## Speedy multiplication

By 'fast multiplication' I mean managing to write so little working that it appears that the result is simply written from the problem.

Fast multiplication by five is, at heart multiplying by 10/2. Shove a zero on the right hand end and divide by two. Very little practice is required to turn this into apparent magic, no visible working.

Fast multiplication by eleven comes from recognising 11=10+1, so all you need to do is add adjacent digits. To labour this, I write numbers with extra space.

374272 x 11  and working from the right (units), copy the 2 and then, working leftwards, add each pair, carrying as necessary.  We will have carry figures either side of the 7, indicated by a dot, which is all the working you need (and can soon be skipped).

. 3.  7.  4  2   7   2                        This is so simple you can write your own examples.
4   1  1  6   9   9   2

The obvious extension is to also see this as multiplication by three in binary, for example

. 1 . 0 .1 .1  1  0  1  0  1  x 11₂
1  0  0  0  1  0  1  1  1  1  1               is a relatively difficult example.

Fast multiplication by nine comes from recognising 11=10-1, so all you need to do is subtract adjacent digits, but the right way around. To detail this, write a subtraction first and then see how to eliminate the 'working'.     975468 x 9

Technique (Algorithm): requires you to recognise the 9-complement; 1 and 8, 2 and 7 and so on.

9 7 5 4 6 8 0      Take the unit digit from ten for the unit answer
9 7 5 4 6 8       For each digit in turn add its  9-complement to its right neighbour
8 7 7 9 2 1 2       For the leftmost digit, subtract one from the leftmost of the target.

So the technique method for 975468 x 9 starts on the right with 10-8=2. Next the complement of 6 is 3, plus 8 is 11, write 1 and carry 1. Complement of 4 is 5 plus 6 (and 1) is 12, write 2 and carry 1. Complement of 5 is 4 plus 4 (and 1) is 9, write 9. Complement of 7 is 2 plus 5 is 7, write 7; complement of 9 is 0 plus 7 is 7, write 7. Last digit is one less than the 9, write 8.  Practice: I multiplied this last answer by 9 several times, reaching 518 403 689 388 successfully.

This may seem laborious in prospect but it really does become easy quite quickly. I found the third iteration was quite sufficient, but I'd need ten or twenty examples to have made this move to being in some sense automatic. A few more cycles on the last answer gave me 3 401 246 606 074 668, well past what my calculator can check. 275 500 975 102 048 108 is my original 975468 x 9¹² = 2.755x10¹⁷.

Speedy creation of squares is explained on several pages, starting here.

There are similar techniques for all the single digits.

To multiply by 12, simply double each digit and add its right neighbour. You need to imagine additional zeros. 975468 x 12 becomes 0975468 and rapidly you get 11705616, then 140 467 392, 1 685 608 704, 20 227 304 448, 242 727 653 376 and so on.

Practice these before attempting more.

Email: David@Scoins.net      © David Scoins 2018