# Ordering of rational numbers (Math 0)

So far while learn­ing about how to com­bine ra­tio­nal num­bers, we have seen ad­di­tion, sub­trac­tion, mul­ti­pli­ca­tion and di­vi­sion. There is one fi­nal ma­jor thing we can do to a pair of ra­tio­nal num­bers: to com­pare them. Once you know what you’re look­ing for, it is very easy to com­pare cer­tain pairs of ra­tio­nal num­bers; in this page, we’ll look at how to ex­tend that.

In­tu­itively, if I gave you an ap­ple in one hand, and four ap­ples in the other hand noteMy hands are rather large., you would be able to tell me that the four-ap­ples hand was hold­ing more ap­ples. This is the kind of com­par­i­son we are try­ing to gen­er­al­ise, and we will do it from one sim­ple ob­ser­va­tion.

The ob­ser­va­tion we will make is that it is very easy to de­ter­mine whether a ra­tio­nal num­ber is nega­tive or not. noteRe­call that “nega­tive” meant “it is ex­pressed in anti-ap­ples rather than ap­ples”. In­deed, we just need to see if we’re hold­ing an anti-thing or not. This might be hard if we have to do some calcu­la­tions first—for ex­am­ple, it’s not im­me­di­ately ob­vi­ous whether $$\frac{16}{107} - \frac{3}{20}$$ is anti- or not—but we’ll as­sume that we’ve already done all the calcu­la­tions to re­duce an ex­pres­sion down to just a sin­gle ra­tio­nal num­ber. (In this ex­am­ple, we can use the sub­trac­tion tech­niques to work out that $$\frac{16}{107} - \frac{3}{20} = -\frac{1}{2140}$$. That’s ob­vi­ously nega­tive, be­cause it’s got a nega­tive sign out the front.)

To sum­marise, then, what I have just as­serted is that it is easy to see whether a ra­tio­nal num­ber is nega­tive or not; we say that a num­ber which is pos­i­tive noteThat is, ex­pressed in ap­ples rather than anti-ap­ples. is “greater than $$0$$”. Re­call that $$0$$ was the name we gave to the ra­tio­nal num­ber which is “no ap­ples at all”; then what this is say­ing is that if I have a pos­i­tive num­ber of ap­ples in one hand, and no ap­ples at all in the other, then ac­cord­ing to the in­tu­ition ear­lier, I have more ap­ples in the first hand than in the no-ap­ples hand. (Hope­fully you see that this is true; if not, let us know, be­cause this is one of those strange ar­eas where it’s very hard for a math­e­mat­i­cian to re­mem­ber not un­der­stand­ing it im­me­di­ately and in­tu­itively, since we’ve each been do­ing this for decades. We’re do­ing our best to re­mem­ber what parts of the maths are gen­uinely difficult and weird, but we might get it wrong.)

Similarly, we say that $$0$$ is less than any pos­i­tive num­ber, and write $$0 < \frac{5}{16}$$, for in­stance. The lit­tler quan­tity always goes on the lit­tler end of the ar­row. (The num­ber $$0$$ it­self is nei­ther nega­tive nor pos­i­tive. It’s just $$0$$. There­fore we can’t write $$0 < 0$$ or $$0 > 0$$; it’s ac­tu­ally the case that $$0=0$$, and this ex­cludes the other two op­tions of $$<) or \(>$$.)

# One weird trick to com­pare any two ra­tio­nals noteMath­e­mat­i­ci­ans hate it!

Now that we can com­pare any ra­tio­nal with $$0$$, we will work out how to com­pare any ra­tio­nal with any other ra­tio­nal.

The key in­sight is that adding the same num­ber of ap­ples noteOr anti-ap­ples. to each hand should not change the rel­a­tive fact of whether there are more ap­ples in one hand or the other. For a real-world ex­am­ple, on a bal­ance scale it doesn’t mat­ter whether you add five grams or even fifty kilo­grams onto each of the two pans; the re­sult of the weight com­par­i­son will be the same. (Now you should prob­a­bly for­get the scales metaphor again, be­cause the weight of an­ti­mat­ter be­haves in a way that doesn’t lend it­self nicely to what we’re try­ing to do.)

## Example

So let’s say we want to com­pare $$\frac{5}{6}$$ and $$\frac{3}{4}$$. Which of the two is big­ger?

Well, what we can do is add $$\frac{3}{4}$$ of an anti-ap­ple to both hands. By the prin­ci­ple that “adding the same amount to each hand doesn’t change their quan­tity rel­a­tive to each other”, the re­sult of the com­par­i­son be­tween $$\frac{5}{6}$$ and $$\frac{3}{4}$$ is just the same as the re­sult of the com­par­i­son be­tween $$\frac{5}{6} - \frac{3}{4}$$ and $$\frac{3}{4} - \frac{3}{4}$$: that is, be­tween $$\frac{1}{12}$$ and $$0$$. That’s easy, though, since we already know how to com­pare $$0$$ with any­thing!

So $$\frac{5}{6}$$ is big­ger than $$\frac{3}{4}$$, since $$\frac{1}{12}$$ is big­ger than $$0$$: we write $$\frac{5}{6} > \frac{3}{4}$$.

## Com­par­i­sons with anti-apples

In this sec­tion, we will just close our eyes, swal­low grimly, and hope for the best.

If we want to com­pare $$-\frac{59}{12}$$ and $$\frac{4}{7}$$, what should hap­pen? If you already have the right in­tu­ition built in, then this will be ob­vi­ous, but be­fore you know how to do it, it’s re­ally not clear at all. After all, $$-\frac{59}{12}$$ is “a large amount of anti-ap­ple” (it’s nearly five whole anti-ap­ples!) but $$\frac{4}{7}$$ is “a small amount of ap­ple” (not even one whole ap­ple).

