A couple questions have bugged me for ages about rational numbers:

1. Why is it that we invert and multiply when dividing two fractions?
2. Why is it that when we solve a proportion, we multiply two elements, and then divide by a third?

For example, with a proportion like:

$\frac {3} {180} = \frac {2} {x}$

the method I was taught was to cross-multiply the two numbers that are diagonally across from each other (2 and 180), and divide by the number opposite x (3). We solve for x with $x = \frac {2 \times 180} {3}$, but why?

These things weren’t explained. We were just told, “When you have this situation, do X.” It works. It produces what we’re after (which school “math” classes see as the point), but once I got into college, I talked with a fellow student who had a math minor, and he told me while he was taking Numerical Analysis that they explained this sort of stuff with proofs. I thought, “Gosh, you can prove this stuff?” Yeah, you can.

I’ve picked up a book that I’d started reading several years ago, “Mathematics in 10 Lessons,” by Jerry King, and he answers Question 1 directly, and Question 2 indirectly. I figured I would give proofs for both here, since I haven’t found mathematical explanations for this stuff in web searches.

I normally don’t like explaining stuff like this, because I feel like I’m spoiling the experience of discovery, but I found as I tried to answer these questions myself that I needed a lot of help (from King). My math-fu is pretty weak, and I imagine it is for many others who had a similar educational experience.

I’m going to answer Question 2 first.

The first thing he lays out in the section on rational numbers is the following definition:

$\frac {m} {n} = \frac {p} {q} \Leftrightarrow mq = np, where\: n \neq 0,\, q \neq 0$

I guess I should explain the double-arrow symbol I’m using (and the right-arrow symbol I’ll use below). It means “implies,” but in this case, with the double-arrow, both expressions imply each other. It’s saying “if X is true, then Y is also true. And if Y is true, then X is also true.” (The right-arrow I use below just means “If X is true, then Y is also true.”)

In this case, if you have two equal fractions, then the product equality in the second expression holds. And if the product equality holds, then the equality for the terms in fraction form holds as well.

When I first saw this, I thought, “Wait a minute. Doesn’t this need a proof?”

Well, it turns out, it’s easy enough to prove it.

The first thing we need to understand is that you can do anything to one side of an equation so long as you do the same thing to the other side.

We can take the equal fractions and multiply them by the product of their denominators:

$nq (\frac {m} {n}) = (\frac {p} {q}) nq$

by cancelling like terms, we get:

${mq = np}$

This explains Question 2, because if we take the proportion I started out with, and translate it into this equality between products, we get:

${3x = 2 \times 180}$

To solve for x, we get:

$x = \frac {2 \times 180} {3}$

which is what we’re taught, but now you know the why of it. It turns out that you don’t actually work with the quantities in the proportion as fractions. The fractional form is just used to relate the quantities to each other, metaphorically. The way you solve for x uses the form of the product equality relationship.

To answer Question 1, we have to establish a couple other things.

The first is the concept of the multiplicative inverse.

For every x (with x ≠ 0), there’s a unique v such that xv = 1, which means that $v = \frac {1} {x}$.

From that, we can say:

$xv = \frac {x} {1} \frac {1} {x} = \frac {x} {x} = 1$

From this, we can say that the inverse of x is unique to x.

King goes forward with another proof, which will lead us to answering Question 1:

Theorem 1:

$r = \frac {a} {b} \Rightarrow a = br, where\; b \neq 0$

Proof:

$b (\frac {a} {b}) = rb$

by cancelling like terms, we get:

${a = br}$

(It’s also true that $a = br \Rightarrow r = \frac {a} {b}$, but I won’t get into that here.)

Now onto Theorem 2:

$r = \frac {\frac {m} {n}} {\frac {p} {q}}, where\; n \neq 0, p \neq 0, q \neq 0$

By Theorem 1, we can say:

$\frac {m} {n} = r (\frac {p} {q})$

Then,

$\frac {m} {n} \frac {q} {p} = r (\frac {p} {q}) \frac {q} {p}$

By cancelling like terms, we get:

$\frac {m} {n} \frac {q} {p} = (r) 1$

$r = \frac {m} {n} \frac {q} {p}$

Therefor,

$\frac {\frac {m} {n}} {\frac {p} {q}} = \frac {m} {n} \frac {q} {p}$

And there you have it. This is why we invert and multiply when dividing fractions.

Edit 1/11/2018: King says a bit later in the book that by what I’ve outlined with the above definition, talking about how if there’s an equality between fractions, there’s also an equality between a product of their terms, and by Theorem 1, it is mathematically correct to say that division is just a restatement of multiplication. Interesting! This does not mean that you get equal results between division and multiplication: $\frac {a} {b} \neq ab$, except when b equals 1 or -1. It means that there’s a relationship between products and rational numbers.

Some may ask, since the mathematical logic for these truths is fairly simple, from an algebraic perspective, why don’t math classes teach this? Well, it’s because they’re not really teaching math…

Note for commenters:

WordPress supports LaTeX. That’s how I’ve been able to publish these mathematical expressions. I’ve tested it out, and LaTeX formatting works in the comments as well. You can read up on how to format LaTeX expressions at LaTeX — Support — WordPress. You can read up on what LaTeX formatting commands to use at Mathematical expressions — ShareLaTeX under “Further Reading”.

HTML codes also work in the comments. If you want to use HTML for math expressions, just a note, you will need to use specific codes for ‘<‘ and ‘>’. I’ve seen cases in the past where people have tried using them “naked” in comments, and WordPress interprets them as HTML tags, not how they were intended. You can read up on math HTML character codes here and here. You can read up on formatting fractions in HTML here.

Related post: The beauty of mathematics denied

— Mark Miller, https://tekkie.wordpress.com