Rational arithmetic all works together

We have seen what the ra­tio­nal num­bers are, and five things we can do with them:

Th­ese might seem like five stan­dalone op­er­a­tions, but in fact they all play nicely to­gether in a par­tic­u­lar way. You don’t need to know the fancy name, but here it is any­way: math­e­mat­i­ci­ans say that the ra­tio­nals form an or­dered field, mean­ing that the five op­er­a­tions above:

• work in the rationals

• slot to­gether, be­hav­ing in cer­tain spe­cific ways that make it easy to calculate

You shouldn’t bother learn­ing this page par­tic­u­larly deeply, be­cause none of the prop­er­ties alone is very in­ter­est­ing; in­stead, try and ab­sorb it as a whole.

We’ve already seen the “in­stant rules” for ma­nipu­lat­ing the ra­tio­nals, and where they come from. Here, we’ll first quickly define the ra­tio­nal num­bers them­selves, and all the op­er­a­tions above, in “in­stant rule” for­mat, just to have them all here in one place. Then we’ll go through all the prop­er­ties that are re­quired for math­e­mat­i­ci­ans to be able to say that the op­er­a­tions on ra­tio­nals “play nicely to­gether” in the above sense; and we’ll be rely­ing on the in­stant rules as our defi­ni­tions, be­cause they’re to­tally un­am­bigu­ous. That way we can be much more sure that we’re not mak­ing some small er­ror. (Rely­ing on an in­tu­ition about how ap­ples work could in the­ory lead us astray; but the rules leave no wig­gle-room or scope for in­ter­pre­ta­tion.)

The let­ters $$a, b, c, d$$ should be read as stand­ing for in­te­gers (pos­si­bly pos­i­tive, nega­tive or $$0$$), and $$b$$ and $$d$$ should be as­sumed not to be $$0$$ (re­call­ing that it makes no sense to di­vide by $$0$$).

• A ra­tio­nal num­ber is a pair of in­te­gers, writ­ten as $$\frac{a}{b}$$, where $$b$$ is not $$0$$, such that $$\frac{a}{b}$$ is viewed as be­ing the same as $$\frac{c}{d}$$ pre­cisely when $$a \times d = b \times c$$. noteThis is just the in­stant rule for sub­trac­tion, be­low, to­gether with the as­sump­tion that $$\frac{0}{x} = \frac{0}{y}$$ for any $$x, y$$ nonzero in­te­gers; in par­tic­u­lar, the as­sump­tion that $$\frac{0}{b \times d} = \frac{0}{1}$$ when $$b, d$$ are not $$0$$, which we will need later.

• Ad­di­tion: $$\frac{a}{b} + \frac{c}{d} = \frac{a \times d + b \times c} {b \times d}$$$• Sub­trac­tion: $$\frac{a}{b} - \frac{c}{d} = \frac{a}{b} + \frac{-c}{d} = \frac{a \times d - b \times c}{b \times d}$$$

• Mul­ti­pli­ca­tion: $$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$$$• Divi­sion (where also $$c$$ is not $$0$$): $$\frac{a}{b} \big/ \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a \times d}{b \times c}$$$

• Com­par­i­son (where we write $$\frac{a}{b}$$ and $$\frac{c}{d}$$ such that both $$b$$ and $$d$$ are pos­i­tive noteRe­mem­ber, we can do that: if $$b$$ is nega­tive, for in­stance, we may in­stead write $$\frac{a}{b}$$ as $$\frac{-a}{-b}$$, and the ex­tra minus-sign we in­tro­duce has now flipped $$b$$ from be­ing nega­tive to be­ing pos­i­tive.): $$\frac{a}{b} < \frac{c}{d}$$ pre­cisely when $$\frac{c}{d}-\frac{a}{b}$$ is pos­i­tive: that is, when $$\frac{b \times c - a \times d}{b \times d} > 0$$$which is in turn when $$b \times c - a \times d > 0$$$

Every ra­tio­nal must have an anti-ra­tio­nal un­der ad­di­tion noteMath­e­mat­i­ci­ans say that ad­di­tion is in­vert­ible.

