Everyday constraints that fix a range rather than a value

0:00Here are two true statements that cannot be written as equations. Every student in this class stands under a hundred and sixty centimetres. And this room holds no more than sixty chairs and tables counted together. Neither one names a height, and neither one names a count. Each fixes a ceiling and leaves everything underneath it open. An equation would name one number. These name a whole range of them, and that is not a weaker claim, it is a different one.

0:35Which raises the question this video is about. There are two ways to write a ceiling, and they are not interchangeable. One shuts the ceiling out. The other lets it in. Choosing between them turns out to be a measurement, not a matter of taste. Sam has two hundred coins and wants rice. The shop sells rice only in sealed packets, and a packet costs thirty coins. Let x be the number of packets. Then the outlay is thirty x, and whatever Sam buys, that outlay cannot exceed two hundred.

1:13So the model is thirty x, then a symbol, then two hundred. The usual move here is to write the symbol down from the wording and move on. We are not going to do that. We are going to work out which symbol the situation actually supports. And the first thing to establish is what happens to the money. Because everything else follows from that. Start with the obvious question. Can Sam spend the whole two hundred?

1:43Two hundred divided by thirty is twenty thirds, which sits between six and seven and is not a whole number. So look at the actual purchases. Six packets cost a hundred and eighty, leaving twenty coins behind. Seven packets cost two hundred and ten, and Sam is ten coins short. Here is what is left in the purse for zero packets up to seven: two hundred, a hundred and seventy, a hundred and forty, a hundred and ten, eighty, fifty, twenty, and then minus ten.

2:20The purse can cover the first seven of those and not the last. And the smallest amount Sam can be left holding is twenty coins. No purchase does better. The boundary here is not merely hard to hit. Nothing hits it. That last claim deserves suspicion, because we only looked at eight purchases. An empty search tells you nothing was found in the range you searched. It does not tell you nothing exists.

2:51So the range has to be one that could not have missed anything. Take the cheapest thing the shop sells, buy one more of it than the range allows, and buy nothing else. Here that is eight packets, costing two hundred and forty, which the purse cannot cover. If the cheapest way to stand just outside the range already overspends, then every basket outside it overspends, whatever it is made of.

3:20As a second check, the same sweep run over a range six times longer returns the same answer: nothing. Now the claim is safe, and we can use it. Now a second shopper, and the reason for her is that her money behaves differently. Mia has a hundred and twenty coins. A notebook costs forty and a pen costs twenty. Two things to buy means two letters. Let x count notebooks and y count pens, so the outlay is forty x plus twenty y.

3:56Same shape as before: an outlay, a ceiling, and a symbol to be decided. But ask Mia's version of the question and the answer comes out the other way. Her purse can be emptied to the last coin, and it can be done in four different ways. No notebooks and six pens. One notebook and four pens. Two and two. Three notebooks and no pens. Every one of those costs exactly a hundred and twenty. Divide through by twenty and each of them satisfies two x plus y equals six.

4:35So now we can stop guessing at symbols and start scoring them. Here is the method. The purse itself decides what is allowed, by what is left in it afterwards, and no symbol is involved in that decision. Then hand in six candidate relations and ask each one how often it disagrees with the money. Below. At most. Exactly equal. Above. At least. And anything but equal. Four of those six are order symbols, and you can find out which by swapping the two amounts over and seeing whose answer changes. The two built on equality cannot tell the sides apart.

5:18A statement setting two numbers, or two expressions, in one of those four relations is what the word inequality names. At Sam's shop the disagreements come out nought, nought, seven, eight, eight, and one. Two of the six describe his situation perfectly. Not one. Two. At Mia's shop the same six give four, nought, twelve, forty-nine, forty-five, and thirty-seven. Exactly one describes hers. The strict symbol gets her situation wrong four times.

5:55That asymmetry is the whole subject, so look at it directly. Ask what the strict statement permits and what the slack statement permits, and compare the two lists. At Sam's shop, both permit zero packets through six. Seven purchases each, and the same seven. The two symbols disagree nowhere. There is no purchase that tells them apart. At Mia's shop the strict statement permits twelve purchases and the slack one permits sixteen.

6:28They differ by four. And the four are not four arbitrary purchases. They are exactly the four that empty her purse: nothing and six, one and four, two and two, three and nothing. The disagreement between the two symbols is not merely the same size as the boundary. It is the boundary. Which lets us say precisely what each symbol claims. The gap between them is a set of purchases, and it is the set that lands exactly on the ceiling.

7:02Mia's set has four things in it. Sam's has nothing in it at all. So the strict symbol is not earned at Sam's shop by being right where the other is wrong. Both are right about every purchase he can make. It is earned because the sentence thirty x equals two hundred is never true. Mia's slack symbol, by contrast, is forced. Write hers strictly and you have just refused four purchases she can actually make.

7:34One situation leaves the choice open and settles it on the boundary. The other has no choice in it. The symbol records whether the ceiling is reachable, and reachable is something you count. There is a cleaner way to see what the slack symbol is holding. It is two statements at once: the strict one, and the equation. At Mia's shop the strict half permits twelve, the equation permits four, and the slack statement permits sixteen.

8:07Twelve and four make sixteen, and nothing is counted twice. No purchase is both under the budget and equal to it. One notebook and two pens costs eighty. That satisfies the strict half and not the equation. Two notebooks and two pens costs a hundred and twenty. That satisfies the equation and not the strict half. Both are purchases Mia can pay for, and only one of them empties her purse.

8:38Run the same split at Sam's shop and you get seven, nothing, and seven. His equation half is empty, which is the same fact from before wearing different clothes. Now a test of whether any of this came from the English rather than the arithmetic. Keep Sam's story word for word and change one number. Packets now cost twenty-five coins. Two hundred divided by twenty-five is eight exactly. Eight packets spend the whole purse.

9:13So the boundary is reached, the two symbols now differ by one purchase, and only the slack symbol fits the money. Same shopper, same shop, same sentence, opposite symbol. Do it to Mia too. Put her pens at thirty coins instead of twenty. Now only two purchases empty her purse rather than four, and the two symbols differ by two instead of four. The symbol is not read off the wording. It is read off the numbers.

9:45One more thing decides these answers, and it is easy to miss. We assumed all along that x counts packets and is therefore a whole number. Suppose instead the shop would sell a third of a packet. Then twenty thirds of a packet costs exactly two hundred coins. Sam's boundary becomes reachable, and his two symbols stop being interchangeable. Measured across three universes, whole numbers, halves and thirds, the gap between Sam's two symbols is nought, nought, and one.

10:21Halves do not help him, because twenty thirds is not a half of anything. Thirds do. Mia's gap over the same three is four, seven, and ten. Her boundary was reachable to begin with, and finer counts only give it more ways to be reached. So being unable to reach a ceiling is never a fact about a price alone. It is a fact about the price and the numbers the counter is allowed to take, together.

10:52Which brings us back to where we started. An equation on Sam's shop, thirty x equals a hundred and eighty, permits exactly one purchase: six packets. His constraint permits seven. Mia's permits sixteen. That is what a constraint hands you. Many values pass, and not one of them is singled out. It is not a vaguer statement than an equation. It says something exact about every value of the letter. It just says yes to more than one of them.

11:26Finding all of them, for a statement in one letter, is the next thing to learn, and it can be done completely. Mia's two-letter statement we have built as a model and will leave as one. Ask of any ceiling whether something reaches it. The answer is a count, and the count picks the symbol.