X is for . . .

This month I'm posting Mad-Cool-Math Nuggets to foster an appreciation of all things mathematical. Then I'm going to teach pigs to fly.

Image courtesy: Wikicommons

X is for . . . the following problem:

Let x = 0.99999 and don't ever stop typing nines, because they will be repeating for all eternity. Wait, we can't do that. How am I going to get the proper notation on Blogger? Usually, you draw a horizontal bar on top of the first 9. But that's going to be tricky. Some places uses parenthesis, such as 0.(9). Ew.

Figulty fum. Let's just agree that 0.9999repeat will be my notation for this repeating decimal today.

So now, I will amaze you by proving that 0.9999repeat is actually equal to 1! Better than pulling a rabbit out of a hat, huh?

Here's how it's done:

Let x = 0.9999repeat

Multiply both sides by 10. In algebra, equality will hold as long as you perform the same (legal) operation to both sides of the equation. Something not legal would be division by zero. Note that multiplying 0.9999repeat by 10 looks like giving this number a little push to left. The decimal is now after the first nine instead of before it.

10x = 9.9999repeat

Now we will write 9.9999repeat as the sum of the following two numbers: (Don't freak out, this is equivalent to breaking 1.5 into the sum 1 + 0.5. No biggie.)

10x = 9 + 0.9999repeat

Next, we will perform a substitution. Since x = 0.9999repeat (go back to step one if you forgot), it is perfectly valid to rewrite the above as:

10x = 9 + x

Are you still with me? Good! Now we will subtract x from both sides of the equation. Does that seem strange? Don't worry, it will make sense in a moment.

10x - x = 9 + x - x

Do you know what 10x - x is? (Just say: 10 rabbits minus 1 rabbit leaves me . . . 9 rabbits!) So, 10x - x = 9x. Along the same vein,  x - x is zero. Nothing. Which means x - x can disappear, like magic. So the above simplifies to:

9x = 9

Now we will divide both sides by 9:

(9x)/9 = 9/9

Now 9 divided by 9 is 1. (FYI, we don't write 1x, because it looks weird. We just write x.) So on the left, the 9s "cancel", leaving us with x. On the right, we have 1. So

x = 1

Wasn't that great! Don't you love algebra! Happy dance! We started with x = 0.9999repeat at the beginning and after 3 different operations to both sides, 1 substitution, and 1 rewrite of a number into the sum of its parts, wah-la! We end up with x = 1.

Or is your reaction more like this fellow's?


Image courtesy: John Benson

No worries. The remaining Mad-Cool-Math Nuggets will be extra light. I think we may all be experiencing blog fatigue.