Monday, November 30, 2015

Strange Square Roots

(Click on the comic if you can't see the full image.)
(C)Copyright 2015, C. Burke.

Do NOT show my students this! They might think it's a rule and not a curiosity!

In fact, as strange as it looks, it isn't the only mixed number that has this property. And there isn't any reason to drop a jaw to the desk to figure out the pattern.

Let x be the whole number and y be the fraction (between 0 and 1, exclusive).

Then the equation says that (x)(sqrt(y)) = sqrt(x + y).
Square both sides we get (x2)(y) = x + y.
Subtract y from both sides (x2)(y) - y = x.
Factor the left side (y)(x2 - 1) = x.
Finally, divide y = x/(x2 - 1).

So you can pick any whole number value of x -- 2, 3, 10, whatever -- and substitute on the right side. You will get a value of y which is the fractional part of the mixed number that makes the "strange" square root work.

On a historical note: This is comic #1066. If you thought I'd do something about The Battle of Hastings ... well, it had crossed my mind, but too complicated and no way to plan in advance with the crazy schedule I'm keeping.

Come back often for more funny math and geeky comics.


Glenn said...

After reading this, I had to graph it to visualize the relationship. There are a couple of mistakes in your reasoning afterwards. One is a missing parenthesis, but the big mistake is not every integer works. 1 won't work, and neither will -1.

That is one seriously odd piece of mathematics that would cause so much confusion for high school learners! Wow. I can just imagine the heads exploding in my classroom.

Glenn said...

Okay, even cooler. When you graph the functions x(sqrt(fx)) and sqrt(x + f(x)) it is true for all real values where x >1, not just integer values. The integers make nice fractions, but the actual function is true over all reals > 1.

Very cool function! Love it. Thank you.

(x, why?) said...

Sorry about the extraneous parenthesis. Noticed it yesterday, but didn't have a chance to get in an edit it out.

I did say to use a Whole number, but, yes, the formula would eliminate 1 from the domain. Negative numbers are right out if we wish to keep this real. And zero wouldn't make any sense.

I only concerned myself with the Integers because those are the ones that looked interesting. Splitting out other pieces wouldn't have made me take notice like when I saw this example.

I actually did graph it because I was curious what it looked like.

Glenn said...

Thank you for posting your comics. I love reading them, and I often spend an hour or so exploring some of the fun math you post about.

I may throw the whole numbers to some learners at some point (college level so they don't get confused) and see what happens. Thank you!

Unknown said...

I'm contemplating writing this on the chalk board next week when I turn in my abstract algebra final exam. Might possibly do this in for my topology class, too... :)