8th August 2011 - 1 Multiplying Numbers Just Below 100

The underlying method I talked about two days ago can be extended so you can multiply numbers just below 100. What do we need to do to adjust it? Wherever we used 10, we replace it by 100. Let's try 96 x 91.

Method

Numbers to Multiply	100 minus left hand column
96	4
91	9
Subtract the number diagonally opposite. It doesn't matter which pair we use as we get the same answer. The pairs are shown in red and blue.	Multiply the numbers in this column
96 - 9 or 91 - 4 = 87	4 x 9 = 36
Multiply this by 100
87 x 100 = 8700	36
Finally, we add the two numbers.
8700 + 36 = 8736

When is it Useful?

So let's consider when this method might be useful. Personally, I think the method for small numbers is interesting but too time consuming compared with the benefit of learning tables up to at least 10 x 10. When the maths gets more advanced, to have to resort to fingers to do 8 x 7 is slow compared with recalling from memory. Small numbers are multiplied fairly often so the investment of time is worth it, even if you struggle to learn things by rote, like I do.

 If you know your tables up to 10 x 10, then any two numbers in the 90s will be easy to multiply together because the differences will be at most 10.

Multiplying 98 by any number is going to be fairly straightforwards if doubling is something you can do easily. For example 98 x 46.

Numbers to Multiply	100 minus left hand column
98	2
46	54
Subtract the number diagonally opposite. It doesn't matter which pair we use as we get the same answer. The pairs are shown in red and blue.	Multiply the numbers in this column
98-54  or I prefer 46 - 2 = 44	2 x 54 = 108
Multiply this by 100
44 x 100 = 4400	108
Finally, we add the two numbers.
4400 + 108 = 4508

If you're happy to double and double again, you'll find you can multiply by 96 as that is 4 less than 100.  Similarly, 80 and 60 aren't too bad since then you're just multiplying by 20 (double and put a 0 on the end) or 40 (double, double again and put a 0 on the end).

And just to make sure, let's check the proof still works out ok.

Proof

Let's apply this method to a and b which are two whole numbers less than 100.

We subtract them from 100 to get 100-a and 100-b. We multiply these (right hand column in the table) to get

 (100 - a)(100 - b) = 10000 - 100a - 100b + ab. (1)

Next we subtract to get a - (100 - b) = a + b - 100 and then multiply  by 100,

100(a + b - 100) = 100a + 100b -10000. (2)

Finally we add (1) and (2) to get


10000 - 100a - 100b + ab+100a + 100b -10000 = ab.

This means that applying the method does indeed multiply a and b as claimed.

Source: http://www.vedicmaths.org/Introduction/Tutorial/Tutorial.asp

Edit: Corrected two typos where I put 10 instead of 100.