The Weird World of Exponents

Exponents are weird. Let's face it. We have these tiny little numbers floating around in the air like gnats or magicians' familiars, and they do all these magical, weird things to whatever they touch...

When exponents are involved, numbers mysteriously get massively bigger, or massively smaller. Sometimes they just disappear into thin air.

Ah Maths...

You see, having come up with a new notation (i.e. xⁿ meaning x multiplied by itself n times) the mathematical community has to decide on certain logical and notational conventions which allow the notation to be used consistently. Here they are:

Or, more "mathematically" (i.e. using a whole lot of funny letters instead of colourful blocks):

When confronted with this list, most people justifiably panic. I mean, that's just weird, right? 

Well actually, it's not as weird as it may seem at first...Allow me to demonstrate.

Okay, so if  xⁿ = x.x.x.x.x.x... (n times), then it makes sense that 

xⁿ.x^m = x.x.x.x.x...(n times).x.x.x.x.x.x.....(m times) = x.x.x.x.x.x...(n+m times)

 (thank you http://www.loisterms.com/powers.htm for this and a couple of other images in this post!)

In which case, it makes sense that 

xⁿ÷x^m = x.x.x.x.x...(n times) ÷ x.x.x.x.x.x.....(m times) = x.x.x.x.x.x...(n - m times)

Bear with me. We're getting to the exciting bits now.

At this point we have to deal with the fact that we could have the same number of xes on the top and bottom. In order for our new system to be consistent with other mathmatical systems involving the multiplicative identity (i.e. 1), we also have to state that

x⁰=1

Okay, great, but what if we have more xes on the bottom of the fraction than on the top? Looking at this problem through the lens of the previous "laws" (or logical corollaries of the original notational statement), we find ourselves with a new statement

x^-n=1/xⁿ

The last few exponent laws follow in a similar manner...why not try justifying them yourself? 

Call me a nerd (yes, fine I'm a nerd) but I find the system extremely satisfying. Each law demands the next law, and if a single law was missing the whole system would come crashing down as fast as the Wall Street in 1929.

Ah Maths...

 
And that's just for fun; a fractal based on exponents :-)