Unlike toys, you should always try to break your mathematics.

An example: at some point, we get used to the idea of powers as being a shorthand for repeated multiplication. What is 3^{4}? It’s three multiplied by itself four times, 3x3x3x3, which comes out to 81. No problem. But things don’t get interesting until we try to break it.

How do we break exponents? It’s no trouble to find 3^{4}, or 3^{99} (if you have the time). But what is 3^{-1}?

A few weeks ago, a teacher and a parent (independently) told me that they’d been approached by students with this very question. Kids trying to break mathematics! The questions is, does the mathematics, being challenged, grow to handle the attack, or does it simply break? That’s up to the kid. I didn’t hear what happened with the teacher, but the parent told me that she started to tell her child the answer, and he stopped her, wanting to figure it out herself. “Is it even something you can figure out?” she thought. “Isn’t it just a definition you learn in math class?”

A little while later, her son came back with the following argument: if you multiply 3^{-1}x3^{2}, you get 3^{1} (recall your rules of how to multiply the same number raised to different powers). That means 3^{-1} times 9 is 3, which means that 3^{-1} must equal 1/3. And indeed, he was right. The parent was astonished, and thrilled. The child was now ready to extend the definition of exponents to any negative number.

In fact, it’s possible to work out all the rules of exponents for yourself, and if you keep trying to break them, you can figure out what 3^{1/2} must be too. It is easy to forget that *definitions come from people trying to break mathematics*, and unless you try to break the math too, it’s hard to see what people were thinking when they came up with the definitions and rules we have today.

So don’t be gentle. Bend it, and see if it will break.