Rational numbers: Intro (Math 0)

In or­der to get the most out of this page, you prob­a­bly want a good grasp of the in­te­gers first.

“Ra­tional num­ber” is a phrase math­e­mat­i­ci­ans use for the idea of a “frac­tion”. Here, we’ll go through what a frac­tion is and why we should care about them.

What is a frac­tion?

So far, we’ve met the in­te­gers: whole num­bers, which can be ei­ther big­ger than \(0\) or less than \(0\) (or the very spe­cial \(0\) it­self). The nat­u­ral num­bers can count the num­ber of cows I have in my pos­ses­sion; the in­te­gers can also count the num­ber of cows I have af­ter I’ve given away cows from hav­ing noth­ing, re­sult­ing in anti-cows.

In this ar­ti­cle, though, we’ll stop talk­ing about cows and start talk­ing about ap­ples in­stead. The rea­son will be­come clear in a mo­ment.

Sup­pose I have two ap­ples. noteI’m ter­rible at draw­ing, so my ap­ples look sus­pi­ciously like cir­cles.

Two apples

What if I chopped one of the ap­ples into two equally-sized pieces? (And now you know why we stopped talk­ing about cows.)

Now what I have is a whole ap­ple, and… an­other ap­ple which is in two pieces.

Two apples, one halved

Let’s imag­ine now that I chop one of the pieces it­self into two pieces, and for good mea­sure I chop my re­main­ing whole ap­ple into three pieces.

Two apples, formed as two quarters, one half, three thirds

I still have the same amount of ap­ple as I started with—I haven’t eaten any of the pieces or any­thing—but now it’s all in funny-sized chunks.

Now I’ll eat one of the small­est chunks. How many ap­ples do I have now?

Two apples, with a quarter eaten from one

I cer­tainly don’t just have one ap­ple, be­cause three of the chunks I’ve got in front of me will to­gether make an ap­ple; and I’ve also got some chunks left over once I’ve done that. But I can’t have two ap­ples ei­ther, be­cause I started with two and then I ate a bit. Math­e­mat­i­ci­ans like to be able to com­pare things, and if I forced you to make a com­par­i­son, you could say that I have “more than one ap­ple” but “fewer than two ap­ples”.

If you’re happy with that, then it’s a rea­son­able thing to ask: “ex­actly how much ap­ple do I have?”. And the math­e­mat­i­cian will give an an­swer of “one ap­ple and three quar­ters”. “One and three quar­ters” is an ex­am­ple of a ra­tio­nal num­ber or frac­tion: it ex­presses a quan­tity that came from di­vid­ing some num­ber of things into some num­ber of equal parts, then pos­si­bly re­mov­ing some of the parts. noteI’ve left out the point that just as you moved from the count­ing num­bers to the in­te­gers, thereby al­low­ing you to owe some­one some ap­ples, so we can also have a nega­tive ra­tio­nal num­ber of ap­ples. We’ll get to that in time.

The ba­sic build­ing block

From a cer­tain point of view, the build­ing block of the nat­u­ral num­bers is just the num­ber \(1\): all nat­u­ral num­bers can be made by just adding to­gether the num­ber \(1\) some num­ber of times. (If I have a heap of ap­ples, I can build it up just from sin­gle ap­ples.) The build­ing block of the in­te­gers is also the num­ber \(1\), be­cause if you gave me some ap­ples note­which per­haps I’ve now eaten so that I owe you some ap­ples, you might as well have given them to me one by one.

Now the ra­tio­nals have build­ing blocks too, but this time there are lots and lots of them, be­cause if you give me any kind of “build­ing block”—some quan­tity of ap­ple—I can always just chop it into two pieces and make a smaller “build­ing block”. (This wasn’t true when we were con­fined just to whole ap­ples, as in the nat­u­ral num­bers! If I can’t di­vide up an ap­ple, then I can’t make any quan­tity of ap­ples smaller than one ap­ple. noteEx­cept no ap­ples at all.)

It turns out that a good choice of build­ing blocks is “one piece, when we di­vide an ap­ple into sev­eral equally-sized pieces”. If we took our ap­ple, and di­vided it into five equal pieces, then the cor­re­spond­ing build­ing-block is “one fifth of an ap­ple”: five of these build­ing blocks makes one ap­ple. To a math­e­mat­i­cian, we have just made the ra­tio­nal num­ber which is writ­ten \(\frac{1}{5}\).

