What "rational" means, and why the denominator cannot be zero

0:00A farmer has one wheat field and three children. Counting will not help here. There is one field, and one does not divide into three by counting. What is needed is a number that measures a share rather than a heap. Half a cup of oil is the same problem. Nothing in the whole numbers, and nothing in the integers either, names half a cup. So the line gets filled in. Between every pair of whole numbers, the fractions arrive.

0:33And once the integers taught us that every quantity has an opposite, every fraction gets one too. Three quarters sits to the right of nought. Minus three quarters sits the same distance to the left, and the two add to nothing at all. Nineteen sevenths and minus nineteen sevenths do the same. Whatever the size, the opposite is the number that cancels it. So the line now has fractions on both sides, and the two halves mirror each other exactly as they did for the whole numbers.

1:11That raises a small question with a surprisingly useful answer. Where does the minus sign go? You can put it in front of the whole fraction. You can put it on the top. You can put it on the bottom. One fifth, negated three different ways, and all three are the same number: minus nought point two. So the sign is a property of the number, not of the digits. It does not matter where you write it, as long as you write it once.

1:42Twice is different. A minus on the top and a minus on the bottom cancel each other, and you are back to positive one fifth. Now the definition, and it is a definition about what a number can be written as. A number is rational if you can write it as one integer over another, with the lower one not nought. Notice what that does not say. It does not say the number has to look like a fraction.

2:13Five is rational, because five over one exists. Minus ten is rational, because minus ten over one exists. Take every integer from minus twenty to twenty and go looking for a writing of each. All forty one of them have one. Every integer you have ever met was already rational before the word arrived. Which leaves the condition attached to the definition. Why can the bottom not be nought? Not because someone forbade it. Because of something we already established.

2:50Division is meant to undo multiplication. Three over one asks: what number, multiplied by one, gives three? One candidate answers: three. So ask the same question of three over nought. What number, multiplied by nought, gives three? Search a thousand and one candidates, from minus five hundred to five hundred. Not one works. And the reason is the rule we already have: multiplying by nought sends every one of those thousand numbers to the same place, and that place is nought. Nothing was ever going to arrive at three.

3:34Now the case people expect to be easier, and it fails worse. Nought over nought. What number, multiplied by nought, gives nought? Search the same thousand and one candidates. Every single one works. All of them. So the two failures point in opposite directions. Three over nought has nothing to give back. Nought over nought has too much. And a value that could be anything is not a value. An answer that does not narrow anything down has not answered.

4:08That is why the condition is on the bottom of every rational number, and why it is one condition rather than two exceptions. Here is something about fractions that is easy to use and hard to state. One point on the line. Minus one third. Now watch how many ways it can be written. Minus two sixths. Minus three ninths. Minus ten thirtieths. Minus two thousand and twenty six over six thousand and seventy eight.

4:42Every one of those is a different pair of integers and the same single point. Search a window of writings for one half alone and you find forty of them. The list does not end; it just runs off the edge of what we bothered to look at. So a rational number has no single writing. That is a genuine problem, and the fix is a decision. One of the writings gets chosen to stand for all of them. The one where the top and the bottom share no factor above one.

5:17Twelve thirtieths becomes two fifths, by dividing both parts by six, the largest factor they share. Take every fraction with parts up to twenty: five hundred and eleven different numbers. Reduce each one by cancelling. Every single number lands on exactly one representative, and every representative is co-prime. That is not tidiness. It is what makes it possible to say the denominator decides something, later on, without having to ask which denominator you meant.

5:56Which raises a practical question. Given two writings, how do you tell whether they are the same number? Cross-multiply. Top of the first times bottom of the second, against top of the second times bottom of the first. Equal products, equal numbers. Two thirds and four sixths: twelve and twelve. Five quarters and ten eighths: forty and forty. Minus three fifths and minus six tenths: agreed. And the fourth one is worth pausing on. Nine thirds against three. Not three over something, just three. Write it as three over one and the test runs anyway.

6:41The test was checked against the actual values over ninety thousand cases, and it never once disagreed. It also says no when the numbers differ, which is the part that makes it a test rather than a ritual. The four operations, which you have met before and can now see the reason for. Adding and subtracting: rewrite both over a shared bottom, then work on the tops. Two fifths and three tenths is seven tenths. Five sixths less a quarter is seven twelfths.

7:18Multiplying: straight across. Two thirds times three tenths is one fifth. Dividing: turn the second one over and multiply. Two thirds divided by three tenths is twenty ninths. And the laws you rely on still hold. Multiplying across a bracket was checked on eight hundred and nineteen different triples of fractions, and it never failed. Now the detail that gets dropped, and it is worth being exact about. Multiplying two fractions carries two conditions: neither bottom is nought. Fair enough.

7:58Dividing carries three. Neither bottom is nought, and the top of the divisor is not nought either. That third one is separate, and it is the one people leave out. It is there because turning the second fraction over puts its top on the bottom, where nought is not allowed. Same rule, arriving from a different direction. Everything in this topic is that one fact, seen from wherever you happen to be standing.

8:30One last question, and it is the one that says whether the job is finished. If you take two rational numbers and do something to them, do you land back among the rationals? Add: eleven thousand and twenty five pairs tested, and not one escaped. Subtract: the same. Multiply: the same. Divide: ten thousand two hundred and ninety pairs, with nought excluded as a divisor, and not one escaped. Put nought back in as a divisor and seven hundred and thirty five of them break immediately. That is the asterisk, and it is not a new fact. It is the forbidden denominator, arriving for the third time in one topic.

9:17Three operations without conditions, one with an exception, and the exception is something we derived rather than accepted.