Look Out! Big Ideas are Hiding in Your Arithmetic | exSTEMsions

Look Out! Big Ideas are Hiding in Your Arithmetic

“Simple” arithmetic can be surprisingly tricky

Little things count in mathematics.  Unfortunately, there are a lot of little things in mathematics that, for some reason or other, our mathematics education often glosses over, doesn’t give us the full explanation for, or misses the big picture entirely.  For example, is 0 even?  This seems to be a controversial question for some, but if students were given a clear definition of the word “even” and the why behind the definition, everyone would be much clearer about the answer to this question.  (Yes, 0 is even.)  

Another topic that seems to be glossed over is the gritty detail lurking behind our friend square root.   If you’re bored at a party sometime, ask a few of your friends what \(\sqrt{4}\) is.  Many will answer 2, but it’s a fair bet that just as many will answer \(\pm\)2.  Who’s right?  Actually, the square root of 4 is just 2.  That might strike some of us as strange, and that’s perfectly fine, as long as we’re willing to dig a little deeper to get the real skinny.  As we talked about in this week’s problem and solution, part of the reason why we might erroneously say that \(\sqrt{4}\) is \(\pm\)2 stems from the way we were taught how to solve quadratic equations in school, and the shortcuts we may have taken to arrive at the answer.

There are really interesting questions in arithmetic!

The fact that the square root of a number is necessarily positive is one of many interesting issues that could be explored somewhere in elementary, middle, or high school, but perhaps falls by the wayside every so often.  That’s a shame, because this interesting topic hides some interesting details.  In fact, there are many “simple” issues and questions, in that they deal with simple numbers, and simple operations, where the math hiding behind these questions is surprisingly not so simple; here are but a few:

Why is a negative times a negative a positive?

What is 0 raised to the power 0, and why?

Why is 0 factorial supposed to be 1?  

Why isn’t 1 prime?

We’re never allowed to divide by 0, but why can’t we allow 0/0 = 1?

These are “only” questions about arithmetic, but truly understanding their answers gives one a wonderful opportunity to do some unexpectedly deep mathematics.  This also reminds us that math can be exciting and interesting at any level.  As such, we will be touching upon a few of these in future problems!

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