Chapter 7
MULTIPLYING LOWER NUMBER
You may have noticed that our method of multiplication doesn’t seem to work with some numbers. For
instance, let’s try 6 × 4.
We use a reference number of 10. The circles go below because the numbers 6 and 4 are below 10.
We subtract crossways, or diagonally.
6 − 6 = 0
or
4 − 4 = 0
We then multiply the numbers in the circles:
4 × 6 =
That was our original problem. The method doesn’t seem to help.
Is there a way to make the method work in this case? There is, but we must use a different reference
number. The problem is not with the method but with the choice of reference number.
Let’s try a reference number of 5. Five equals 10 divided by 2, or half of 10. The easy way to
multiply by 5 is to multiply by 10 and halve the answer.
Six is higher than 5 so we put the circle above. Four is lower than 5 so we put the circle below. Six is
1 higher than 5 and 4 is 1 lower, so we put 1 in each circle.
We add or subtract crossways:
6 − 1 = 5
or
4 + 1 = 5
We multiply 5 by the reference number, which is also 5.
To do this, we multiply by 10, which gives us 50, and then divide by 2, which gives us 25. Now we
multiply the numbers in the circles:
1 × −1 = −1
Because the result is a negative number, we subtract it from our subtotal rather than add it:This is a long and complicated method for multiplying low numbers, but it shows we can make the