Robert Recorde and an Early Multiplication Method

16th century mathematician, Robert Recorde, wrote a books on mathematics. Roman Numerals were used commonly up to his time and he wrote of the advantages of used our current Hindu-Arabic numerals. He introduced a method of multiplication. The method works best for multiplying single-digit numbers, but it also works for higher numbers. 

For example, consider 9 x 7:

Write down the following diagram. The numbers being multiplied are written on the left-hand side of the X; their 10s complements are written on the right-hand side (10-9=1 and 10-7=3):

The right-hand side gives the value of the ones digit of the answer: the product of 1x3. The value of the tens digit is given by the difference of numbers on opposite sides of the X. Both 7-1=6 and 9-3=6, so it doesn't matter which difference is used. Therefore, the final answer is 63 (6x10 +3).

Here is an example with higher numbers. Consider 13 x 7. On the right-hand side of the X, write -3 and 3 (10-13= - 3, and 10-7=3):

The ones unit is -3 x 3 = -9 and the tens digit is 10 which is found by the difference of 13 and 3. So the final answer is 10 tens or 100 -9 = 91.

A little algebra will explain how this X method works. Let the two numbers being multiplied be a and b. The left-hand side of the X becomes a and b, while the right-hand side is 10-a and 10-b.

  

Multiplying the right-hand side of the X, (10-a)x(10-b)=100-10a-10b+ab.

Next take the difference of the opposite sides of the X. This gives us a-(10-b) for the tens unit so we multiply this difference by 10 for 10a-100+10b. Adding this result to the product of the right-hand side gives: 10a-100+10b+100-10a-10b+ab. Note the following terms cancel: 10a & -10a, -100 & 100, and 10b & -10b, leaving the ab as the answer to a times b. 

The math series, Numberphile, recently had a video with Rob Eastaway. He shows one of Recorde's books and walks through the X multiplication method: https://www.youtube.com/watch?v=21Ho32fAEEM.

Also see: Ethiopian Multiplication