Con­cretely, ev­ery ra­tio­nal $$\frac{a}{b}$$ must have some ra­tio­nal $$\frac{c}{d}$$ such that $$\frac{a}{b} + \frac{c}{d} = \frac{0}{1}$$. We’ve already seen that we can define $$-\frac{a}{b}$$ to be such an “anti-ra­tio­nal”, but re­mem­ber that in or­der to fit in with our in­stant rules above, $$-\frac{a}{b}$$ is also not a pair of in­te­gers; we should in­stead use $$\frac{-a}{b}$$.

Then $$\frac{a}{b} + \frac{-a}{b} = \frac{a \times b + (-a) \times b}{b \times b} = \frac{0}{b \times b}$$$Fi­nally, we want $$\frac{0}{b \times b} = \frac{0}{1}$$; but this is im­me­di­ate by the in­stant-rule defi­ni­tion of “ra­tio­nal num­ber”, since $$0 \times 1 = 0 \times (b \times b)$$ (both be­ing equal to $$0$$). Ad­di­tion doesn’t care which way round we do it noteMath­e­mat­i­ci­ans say that ad­di­tion is com­mu­ta­tive. Con­cretely, this is the fact that $$\frac{a}{b} + \frac{c}{d} = \frac{c}{d} + \frac{a}{b}$$$ This is in­tu­itively plau­si­ble be­cause “ad­di­tion” is just “plac­ing ap­ples next to each other”, and if I put five ap­ples down and then seven ap­ples, I get the same num­ber of ap­ples as if I put down seven and then five.

But we have to use the in­stant rules now, so that we can be sure our defi­ni­tion is com­pletely wa­ter­tight.

So here we go: $$\frac{a}{b} + \frac{c}{d} = \frac{a \times d + b \times c}{b \times d} = \frac{c \times b + d \times a}{d \times b} = \frac{c}{d} + \frac{a}{b}$$$where we have used the fact that mul­ti­pli­ca­tion of in­te­gers doesn’t care about the or­der in which we do it, and similarly ad­di­tion of in­te­gers. Ad­di­tion doesn’t care about the group­ing of terms noteMath­e­mat­i­ci­ans say that ad­di­tion is as­so­ci­a­tive. Here, we mean that $$\left(\frac{a}{b} + \frac{c}{d}\right) + \frac{e}{f} = \frac{a}{b} + \left( \frac{c}{d} + \frac{e}{f} \right)$$$ This is an in­tu­itively plau­si­ble fact, be­cause “ad­di­tion” is just “plac­ing ap­ples next to each other”, and if I put down three ap­ples, then two ap­ples to the right, then one ap­ple to the left, I get the same num­ber ($6$) of ap­ples as if I had put down one ap­ple on the left, then three ap­ples in the mid­dle, then two on the right.

But we have to use the in­stant rules now, so that we can be sure our defi­ni­tion is com­pletely wa­ter­tight. So here goes, start­ing from the left-hand side:

$$\left(\frac{a}{b} + \frac{c}{d}\right) + \frac{e}{f} = \frac{a \times d + b \times c}{b \times d} + \frac{e}{f} = \frac{(a \times d + b \times c) \times f + (b \times d) \times e}{(b \times d) \times f}$$

Since ad­di­tion and mul­ti­pli­ca­tion of in­te­gers don’t care about group­ing of terms, this is just $$\frac{a \times d \times f + b \times c \times f + b \times d \times e}{b \times d \times f}$$$Now go­ing from the right-hand side: $$\frac{a}{b} + \left( \frac{c}{d} + \frac{e}{f} \right) = \frac{a}{b} + \frac{c \times f + d \times e}{d \times f} = \frac{a \times (d \times f) + b \times (c \times f + d \times e))}{b \times (d \times f)}$$ and similarly we can write this $$\frac{a \times d \times f + b \times c \times f + b \times d \times e}{b \times d \times f}$$$ which is the same as we got by start­ing on the left-hand side.

Since we showed that both the left-hand and the right-hand side are equal to the same thing, they are in fact equal to each other.

How mul­ti­pli­ca­tion behaves

Mul­ti­pli­ca­tion always spits out a ra­tio­nal num­ber noteMath­e­mat­i­ci­ans say that the ra­tio­nals are closed un­der mul­ti­pli­ca­tion.

