They're Not So Different!

What we call mathematics is actually a really wide subject with a lot of branches, categories, sub-categories which are sometimes so different from one another that it's hard to believe that they are all coming from the same subject, mathematics. But when you start seeing the same thing from different perspectives, you'll see that somethings in mathematics, no matter how distant they seem, are actually equivalent. You'll see that different things are actually saying the same thing. To demonstrate that we'll see a visual proof of the AM-GM inequality for two variables. We'll learn that things aren't so different after all.

Before we get to the amazing proof of the AM-GM inequality, we'll need to learn a couple of things. First we're going to see how some geometrical constructions are equivalent to mathematical operations. We're going to learn to add, subtract, multiply, divide and take the square root of a number with a compass and a straightedge.

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The image above demonstrates addition, subtraction, multiplication and division through geometric construction. Also pay attention to the colors. Addition and subtraction are pretty straightforward. Multiplication and division use the properties of similar figures. So we are able to perform the four basic arithmetic operations with a straightedge and compass. And these constructions aren't that difficult.

Taking the square root is a bit more complicated and it looks like this:

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If you want to know why it works, here's the proof:

$ADC$ is a right angled triangle because all triangles inscribed in a semi-circle are right angled. A little bit of angle-chasing [and the picture shows how to do that] reveals to us that triangles $ABD$ and $DBC$ are similar. Thus we have,

$\frac{BD}{AB}=\frac{BC}{BD}$

Or, $BD^2=AB\times BC=x$

So, $BD=\sqrt{x}$.

Now that we've equipped ourselves with the knowledge required to prove the two variable case of the AM-GM inequality, let's get on with it.

The two variable case of the Arithmetic mean-Geometric mean inequality tells us given two positive numbers that are not equal, their arithmetic mean is always greater than their geometric mean. In other words, if $a$ and $b$ are two unequal positive numbers, then $\frac{a+b}{2}>\sqrt{ab}$.

If you've understood everything up to this point, this proof's going to be really easy for you. I'm not going to use any words for this! Behold:

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The best part about this proof is that you can actually see that $\sqrt{ab}$ is smaller than $\frac{a+b}{2}$. You really don't need words for this [not entirely true, but the words needed were discussed earlier, so you know what I'm saying].

This was just an example [which is one of my favorites] to demonstrate that you could use different ways to say the same thing.

Note that I said nothing about taking the cubic root or fourth root or higher roots. This is because you can't do that with a straightedge and a compass. That is why you can't trisect an arbitrary angle or double a cube. But we'll discuss about this later someday.

Sources: The first two pictures were found through a Google search and then shamelessly stolen and modified. :)