Writing a situation as an expression before computing it

0:00Here are four things you have been writing down since you were small. Thirteen plus two. Twenty minus four. Twelve times five. Eighteen divided by three. Nothing new there at all. You have been doing these since primary school. But they have a name, and the name turns out to be worth having. Each one of those is an arithmetic expression. An expression is a written phrase built out of numbers and the operation signs.

0:31And here is the part that changes how you read every one of them. An expression is not an order to calculate. It is a name for a number, which also happens to record how that number was built. That one distinction is the whole of this video. Start with the simplest fact about an expression. It stands for exactly one number, and that number is called its value. Thirteen plus two has the value fifteen.

1:03Not usually fifteen. Not fifteen if you are careful. Fifteen, always. That is what makes it a name rather than a suggestion. Twenty minus four has the value sixteen. Twelve times five has the value sixty. Eighteen divided by three has the value six, and it divides exactly. Four expressions, four values, one apiece. Now watch what having a value lets us write. We write thirteen plus two equals fifteen.

1:38And most people read that sign as a button. Do the sum, and the answer appears over here. It is not a button. It is a claim. It says that the thing on the left and the thing on the right are names for the same number. Which means the sign does not care what shape either side is. Look at this one. Thirteen plus four, equals, blank, plus six.

2:03There is no answer on the right at all. There is another expression. And that is a perfectly sensible thing to write down, because the sign is comparing two values. The blank has to be eleven. Nothing else fits at all, and that is worth checking rather than assuming. If equals really meant, and the answer is, that line would be gibberish. It is not. There is a second thing hiding in an expression, and it is about how you say it out loud.

2:36Take thirteen plus two again. You can read it as an instruction. Thirteen, plus two. Something to be done. Or you can read it as a name. The sum of thirteen and two. Something that simply is. Both of those mouthfuls point at the number fifteen. Same with twelve times five. You can say twelve, times five. Or you can say the product of twelve and five. The first version sounds like work. The second sounds like a thing.

3:08And the second reading is the one that makes everything after this make sense. Because a thing can be handed to somebody as it is. A piece of work has to be finished first. So here is a situation. You practise something for twenty five minutes every day, Monday to Friday. How much practice is that across the week? Now, the reflex is to go straight for the total, and you can feel yourself doing it.

3:38Hold off for a moment, because the question I actually want is a slightly different one. Write down an expression for the week's practice. Not the answer. The expression. Five days, twenty five minutes on each of them. Five times twenty five. That is it. That is a complete answer to the question I asked. And I am going to leave it sitting there exactly like that. Which probably feels like stopping half way, so let me show you what you would be throwing away.

4:13Five times twenty five, and one hundred and twenty five, are the same number. The same number, but not the same amount of information. Look at five times twenty five. You can still see the five days in it. You can still see the twenty five minutes. Now look at one hundred and twenty five. Where are the days? Gone. Where is the amount you did each day? Also gone.

4:39The total is a perfectly good number that has forgotten where it came from. So which of the two is the better answer depends entirely on what you were asked for. They are both right. They are simply right about two different questions. Now turn the whole thing around. Instead of starting with an expression, start with a number. Twelve. Ten plus two is twelve. Fifteen minus three is twelve.

5:10Three times four is twelve. Twenty four divided by two is twelve. Four different expressions, and one value between them. And notice that this does not make twelve ambiguous in the slightest. Each of those expressions still has exactly one value. It just happens to be the same one. Many names, one number. Rather like a person with a name and a nickname — both pick out the same person, and neither of them is the real one.

5:44Which raises a question. How many names does twelve actually have? Let us count them properly. Two numbers, one operation, and both numbers under a hundred. Additions first. Ten plus two, one plus eleven, and so on down the line. There are eleven of those. Subtractions: rather more, because you can start high and come back down. Eighty seven of them. Multiplications: only six, because twelve does not have very many factors.

6:18Divisions that come out exactly: eight. Add all of that up and you get one hundred and twelve names for one small number. And that was under a rule that only allowed two numbers and one sign. Pick a number of your own and try it. Any number you like — seven, forty, ninety nine — every one of them comes out crowded. This is the moment the idea stops being a definition and starts being something you have seen.

6:49Which brings us to something slightly strange about all this. Go from an expression to its value and there is nothing whatsoever to decide. Ten plus two goes to twelve. Every time, for anybody, in any mood. It is automatic. A machine can do it, and machines do. Now try going the other way. Start at twelve and get back to the expression. Which one? There were a hundred and twelve of them, and that was with the rules turned right down.

7:21There is no way back, because the arrow was never reversible in the first place. Working out a value throws information away, and thrown-away information does not come back. That is not a flaw in anything. It is what evaluating means. It is simply worth knowing which direction you are travelling in, and what you are spending to get there. One last thing, and it is a warning rather than a lesson.

7:51Everything so far has used exactly two numbers and one sign, and that restriction has quietly been doing a great deal of work. Watch what happens the moment there are three numbers and two signs. Thirty, plus five, times four. Read that strictly left to right. Thirty plus five is thirty five, and thirty five times four is one hundred and forty. Now read it doing the multiplication first. Five times four is twenty, and twenty plus thirty is fifty.

8:26One hundred and forty, or fifty. Those are ninety apart. And neither of them is an arithmetic mistake. Both of those sums are done correctly. What is undecided is not the arithmetic at all. It is which expression was written down. So a line like that is not yet a name for anything, until we agree how to read it. Which is exactly where we go next.