... And a Product of 2 Primes Later...

Dear Mathman,

You said "next week". What calendar are you on? It's been a month and 4 days since you last posted! Come on, your fan can't wait!

Sincerely,
Your Fan


Dear Fan,

Thanks for paying attention. I have no good calendar explanation other than it's been the first prime number shy of the 6th perfect square since I've posted. The perfect square is the perfect vehicle for our next topic anyways: Roots.

Roots are the BFFs of the exponents. I know some may say "BFF"- that's SOOOO 2007 I can't believe you'd use that term Mathman; After all, you are incredibly hip and relevant. Well, most people still know what it means so we're going with it. With me? Groovy.

Let's review exponents: shortcut for repeated multiplication where the base is the number that is repeatedly multiplied and the exponent is the little number high and to the right that tells us how many times we repeatedly multiply.

Ex 4^2 (the ^ means raise that number) = 4x4 = 16. Now you may have heard that 4^2 can be read as "4 squared." NWDTCF? (Now Where Did That Come From) If you remember finding the area of a square, you remember you multiply the side by itself (since they're all the same it doesn't really matter which side you pick). When we get the answer, we measure it in square units (which are tiny squares that fill up the space). Extending this, we get the "Perfect Squares" (made up of squares with side lengths that are counting numbers):

1^2 = 1x1=1
2^2 = 2x2=4
3^2 = 3x3=9
4^2 = 4x4=16
5^2 = 5x5=25
6^2 = 6x6=36

1,4,6,16,25 and 36 are the first Perfect Squares.

Next awesome math term: Square Root. Simply put, that's what the length of the side would be for a square with a given area. Looks like:
___
V

So the easy ones are the square roots of 1,4,9,16,25,36,49,64,81,100 - their square roots are 1,2,3,4,5,6,7,8,9,10 respectively.

Let's tie the whole room together: Pick a positive whole number (no fraction or decimal). Square it (multiply it by itself). Now find the square root of that result. You should get your original number. So the square root is the inverse to ^2.

Take this further: any number raised to a power has a root that "undoes" what it just did. Trying to show this on this blog is next to impossible since I get a "typewriter" interface. And you thought BFF was an old term...

Exponents and Roots - BFFs.

Stay radical (or at least learn a bit more about them)

- Mathman.

Yes, Supersize Those Fries.

Last entry we had determined that: 1. PEMDAS as a saying doesn't make a whole lot of sense, 2. The MDAS is ok, but ignores the "I'll write my name with <3 on the back of my hand" relationship between Multiply and Divide (also Add and Subtract, but not Multiply and Subtract), and 3. Sheesh Mathman, you're pretty loquacious.

What we didn't deal with was: What about the PE a.k.a. Parenthesis and Exponents part of the Order of Operations?

First, the Parenthesis are always first. DO IT NOW! is their mantra. Don't ask questions. Just Do It. (Actually just saying Parenthesis are the only bossy symbols is not a correct statement. Brackets also mean the same thing.)

I don't know about you, but I'm not a fan of people constantly bossing me around. The grouping symbols, ( ), [ ], and { }, kind of have their own thing going on. They live in their own little dreamworld and don't really have anyone else to hang out with since they're so full of themselves and don't really care about others- unlike Multiply <3 Divide or Subtract <3 Add. Ahh, special relationships...

So the pressing question is now, what about E? I mean, are Exponents to be left forever by themselves? In PEMDAS they're right after the bossy ones and before the googly eyed pairs. Don't they have any prospects for a long term relationship?

First up, Exponents deserve to be right after the bossy ones. Exponents are a shortcut for repeated multiplication which was at the forefront of the MDAS (but may be second to divide). An Exponent- like 2^5 is a shorter way of writing 2*2*2*2*2, just like 2*5 is a shorter way of writing 2+2+2+2+2 (multiplication is a shortcut for repeated addition). I guess that makes exponents kind of the big brother to multiplication and the even bigger brother to addition. Does that mean exponents are repeated repeated addition?

But what about their prospects for BFFs or eternal love? They're at the top (not counting the brats) of the Order of Operations, but should they have to be alone?

In short, no.

Exponents do have a special relationship going on, but they're so secure in it that they don't really advertise it on the back of their hand or with a sketchy tattoo. They've got roots. No really, they've got roots.

Tune in next week to find out more.

Until then: Stay Radical. (Ooh, That's the perfect tagline in this case.)