# A method of mental multiplication

by Bruce Tonkin
Tags: mental, method, multiplication
P: 834
cmb has bumped the thread, so I'll guess I'll critic.

 Quote by Bruce Tonkin A TECHNIQUE FOR MENTAL MULTIPLICATION (ALSO USEFUL FOR COMPUTER ARITHMETIC) So I set out to find out how to compress the steps I’ve outlined above. The method I came up with is mathematically completely equivalent to the long form of multiplication I mentioned. As a side benefit, it will work in any number base. I have shown this method to very many people over the past 50 or more years, and no one has ever said “oh, this is just like the XYZ method”. In fact, everyone has told me that my method is unique. I find that very difficult to believe, considering how easy it is. If you have heard of something similar, please let me know. I am ready to hear that my technique is at least 500 years old, if not older.
The technique doesn't have a name per se, but it is listed on Wiki. The technique is nothing more than a clever way to write down
$74 \times 36 = 100 \times (7 \times 3) + 10 \times (4 \times 3 + 7 \times 6) + (4 \times 6)$,
so it doesn't scale nicely as digits increase. This makes it inferior to the Trachtenberg System, which not only uses less multiplications per step, but scales very easily.

At its core, the technique boils down to the log space version of long multiplication.

 It is possible to multiply extremely large numbers using this method, to practically any number of digits in the result. A multiplication of two n-digit base 256 numbers would take 2n (one byte) integer multiplications and 2n – 1 additions to complete (plus perhaps 2n shift and AND operations to store the digits of the result).
Firstly, increasing the base does not change complexity. It only hides it. 1005 x 1011 can be thought of as multiplying two 4 digit numbers or multiplying two one-base-1028-digit numbers. The latter trivially consists of only one multiplication. All it does is require a larger look up table.

Secondly, any algorithm equivalent to long multiplication is useless when we are talking about thousands or tens of thousands of digits. Every such algorithm will have n2 complexity, not 2n as you suggest.

P: 3,297
A method of mental multiplication

 Quote by Bruce Tonkin I have shown this method to very many people over the past 50 or more years, and no one has ever said “oh, this is just like the XYZ method”.
You should compare it to the Trachtenberg System. (I, myself, haven't studied either method so I don't know if they are similar.)
P: 28
 P: 1 As a mental math system, this looks like a rephrasing of the "Vertically and Crossways" (wiki link) Vedic method, which is really very similar to the grade school algorithm. (It just does things in a different order, and saves some time by lumping together the same-magnitude multiplications.) There are several mental multiplication methods that seem easier to me. Someone already mentioned Trachenberg. You could also do this with base numbers: $74 \cdot 63 = 70\cdot 67 - 28 = 4662$. (See the wiki page or look up Ars Calcula for an explanation.) You could also just use the distributive law: $74 \cdot 63 = 70\cdot 63 + 4 \cdot 63 = 4410 + 252 = 4662$ (this is a bit harder to do mentally). Both of these seem faster / easier than the Vedic method (to me, at least). As for the algorithmics: I agree with the other poster. This looks like O(n^2) time to me, not O(n).