Multiplying two digit numbers

We were chatting at dinner the other day about mental arithmetic and Navid had a neat trick for multiplying numbers in his head.   He’s written it out below.

Mentally multiplying two digit numbers is difficult for the majority of people. This is probably due to our limited working memory overflowing when we try to store the partial products, leading to errors.

In many cases, I use a trick to simplify the problem, which is the subject of this blog post. It requires a bit of investment, but lets us compute products such as 84*78 both accurately and (in due time) quickly.

 

Theory

This trick works for a * b when a + b is even, that is to say either a and b are both even or both odd, for example 65*43 and 88*78. Furthermore, it works best when a and b are close together, but as your proficiency increases this becomes less important.

The essence of the trick is in the difference of two squares, namely, we want to write a * b =       (c + d) * (c – d) = c^2 – d^2                  

We quickly find that. c = (a + b) / 2 and d =|a – b| / 2

In other words c is the average of a and b, whereas d is the difference between the average and a or b. Note that the condition that a + b is even ensures that both c and d are integers.

You are probably wondering what the point of this complication is. Well, in many cases computing c^2 – d^2 is easier than working out a * b. For example, take 43*57, it is easy to see their average is 50 and the difference is 7. Hence, the product reduces to 502 – 72 = 2500 – 49 = 2451. Try it out on the following examples:

a) 66*62                             b) 98*84                             c) 39*33                             d) 44*76

 

Tips and Conclusion

After doing a few problems you will find that most of the time d is small enough to recall d^2 from memory. The trickier part is calculating c^2, fortunately, we can use the fact                 (x + y)^2 = x^2 + 2xy + y^2 to aid us. For example, writing 82 = 80 + 2 we find that 822 = 802 + 2*2*80 + 4 = 6400 + 324 = 6724. Personally, I found that this speedy computation of c^2, removed the major bottleneck from the method.

Some practice is required but soon you will be able to multiply the majority of numbers of this form in under ten seconds.

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