7th August 2011 - 1 Multiplying Single Digits Part 2 - Using Fingers

The Method

Using the principle covered in yesterday's post, here are two videos showing how to use your fingers to apply it. The first touches fingers together

and the second folds some down.

I think the videos explains the methods clearly, but I do have one minor issue with the second and that is the comment "To show it's no fluke, let's try another multiplication." Unfortunately, two examples do not show it is not a fluke. Checking all possibilities would, or a proof such as the one I gave yesterday. The proof needs a slight change from yesterday's to apply to this method.

The Proof

Let's look at what is happening with the numbers first, and then generalise to show how the method works for any two numbers a and b, between 5 and 10 inclusive.

The circled fingers in the first picture correspond to the fingers in both circles in the second picture.

Circled Fingers

On my left hand I have 3 fingers folded down because 8 - 5 = 3. On the right hand it's 7 - 5 = 2. These are added, 3 + 2 = 5. This corresponds to the number of fingers circled in the first picture. This number is multiplied by 10 to give 50. In general, on the left hand there are a - 5 fingers circled, and on the right b - 5. Summing these gives

(a - 5) + (b - 5) = a - 5 + b - 5 = a + b - 10.

This is multiplied by 10 to give

10(a + b - 10) = 10a + 10b - 100,

which is the same as we had yesterday. We need to work out the value obtained from the non-circled fingers and then add that to this value.

Non-circled Fingers

The number of non-circled fingers is 10 - 8 = 2 on the left hand and 10 - 7 = 3 on the right hand. We multiply these two numbers together to give 6 and add it to the answer from the circled fingers, calculated above.

On the left hand we have 10 - a non-circled fingers, and on the right 10 - b. Multiplying these together gives

(10 - a)(10 - b) = 100 -10a - 10b + ab.

Finally, adding this to the circled fingers answer, and simplifying, gives

10a + 10b - 100 + 100 -10a - 10b + ab = ab,

showing that both methods do correctly multiply a and b together.