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.

O, you just hurt my brain!

ReplyDeleteWhoa! You lost me. I'll have take your word for it. ;)

ReplyDeleteWhoa! You lost me. I'll have take your word for it. ;)

ReplyDeleteYou lost me at X. No, seriously! Haha:)

ReplyDeleteThat's cool. I guess you can resolve the apparent mystery defining

Deletex=1-epsilon

and do lim elspilon->0 in the end, cause thats what your 0.99repeat does.

It's hard to write proper math in blogger, I agree, or in MS Word or Powerpoint (the math fonts are terrible). I refuse to use anything but LaTeX, which you have probably used yourself. I was very happy when I found a LaTeX plugin for Powerpoint on the Internet >:)

Cold As Heaven

Yep, I wrote my dissertation in Latex many years ago.

DeleteRight now I'm imitating the rabbit! You totally lost me (not for the first time this month!) but it was very entertaining just the same!

ReplyDeleteKeith's Ramblings

Whoa, I do not miss algebra at all.

ReplyDeleteThat was my reaction. I never liked Algebra.

ReplyDeleteI love Algebra! Yeah Algebra!!!

ReplyDeleteStephen Tremp

A-Z Co-host

X is for Xenoglossy

www.stephentremp.com

Then you're one of the chosen few! Good for you.

DeleteYou had me all the way up to "multiply both sides by 10"! I have a healthy respect for algebra... so I'll leave it alone, as I would just end up performing an illegal operation. I'm too pretty for prison!

ReplyDeleteUh...okay but only if you say so. I think teaching pigs to fly would be easier but I have a way with pigs. And a whole other way with numbers. I think you could express this whole equation by saying "think nothing of it!"

ReplyDeleteHaha! You finally made my head spin and all I can say is xoxo :)

ReplyDeleteI love that calculation. It blew my mind the first time I saw it.

ReplyDeleteAs for repeating, how about 0.9999... ?

That is an option, but sometimes it is only part of the number that repeats such as 8.7634343434 and the dot, dot, dot doesn't tell which digits repeat.

DeleteThat is an option, but sometimes it is only part of the number that repeats such as 8.7634343434 and the dot, dot, dot doesn't tell which digits repeat.

DeleteThat was my favorite mind boggle in high school! I literally sat there and thought about it for a very long time trying to stretch my mind around the idea of it. Math has some awesome loopholes.

ReplyDeleteWell that was something different than coffee to wake up my brain. :)

ReplyDeleteI'm better at math than I thought, though it takes awhile for the concepts and knowledge to come back. I never figured out how to write the line over a repeating numeral either, though some keyboards probably have it hidden somewhere.

ReplyDeleteLove it. I like algebra, even when (as here) it doesn't seem to be reasonable.

ReplyDeleteRebecca at The Ninja Librarian

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