December 07, 2022 @ 21:21
I recently came across this video that teaches how to perform some quick mental multiplication. It's called the Base Method of Multiplication. I think it's also something known as Vedic Mathematics.
It's a useful trick but it's not immediately obvious why it works. I thought I'd derive it using the familiar form of multiplication (that I learned at school). I feel the need to convince myself that the Base Method of Multiplication works before I commit it to memory.
First I want to talk about the way I learned to handle multiplication when I was at primary school.
Given two numbers each of two digits, \(ab * cd \).
Let this be:
$$ \begin{matrix} a & b & \\ c & d & *\end{matrix} $$Or:
$$ (10a + b) (10c + d) $$And if we expand this then it becomes:
$$ 100ac + 10ad + 10bc +bd $$Here's an example for extra clarity: \(33 * 10 \).
$$ \begin{matrix} 3 & 3 & \\ 1 & 0 & *\end{matrix} $$Or:
$$ (10(3) + 3) (10(1) + 0) $$ $$ 100(3)(1) + 10(3)(0) + 10(3)(1) +(3)(0) $$This evaluates to \( 330 \), as you might expect it to.
I'll assume you have watched the video.
Using the notation introduced on this page earlier, the method the video describes is this:
$$ 10c(( (10c + d) -10c ) + (10a + b)) + (10a + b -10c)(10c + d -10c) $$Where \(10c\) is the base. I'm not implying here that the base is \( 10 \), you could have any \(c\).
Let's simplify:
$$ 10c(d + (10a + b)) + (10a + b -10c)d $$$$ 10cd + 100ac + 10bc + 10ad + bd -10cd $$$$ 100ac + 10ad + 10bc + bd $$The above is the equation I learned at school. Now I've convinced myself it's true and I'll commit this to memory.