two multiplication methods used by scribes, was a systematic method for multiplying two numbers that does not require the multiplication table, only the ability to multiply and divide by 2, and to add. It decomposes one of the multiplicands (generally the larger) into a sum of powers of two and creates a table of doublings of the second multiplicand. This method may be called mediation and duplation, where mediation means halving one number and duplation means doubling the other number. It is still used in some areas.

The second Egyptian multiplication and division technique was known from the hieratic Moscow and Rhind Mathematical Papyri written in the seventeenth century B.C. by the scribe Ahmes.

Although in ancient Egypt the concept of base 2 did not exist, the algorithm is essentially the same algorithm as long multiplication after the multiplier and multiplicand are converted to binary. The method as interpreted by conversion to binary is therefore still in wide use today as implemented by binary multiplier circuits in modern computer processors. *wik
This tablet, from about 2000 BC, was a school math book for teaching kids how to calculate inheritance. The problem asks how much each of seven boys would get when their father died, according to Babylonian law. Apparently the law said they should each get a different proportion, with the oldest getting the most and the younger kids less and less. Whoever did the math worked up from the bottom (which was not normal), and also made a mistake in his or her calculations!
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