Here’s where the “close our eyes” hap­pens. Let’s just go by the prin­ci­ple that adding the same amount of ap­ple to both hands shouldn’t change any­thing, and we’ll add $$\frac{59}{12}$$ ap­ples to both sides.

Then the $$-\frac{59}{12}$$ be­comes $$0$$, and the $$\frac{4}{7}$$ be­comes the rather grue­some $$\frac{461}{84}$$. noteThis is all good prac­tice for you to get fluent with adding and sub­tract­ing. But we already know how to com­pare $$0$$ with things, so we can see that $$\frac{461}{84}$$ is big­ger than $$0$$.

There­fore we must have $$\frac{4}{7}$$ be­ing big­ger than $$-\frac{59}{12}$$.

By the same to­ken, any amount of anti-ap­ple is always less than any amount of ap­ple, and in­deed any amount of anti-ap­ple is always less than $$0$$.

## Another example

How about com­par­ing $$\frac{-3}{5}$$ and $$\frac{9}{-11}$$? The first thing to do is to re­mem­ber that we can take the minus signs out­side the frac­tions, be­cause an anti-chunk of ap­ple is the same as a chunk of anti-ap­ple.

That is, we are try­ing to com­pare $$-\frac{3}{5}$$ and $$-\frac{9}{11}$$.

Add $$\frac{3}{5}$$ to both, to com­pare $$0$$ and $$-\frac{9}{11} + \frac{3}{5} = -\frac{12}{55}$$.

Add $$\frac{12}{55}$$ to both again, to com­pare $$\frac{12}{55}$$ and $$0$$.

Clearly the $$\frac{12}{55}$$ is big­ger, so it must be that $$\frac{-3}{5}$$ is big­ger than $$\frac{9}{-11}$$.

## In­stant rule

Just as we had an in­stant rule for ad­di­tion, so we can make an in­stant rule for com­par­i­son.

If we want to see which of $$\frac{a}{b}$$ and $$\frac{c}{d}$$ is big­ger, it is enough to see which of $$0$$ and $$\frac{c}{d} - \frac{a}{b}$$ is big­ger.

But

$$\frac{c}{d} - \frac{a}{b} = \frac{c \times b - a \times d}{b \times d}$$

Sadly from this point there are ac­tu­ally two cases to con­sider, be­cause we might have pro­duced some­thing that looks like any of the fol­low­ing:

• $$\frac{5}{6}$$

• $$\frac{-4}{7}$$

• $$\frac{3}{-8}$$

• $$\frac{-2}{-9}$$

(That is, there could be minus-signs scat­tered all over the place.)

How­ever, there is a way to get around this, and it hinges on the fact from the Divi­sion page that $$\frac{-1}{-1} = 1$$.

If $$b$$ is nega­tive, then we can just write $$\frac{a}{b} = \frac{-a}{-b}$$, and now $$-b$$ is pos­i­tive! For ex­am­ple, $$\frac{5}{-6}$$ has $$b=-6$$; then that is the same as $$\frac{-5}{6}$$. Similarly, $$\frac{-7}{-8}$$ is the same as $$\frac{7}{8}$$.

Like­wise we can always write $$\frac{c}{d}$$ so that the nu­mer­a­tor is pos­i­tive: as $$\frac{-c}{-d}$$ if nec­es­sary.

So, we have four cases:

• If $$b, d$$ are both pos­i­tive, then $$\frac{c \times b - a \times d}{b \times d}$$ is pos­i­tive pre­cisely when $$c \times b - a \times d$$ is pos­i­tive as an in­te­ger; i.e. when $$cb > ad$$.

• If $$b$$ is pos­i­tive and $$d$$ is nega­tive, then

$$\frac{c}{d} - \frac{a}{b} = \frac{-c}{-d} - \frac{a}{b} = \frac{(-c) \times b - a \times (-d)}{b \times (-d)}$$
where the de­nom­i­na­tor noteRe­mem­ber, that’s the thing on the bot­tom of the frac­tion: in this case, $$b \times (-d)$$. is pos­i­tive. the rest of this sec­tion and bul­let point

## Why did I say “Math­e­mat­i­ci­ans hate it”?noteAside from par­o­dy­ing In­ter­net ban­ner ads, that is. A di­ver­sion on pedagogy

The way I’ve done this is all com­pletely cor­rect, but it’s slightly back­wards from the way a math­e­mat­i­cian would usu­ally pre­sent it. (Not suffi­ciently back­wards that math­e­mat­i­ci­ans should hate it, but I couldn’t re­sist.)

Usu­ally, when find­ing a way of com­par­ing ob­jects, math­e­mat­i­ci­ans would prob­a­bly do what we’ve done above: find an easy way of com­par­ing some of the ob­jects, and then try to ex­tend it to cover all the ob­jects.

But if a math­e­mat­i­cian already knows a way of com­par­ing ob­jects and is just writ­ing it down for the benefit of other math­e­mat­i­ci­ans, they would usu­ally write down the com­plete “cover all the ob­jects” method right at the start, and would then go on to show that it does in­deed cover all the ob­jects and has all the right prop­er­ties.

This has the benefit of pro­duc­ing very terse de­scrip­tions with the min­i­mum nec­es­sary amount of writ­ing; but it’s very bad at helping other peo­ple un­der­stand where it came from. If you know where some­thing came from, you stand a bet­ter chance of be­ing able to recre­ate it your­self if you for­get the bot­tom line, and you might well re­mem­ber the bot­tom line bet­ter, too. This is why we’ve done it slightly back­wards here.

Parents:

• Mathematics

Math­e­mat­ics is the study of num­bers and other ideal ob­jects that can be de­scribed by ax­ioms.