This one is very easy to show: $$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$$, but $$b \times d$$ is not zero when $$b$$ and $$d$$ are not zero, so this is a valid ra­tio­nal num­ber.

There must be a ra­tio­nal $$1$$ such that mul­ti­ply­ing a thing by $$1$$ doesn’t change the thing noteMath­e­mat­i­ci­ans say that mul­ti­pli­ca­tion has an iden­tity el­e­ment.

From our mo­ti­va­tion of what mul­ti­pli­ca­tion was (“do unto $$\frac{a}{b}$$ what you would oth­er­wise have done unto $$1$$”), it should be clear that the num­ber $$1$$ ought to work here. How­ever, $$1$$ is not strictly speak­ing a ra­tio­nal num­ber ac­cord­ing to the let­ter of the “in­stant rules” above; so in­stead we use $$\frac{1}{1}$$.

Mul­ti­pli­ca­tion doesn’t care which way round we do it noteMath­e­mat­i­ci­ans say that mul­ti­pli­ca­tion is com­mu­ta­tive.

This had its own sec­tion on the mul­ti­pli­ca­tion page, and its jus­tifi­ca­tion there was by ro­tat­ing a cer­tain pic­ture. Here, we’ll use the in­stant rule:

$$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} = \frac{c \times a}{d \times b} = \frac{c}{d} \times \frac{a}{b}$$
where we have used that mul­ti­pli­ca­tion of in­te­gers doesn’t care about or­der.

Mul­ti­pli­ca­tion doesn’t care about the group­ing of terms noteMath­e­mat­i­ci­ans say that mul­ti­pli­ca­tion is as­so­ci­a­tive.

This is much harder to see from our in­tu­ition, be­cause while ad­di­tion is a very nat­u­ral op­er­a­tion (“put some­thing next to some­thing else”), mul­ti­pli­ca­tion is much less nat­u­ral (it boils down to “make some­thing big­ger”). In a sense, we’re try­ing to show that “make some­thing big­ger and then big­ger again” is the same as “make some­thing big­ger by a big­ger amount”. How­ever, it turns out to be the case that mul­ti­pli­ca­tion also doesn’t care about the group­ing of terms.

Us­ing the in­stant rule for mul­ti­pli­ca­tion, it’s quite easy to show that $$\frac{a}{b} \times \left(\frac{c}{d} \times \frac{e}{f} \right) = \left(\frac{a}{b} \times \frac{c}{d} \right) \times \frac{e}{f}$$$In­deed, work­ing from the left-hand side: $$\frac{a}{b} \times \left(\frac{c}{d} \times \frac{e}{f} \right) = \frac{a}{b} \times \frac{c \times e}{d \times f} = \frac{a \times (c \times e)}{b \times (d \times f)}$$ which is $$\frac{a \times c \times e}{b \times d \times f}$$ be­cause mul­ti­pli­ca­tion of in­te­gers doesn’t care about the group­ing of terms. On the other hand, $$\left(\frac{a}{b} \times \frac{c}{d} \right) \times \frac{e}{f} = \frac{a \times c}{b \times d} \times \frac{e}{f} = \frac{(a \times c) \times e}{(b \times d) \times f} = \frac{a \times c \times e}{b \times d \times f}$$$

Th­ese two are equal, so we have shown that the left-hand and right-hand sides are both equal to the same thing, and hence they are the equal to each other.

How ad­di­tion and mul­ti­pli­ca­tion interact

Mul­ti­pli­ca­tion “filters through” ad­di­tion noteMath­e­mat­i­ci­ans say that mul­ti­pli­ca­tion dis­tributes over ad­di­tion.

