On Not Being an Idiot Savant

The term, idiot savant, was introduced to explain computing prodigies. There are people who can perform long, demanding arithmetic calculations in their head very quickly. This talent or capability apparently does not depend on general intelligence; one can be an idiot otherwise and still have the talent. There were some spectacular instances of this in the last century that got quite a bit of attention; the term, idiot savant, was introduced to describe these people.

It seems to be a definite specialized capability. If one is facile with numbers one can do quick arithmetic in an impressive way. When I was young I was taken with this and trained myself in the art - there are some tricks of the trade one can use and it is something you can practice. I could, for example, write 2 five digit numbers down and then write their product down in a fairly smooth motion. (It's much harder if you are not allowed to write the answer down and have to speak it.) This will startle people when they see you do it. But it's actually a trick - one cross multiplies and keeps running sums so that you are only doing one simple calculation at a time. There are several books on techniques for doing fast mental arithmetic; one of the best and most interesting was by a man name Trachtenberg (sp?) who worked out a whole system of techniques, supposedly while he was a prisoner during world war II.

Fast mental arithmetic was important in the days before calculators and computers. In college I always had an edge in the physics courses because I didn't need a slide rule.

However the techniques for fast mental arithmetic do not explain the "idiot savant" talent. My take on this is that the talent really involves the ability to keep an extraordinary number of numbers in short term memory. To illustrate: If you give me two three digit numbers and ask for their product I can do it. But the technique I use generates the digits of the answer from back to front. If I am doing this all in my head I have to keep in mind the two numbers and the digits of the answer while at the same time running through the calculations. This I can do, even now. Four digit numbers I could do while in practice. Five digit numbers I couldn't do - there were too many digits to keep track of. But people with the idiot savant talent can multiply 10 digit numbers in their head and give the answer.

Early in this century there was an efficiency expert, Frank Galbreth, who sired a dozen children. Two of his children wrote Cheaper by the dozen which was a very entertaining account of an industrial efficiency expert bringing to bear the techniques of industrial efficiency analysis to the rearing of a large family. Their account of Daddy teaching the children techniques for fast arithmetic sparked my interest.


The cross multiplication technique

This is an easy and useful way to multiply numbers. The underlying idea is that decimal notation expresses a number as a polynomial in powers of ten. To illustrate this lets look at the algebra for multiplying two three digit numbers. We have

def = 100*d + 10*e + f
abc = 100*a + 10*b + c
adc*def = 10000*ad + 1000*(ae+bd) + 100*(af+be+cd) + 10*(bf+ce) + cf

What we are going to do is calculate the digits of the answer from back to front, i.e., from the low order digit to the high order digit. Start with c*f, the product of the last two digits. Write down (remember) the last digit of their product and carry the first digit, if any. This will be the start of a running sum. Now slide over one digit in the second number (that's b) and multiply it by the last digit of the first number (that's f). Take their product and add it to the carry number. Now slide right in the the second number and left in the first number and add that product to the carry number. Strip off the last digit, write it down, and carry the remaining digit(s). After you get to the middle you will be starting with the first digit of the second number and will keep sliding to the left for the starting point in the second number. Here is a numerical example:

197
321
---
?

Our first term is 1*7=7. Write down the seven and carry 0.

197
321
---
?7

The next term is 2*7=14 which starts our running sum. Then we add 1*9=9 to get 23. Write down the 3 and carry 2 to start our next running sum.

197
321
---
?37

In the next term we start with our carry, 2, add 3*7=21 to get 23, 2*9=18 to get 41, and 1*1 to get 42. Write down the 2 and carry 4.

197
321
---
?237

We now are at the point where we start with the beginning of the second number and start sliding left in the first number. Start with the carry 4, add 3*9=27 to get 31 and 2*1 to get 33. Write down 3 and carry 3.

197
321
---
?3237

We're at the end. Take the carry, 3, and add 1*3 to get the remaining digits, 6 in this case. Write it down.

197
321
---
63237

The explanation may seem cumbersome. However all the process requires is that you be able to keep a running sum of two digit numbers in your head and that you know your multiplication table. The process is very mechanical; it takes a bit of practice to get proficient at it. If you are going to multiply longer numbers you will need to be able to keep a running sum of three digit numbers in your head.

Learn this trick and amaze and startle your friends. Seriously, is it worth learning? It all depends. Nowadays the calculator and the computer have taken away most of the need for being able to do quick arithmetic. Still, if you work with numbers a lot, it can be a useful knack to have.


This page was last updated February 7, 1998.