On the way home today, my daughter told me she needed a scientific calculator for class. After a heavy sigh on my part, we stop and get one. Nine bucks. I can remember when those things were in the hundreds of dollars.
As we head out, I ask her what class it’s for.
Wait, what? Okay, maybe I’m an old fogie but what in the world would she need a calculator for for an algebra class? I could see it for something like high school physics (which she’s also taking) where the numerical answer matters, but algebra? This isn’t a graphing calculator which I could also see, but instead is there for simple number crunching. It has nothing to do with the symbolic manipulation which is at the core of algebra as opposed to simple arithmetic.
I was tempted, really tempted to say all of that but, I refrained. I am however rather disappointed that the algebra teacher feels the need to require a calculator in the class. This suggests that the actual algebra has been watered down and is padded with numeric manipulation. I mean writing the square root of two in symbolic notation should be a perfectly acceptable answer like this:
See? They don’t need 1.4142… It’s a waste of time and a distraction from learning algebra.
Still, since complaining, especially to my daughter, will accomplish nothing I kept my mouth shut. But it got me thinking about algebra and numbers. Back when I was in sixth grade we learned a process for converting a repeating decimal into a fraction. Now, I’d long since forgotten that procedure having used it exactly never in the intervening years. But, since I now know at least a little algebra I thought I’d be able to derive it again. And…as it happens.
First, understand that a rational number is one that is the ratio of two integers: 1/2, 2/7, 438/926 and so on. One thing about rational numbers is that they always produce a terminating or repeating pattern of digits. 1/2 is 0.5 and stops there exactly. 2/7 is 0.285714 with the string “0.285714” repeated to infinity. 4/2 = 2 exactly. You can think of terminating ratios as also being repeating decimals just that what’s repeating is zeros. 1/2 is 0.500000000… and so on with zeros to infinity.
Going from ratios to decimals is fairly straightforward. You simply divide. But what about going the other way. Let’s try one.
Let’s start with a number. Call it X, where X is oh something like this:
x = 76.4879838383…
with the “83” repeated out to infinity.
We can see that:
We can also see that:
1000000X = 76487983.83838383…
Note that the points after the decimal point are now the same on the two. Since we can subtract the same thing from both sides of an equation and still have the equation be true we can do the following:
1000000X – 10000X = 76487983.83838383… – 764879.8383838383…
Everything after the decimal point subtracts out leaving us with only:
990000X = 76487983 – 764879 = 75723104
X = 75273104/990000
We can reduce that a little bit, dividing both top and bottom by 16:
X = 4723694/61875
And there you have it, a repeating decimal converted to a rational number in lowest terms.