Similarly, if we di­vided our ap­ple in­stead into six equal pieces, and take just one of the pieces, then we have made the ra­tio­nal num­ber which is writ­ten \(\frac{1}{6}\).

The (pos­i­tive) ra­tio­nal num­bers are just what­ever we could make by tak­ing lots of copies of build­ing blocks.

Examples

  • \(1\) is a ra­tio­nal num­ber. It can be made with the build­ing block that is just \(1\) it­self, which is what we get if we take an ap­ple and di­vide it into just one piece—that is, mak­ing no cuts at all. Or, if you’re a bit squeamish about not mak­ing any cuts, \(1\) can be made out of two halves: two copies of the build­ing block that re­sults when we take an ap­ple and cut it into two equal pieces, tak­ing just one of the pieces. (We write \(\frac{1}{2}\) for that half-sized build­ing block.)

  • \(2\) is a ra­tio­nal num­ber: it can be made out of two lots of the \(1\)-build­ing-block, or in­deed out of four lots of the \(\frac{1}{2}\)-build­ing-block.

  • \(\frac{1}{2}\) is a ra­tio­nal num­ber: it is just the half-sized build­ing block it­self.

  • If we took the ap­ple and in­stead cut it into three pieces, we ob­tain a build­ing block which we write as \(\frac{1}{3}\); so \(\frac{1}{3}\) is a ra­tio­nal num­ber.

  • Two copies of the \(\frac{1}{3}\)-build­ing-block makes the ra­tio­nal num­ber which we write \(\frac{2}{3}\).

  • Five copies of the \(\frac{1}{3}\)-build­ing-block makes some­what more than one ap­ple. In­deed, three of the build­ing blocks can be put to­gether to make one full ap­ple, and then we’ve got two build­ing blocks left over. We write the ra­tio­nal num­ber rep­re­sented by five \(\frac{1}{3}\)-build­ing-blocks as \(\frac{5}{3}\).

Notation

Now you’ve seen the no­ta­tion \(\frac{\cdot}{\cdot}\) used a few times, where there are num­bers in the places of the dots. You might be able to guess how this no­ta­tion works in gen­eral now: if we take the blocks re­sult­ing when we di­vide an ap­ple into “di­videy-num­ber”-many pieces, and then take “lots” of those pieces, then we ob­tain a ra­tio­nal num­ber which we write as \(\frac{\text{lots}}{\text{dividey-number}}\). Math­e­mat­i­ci­ans use the words “nu­mer­a­tor” and “de­nom­i­na­tor” for what I called “lots” and “di­videy-num­ber”; so it would be \(\frac{\text{numerator}}{\text{denominator}}\) to a math­e­mat­i­cian.

Exercises

Can you give some ex­am­ples of how we can make the num­ber \(3\) from smaller build­ing blocks? (There are lots and lots of ways you could cor­rectly an­swer this ques­tion.)

You already know about one way from when we talked about the nat­u­ral num­bers: just take three copies of the \(1\)-block. (That is, three ap­ples is three sin­gle ap­ples put to­gether.)

Another way would be to take six half-sized blocks: \(\frac{6}{2}\) is an­other way to write \(3\).

Yet an­other way is to take fif­teen fifth-sized blocks: \(\frac{15}{5}\) is an­other way to write \(3\).

If you want to mix things up, you could take four half-sized blocks and three third-sized blocks: \(\frac{4}{2}\) and \(\frac{3}{3}\) to­gether make \(3\). Three apples: four halves and three thirds <div><div>

If you felt deeply un­easy about the last of my pos­si­ble solu­tions above, there is a good and perfectly valid rea­son why you might have done; we will get to that even­tu­ally. If that was you, just for­get I men­tioned that last one for now. If you were com­fortable with it, that’s also nor­mal.

How about mak­ing the num­ber \(\frac{1}{2}\) from smaller blocks?

Of course, you could start by tak­ing just one \(\frac{1}{2}\) block.

For a more in­ter­est­ing an­swer, you could take three copies of the sixth-sized block: \(\frac{3}{6}\) is the same as \(\frac{1}{2}\).

