Did you know . . .?

Arabic numbers were a great labor saver when they began to enter into medieval Europe ca. 1200!

Previously, medieval European mathematics were based on Roman numerals, which are straightforward enough to add (at least for small quantities), but extremely difficult to multiply or divide. Such calculations were traditionally performed using an abacus and counters. Arithmetic was a complex subject, taught to mature young men, not schoolboys.

With the introduction of Arabic numerals, one begins to see Europeans undertaking more and more complex calculations. One example is long multiplication, for which the jalousia (or gelosia) method was used. The method is so called after the iron grill, or jalousia, Italian men would place over their windows to keep strangers from staring at their wives. It is also called lattice multiplication.

Illustration from Alexander Murray, Reason and Society in the Middle Ages (Oxford: Clarendon Press, 1978), reproducing Oxford, Bodleian Library, MS Digby 190, fo. 75r [Tractatus de minutis philosophicis et vulgaribus (A Treatise on Small Measurements, Scientific and General)].

Above is an example, from around 1300, of how to multiply4,569,202 (across the top of the grid) by 502, 403 (down the right side). To begin, each digit of the multiplicand is multiplied separately with each digit of the multiplier, and the product is recorded in the corresponding split square. To complete the calculation, you simply add the diagonal columns from top right to bottom left to yield 2,295,570,802,406.

Try it yourself! You might prefer this method to the one you were taught.

Here is how to multiply 469 x 37.

First write the numbers on a grid:

Then multiply each pair of digits:

Finally, total the diagonals:

The final product is 17,353.

(All illustrations from: http://forum.swarthmore.edu/dr.math/problems/susan.8.340.96.html)

Lattice multiplication first was introduced to Europe by Fibonacci (Leondardo of Pisa), whose 1202 treatise Liber Abacii (Book of the Abacus) was the most sophisticated work on arithmetic and number theory written in medieval Europe.

The same principle lies behind Napier's bones.

In his own words:

When my father, who had been appointed by his country as public notary in the customs at Bugia acting for the Pisan merchants going there, was in charge, he summoned me to him while I was still a child, and having an eye to usefulness and future convenience, desired me to stay there and receive instruction in the school of accounting. There, when I had been introduced to the art of the Indians' nine symbols [that is, Arabic numerals] through remarkable teaching, knowledge of the art very soon pleased me above all else and I came to understand it, for whatever was studied by the art in Egypt, Syria, Greece, Sicily and Provence, in all its various forms. --Fibonacci, Liber Abaci (1202)

Russian peasant multiplication: http://forum.swarthmore.edu/dr.math/faq/faq.peasant.html

Ancient Egyptian multiplication: http://forum.swarthmore.edu/dr.math/problems/shelley.6.26.96.html