In my recent post on using abaci, I mentioned that you can calculate square and cube roots with them. Although actually including the method was way too much for one blog post, I did learn how to calculate square roots for the first time, and, dear reader, it’s so cool!
I decided it was worth it to devote a separate post to this, and I hope it’ll be a worthy addition to your list of ways to show off with extra napkins at bars. You too can become a human computer!
First, there are a few methods for finding square roots, and all of them involve being pretty familiar with perfect squares. If you got your times tables drilled into you well enough that you still remember up to 12 x 12, that’s a great start, and you can just do others by hand if you need to.
For higher numbers that you don’t have memorized, it’s helpful to take multiples of 10 (20, 30, and so on) and remember that their squares are just the first number squared, plus double the number of zeroes. So 202 = 400 and 302 = 900, and anything between those two numbers is a square of something between 20 and 30.
Next, let’s focus on the method I have in mind, which is the binomial method. If you remember anything from algebra, you may have a vague recollection of squaring binomials, which involves multiplying terms to get
(a+b)2 = a2 + 2(a x b) + b2
This is the foundation of the binomial square root calculation. It’s important to me to understand where formulas come from instead of just taking them for granted, and this one isn’t too complicated, so here are two different ways to get to that expanded form. The first is the classic FOIL, which reminds you to multiply all your terms together.
If you prefer visuals, you can get the same result by drawing physical squares to represent the numbers.
Now that we’re solid on binomial squaring, we can apply that idea to finding square roots for any number. The trick is to realize that you can treat any number as the sum of two other numbers. That means that if you’re finding the square root of a number y, you’re dealing with
y = x2
and you can say x is the sum of two unknown numbers (a+b) so
y = x2 = (a+b)2 = a2 + 2ab + b2
Putting it in this form means that you have the power to pick a number a that you know the square of, and then subtract a2 from your original number y and see what you have left. Let’s walk through it step by step.
(Etymology note: the square root sign is called a radical and the number we’re solving is the radicand, from the Latin word for “root” that also gives us radish! That makes calculating a square root roughly equivalent to one of your vegetable servings for the day.)
Given that this particular square root was pretty painless (what a coincidence!), it’s easy to see why most people use calculators instead. I still think it’s good for my brain to learn things like this, though, and I enjoy it even more knowing that I can use a computer most of the time. Like so many other things, it’s a lot more fun as an optional game than as a chore.
Keep stretching your brain! Here’s another article on the binomial method of finding square roots, and a different article on Newton’s method. If you’re inspired to try it on the abacus, this article takes you through the steps with a square root, and here’s one for cube roots. Enjoy!