A half, expressed in sixths

Alter­na­tively, five copies of the tenth-sized block: \(\frac{5}{10}\) is the same as \(\frac{1}{2}\).

A half, expressed in tenths <div><div>

The way I’ve drawn the pic­tures might be sug­ges­tive: in some sense, when I’ve given differ­ent an­swers just now, they all look like “the same an­swer” but with differ­ent lines drawn on. That’s be­cause the ra­tio­nal num­bers (“frac­tions”, re­mem­ber) cor­re­spond to an­swers to the ques­tion “how much?”. While there is always more than one way to build a given ra­tio­nal num­ber out of the build­ing blocks, the way that we build the num­ber doesn’t af­fect the ul­ti­mate an­swer to the ques­tion “how much?”. \(\frac{5}{10}\) and \(\frac{1}{2}\) and \(\frac{3}{6}\) are all sim­ply differ­ent ways of writ­ing the same un­der­ly­ing quan­tity: the num­ber which rep­re­sents the fun­da­men­tal con­cept of “chop some­thing into two equal pieces”. They each ex­press differ­ent ways of mak­ing the same amount (for in­stance, out of five \(\frac{1}{10}\)-blocks, or one \(\frac{1}{2}\)-block), but the amount it­self hasn’t changed.

Go­ing more general

Re­mem­ber, from when we treated the in­te­gers us­ing cows, that I can give you a cow (even if I haven’t got one) by cre­at­ing a cow/​anti-cow pair and then giv­ing you the cow, leav­ing me with an anti-cow. We count the num­ber of anti-cows that I have by giv­ing them a nega­tive num­ber.

We can do the same here with chunks of ap­ple. If I wanted to give you half an ap­ple, but I didn’t have any ap­ples, I could cre­ate a half-ap­ple/​half-anti-ap­ple pair, and then give you the half-ap­ple; this would leave me with a half-anti-ap­ple.

We count anti-ap­ples in the same way as we count anti-cows: they are nega­tive.

See the page on sub­trac­tion for a much more com­pre­hen­sive ex­pla­na­tion; this page is more of a whis­tle-stop tour.

Limitations

We’ve had the idea of build­ing-blocks: as \(\frac{1}{n}\), where \(n\) was a nat­u­ral num­ber. Why should \(n\) be just a nat­u­ral num­ber, though? We’ve already seen the in­te­gers; why can’t it be one of those? noteThat is, why not let it be nega­tive?

As it turns out, we can let \(n\) be an in­te­ger, but we don’t ac­tu­ally get any­thing new if we do. We’re go­ing to pre­tend for the mo­ment that \(n\) has to be pos­i­tive, be­cause it gets a bit weird try­ing to di­vide things into three anti-chunks; this ap­proach doesn’t re­strict us in any way, but if you are of a cer­tain frame of mind, it might just look like a strange and ar­tifi­cial bound­ary to draw.

How­ever, you must note that \(n\) can­not be \(0\) (what­ever your stance on di­vid­ing things into anti-chunks). While there is a way to fi­nesse the idea of an anti-chunk noteAnd if you sit and think re­ally hard for a long time, you might even come up with it your­self!, there is sim­ply no way to make it pos­si­ble to di­vide an ap­ple into \(0\) equal pieces. That is, \(\frac{1}{0}\) is not a ra­tio­nal num­ber (and you should be very wary of call­ing it any­thing that sug­gests it’s like a num­ber—like “in­finity”—and un­der no ac­count may you do ar­ith­metic on it).

Summary

So far, you’ve met what a ra­tio­nal num­ber is! We haven’t gone through how to do things with them yet, but hope­fully you now un­der­stand vaguely what they’re there for: they ex­press the idea of “di­vid­ing some­thing up into parts”, or “shar­ing things out among peo­ple” (if I have two ap­ples to split fairly among three peo­ple, I can be fair by chop­ping each ap­ple into three \(\frac{1}{3}\)-sized build­ing blocks, and then giv­ing each per­son two of the blocks).

Next up, we will see how we can com­bine ra­tio­nal num­bers to­gether, even­tu­ally mak­ing a very con­ve­nient short­hand. noteThe study of this short­hand is known as “ar­ith­metic”.

Parents:

  • Rational number

    The ra­tio­nal num­bers are “frac­tions”.

    • Mathematics

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