Dear Uncle Colin,
In Basic Maths For Dummies, you mention a method for multiplying numbers from 6 × 6 to 10 × 10 on your fingers. It's almost magical! Why does it work?
-- Does It Guarantee Interesting Times Sums?
Hello, DIGITS, I'm glad you're finding the book helpful!
The method works because of algebra. If you call the number of the finger on your right hand $a$ and the number on your left-hand $b$, you're working out $(5+a) \times (5+b)$ -- for example, to get 6 × 8, you use your thumb ($a=1$) and third finger ($b=3$). When you multiply out the brackets, you get $25 + 5 × a + 5 × b + a × b$.
The method gives you ($a+b$) tens from the fingers 'below' the touching ones, and -- a bit more complicatedly -- $(5-a) × (5-b)$ units from the fingers 'above'. (Check it with $a=1$ and $b=3$, or any other combination and convince yourself it works!) Altogether, you have $10 × (a+b) + (5 - a) × (5 - b) = 10 × a + 10 × b + 25 - 5 × a - 5 × b + a \times b$, or $25 + 5 × a + 5 × b + a \times b$, the same as before -- so it works, whatever values you pick for $a$ and $b$ :o)
-- Uncle Colin