What we will show is that $$\left(\frac{c}{d} + \frac{e}{f}\right) \times \frac{a}{b} = \left(\frac{a}{b} \times \frac{c}{d}\right) + \left(\frac{a}{b} \times \frac{e}{f}\right)$$$This is in­tu­itively true: the left-hand side is “make $$\frac{a}{b}$$, but in­stead of start­ing out with $$1$$, start with $$\left(\frac{c}{d} + \frac{e}{f}\right)$$”; while the right-hand side is “make $$\frac{a}{b}$$, but in­stead of start­ing out with $$1$$, start with $$\frac{c}{d}$$; then do the same but start with $$\frac{e}{f}$$; and then put the two to­gether”. If we draw out di­a­grams for the right-hand side, and put them next to each other, we get the di­a­gram for the left-hand side. (As an ex­er­cise, you should do this for some spe­cific val­ues of $$a,b,c,d,e,f$$.) We’ll prove it now us­ing the in­stant rules. The left-hand side is $$\left(\frac{c}{d} + \frac{e}{f}\right) \times \frac{a}{b} = \frac{c \times f + d \times e}{d \times f} \times \frac{a}{b} = \frac{(c \times f + d \times e) \times a}{(d \times f) \times b}$$$ which is $$\frac{c \times f \times a + d \times e \times a}{d \times f \times b}$$.

How com­par­i­son in­ter­acts with multiplication

The aim here is ba­si­cally to show that we can’t mul­ti­ply two things and get an anti-thing. Writ­ten out in the no­ta­tion, we wish to show that if $$0 < \frac{a}{b}$$ and if $$0 < \frac{c}{d}$$ then $$0 < \frac{a}{b} \times \frac{c}{d}$$, since the test for anti-ness is “am I less than $$0$$?”.

Since $$0 < \frac{a}{b}$$, there are two op­tions: ei­ther both $$a$$ and $$b$$ are pos­i­tive, or they are both nega­tive. (If one is nega­tive and one is pos­i­tive, then the frac­tion will be nega­tive.)

Like­wise ei­ther both $$c$$ and $$d$$ are pos­i­tive, or both are nega­tive.

So we have four op­tions in to­tal:

• $$a, b, c, d$$ are positive

• $$a, b, c, d$$ are negative

• $$a, b$$ are pos­i­tive; $$c, d$$ are negative

• $$a, b$$ are nega­tive; $$c, d$$ are pos­i­tive.

In the first case, we have $$\frac{a \times c}{b \times d}$$ pos­i­tive, be­cause all of $$a, b, c, d$$ are.

In the sec­ond case, we have $$a \times c$$ pos­i­tive and $$b \times d$$ also pos­i­tive (be­cause both are two nega­tive in­te­gers mul­ti­plied to­gether); so again the frac­tion $$\frac{a \times c}{b \times d}$$ is pos­i­tive.

In the third case, we have $$a \times c$$ nega­tive and $$b \times d$$ also nega­tive (be­cause both are a pos­i­tive num­ber times a nega­tive num­ber); so the frac­tion $$\frac{a \times c}{b \times d}$$ is a nega­tive di­vided by a nega­tive. There­fore it is pos­i­tive, be­cause we can mul­ti­ply the nu­mer­a­tor and the de­nom­i­na­tor by $$-1$$ to turn it into a pos­i­tive di­vided by a pos­i­tive.

We’ll con­sider $$\frac{1}{3}$$ and $$\frac{-2}{-5}$$. Then the product is $$\frac{1}{3} \times \frac{-2}{-5} = \frac{-2}{-15}$$; but that is the same as $$\frac{2}{15}$$.

In the fourth case, we can do the same as the above, or we can be a bit sneaky: us­ing the fact from ear­lier that mul­ti­pli­ca­tion doesn’t care about or­der, we can note that $$\frac{a}{b} \times \frac{c}{d}$$ is the same as $$\frac{c}{d} \times \frac{a}{b}$$. But now $$c$$ and $$d$$ are pos­i­tive, and $$a$$ and $$b$$ are nega­tive; we’ve already shown (in the pre­vi­ous case) that if the first frac­tion is “pos­i­tive di­vided by pos­i­tive”, and the sec­ond frac­tion is “nega­tive di­vided by nega­tive”. By swap­ping $$a$$ for $$c$$, and $$b$$ for $$d$$, we can use a re­sult we’ve already proved to ob­tain this fi­nal case.

Hence we’ve shown all four pos­si­ble cases, and so the fi­nal re­sult fol­lows.

Parents:

• Do you think we could have the ac­tual names of the rules as sub­head­ings or as foot­notes? Like, at the end of the “Mul­ti­pli­ca­tion doesn’t care about the group­ing terms” we could write “Math­e­mat­i­ci­ans call this the as­so­ci­a­tive prop­erty of mul­ti­pli­ca­tion”.