The dropped multiplication sign, and what it silently means

0:00Here is a list. Four, eight, twelve, sixteen, twenty, twenty four, twenty eight. The multiples of four, in order, and it is obvious how it carries on. So here is a question you can answer with no algebra at all. What is the third entry? Twelve, and you did not count your way to it. You worked out four times three, because the third entry is three fours. The twenty ninth entry is four times twenty nine, which is a hundred and sixteen.

0:33Every entry is four times its own position in the list. That is the whole pattern. But now ask for the entry at some position nobody has named yet. You need a way to write four times a position you have not been told. Call that position n. It is only a name for a number nobody has said out loud. Then the entry sitting at that position is four times n.

1:01Write it with the multiplication sign in, exactly as you would with any number. Four times n. One line holding the whole list, every position at once. Put three in for n and it says four times three. Twelve. Put twenty nine in and it says four times twenty nine. A hundred and sixteen. Nothing clever has happened yet. That is still ordinary multiplication. But this expression is about to be written differently, and the difference matters.

1:35Because one of those symbols is about to disappear. The multiplication sign comes out. Four times n is written four n. The two symbols close up, and the cross between them is simply gone. There is an order to it as well, and the order is not optional. The number goes first and the letter after it. Four n, never n four. So four n is not a new object, and it is not a new kind of quantity.

2:07It is four times n, with a symbol you are expected to put back. Every single time you read it. That is the entire deal. And a convention that asks you to restore something silently is worth being suspicious of. So the first real question is why it is safe at all. The reason is simpler than you might expect, and it is the whole foundation. Exactly one operation is ever allowed to vanish. Multiplication, and nothing else.

2:41A plus is always written. A minus is always written. Nothing else is ever left out, so nothing else can be hiding in that gap. See a number and a letter sitting together with no sign between them. There is exactly one thing you could put back, so there is nothing to decide. If plus could be dropped too, four n would be genuinely unreadable. It might mean four times n, or four plus n, and nothing anywhere would tell you which.

3:12The convention works because it is stingy. One operation, and no more than one. So reading these is a habit, and the habit is to restore the sign first. Take seven k, and suppose k is four. Write it out. Seven times four. Then compute. Twenty eight. Two steps, and the first is the one people skip. Now something slightly harder. Five m plus three, with m equal to two.

3:44Restore the sign, and it reads five times m, plus three. Multiplication happens before addition, exactly as it always did. Five times two is ten, and ten plus three is thirteen. Add first and you would get twenty five, and the sign being invisible never made that allowed. Which brings us to an exercise that is not the one you expect. Nine finished substitutions. Somebody has already done all the work.

4:17Each one names a value for its letter, and states an answer. Your job is not to compute them. It is to audit them. That is a genuinely different skill, and it is the one that gets tested. Before you start, one warning that changes how you read the whole board. Eight of these nine are wrong. Exactly one of them is already correct. Go looking for nine mistakes and you will invent one, which is worse than missing one.

4:48The wrong eight fall into three families, and every family deserves a name. Family one, and it is half of them. Nothing restored into the gap at all. Three d, with d equal to six, answered as thirty six. The three and the six have been read as two digits standing side by side. Not three times six, which is eighteen. Just thirty six, as a numeral. Two r plus one, with r equal to eight, answered as twenty nine.

5:22The same error dressed up: two and eight became twenty eight, and then one was added. It happens twice more, and one of those loses an entire term along the way. Now here is what makes this family worth naming out loud. It is never accidentally right, and it always gives too much, so a surprisingly large answer is the first place to look. Family two. The sign does get restored, but it is given the wrong reach.

5:54Three s minus two, with s equal to seven, answered as fifteen. Fifteen is three times five, so the three was made to multiply all of s minus two. There is no bracket there, so the three multiplies s and nothing else. Three times seven is twenty one, and twenty one minus two is nineteen. Four out, purely from a mistake about what the multiplication touches. The same family catches a harder one. Three, bracket, m plus one, with m equal to minus six.

6:31Answered as nineteen, which is three times six plus one: the bracket ignored, the minus discarded. Honour the bracket and m plus one is minus five, and three of those is minus fifteen. Family three. A negative that did not survive the journey. Ten minus a, with a equal to minus four, answered as six. Six is ten minus four. The minus sign on the value has quietly gone missing.

7:04But you are taking away minus four, and taking away a negative adds. Ten take away minus four is fourteen. The last one is the same idea, hiding inside a bracket. h minus, bracket, three minus n, with h equal to five and n equal to six. The bracket is three minus six, which is minus three, and taking away minus three adds three. So it is eight, and notice that every one of these was settled the same way, by working it out at the value.

7:40Which is exactly the technique for a question people get genuinely stuck on. Is five u the same thing as five plus u? They look similar. A five, a u, and not much else in either of them. And you do not need an argument to settle it. You need one value. Let u be eleven. Five u is fifty five. Five plus u is sixteen. That is enough. One value where they disagree closes the question forever.

8:11Try more anyway, because watching them stay apart is worth something. u is two, ten against seven. u is eight, forty against thirteen. u is five, twenty five against ten. No whole number ever makes them meet, and one disagreement was always all it took. One last pair, and this one is sharper. Ten y minus three, against ten, bracket, y minus three. The same three symbols. The only difference is a pair of brackets.

8:46In the first, the ten multiplies y, and then three comes off the result. In the second, three comes off y first, and the ten multiplies what is left. Let y be two. The first gives seventeen. The second gives minus ten. y is nought, minus three against minus thirty. y is seven, sixty seven against forty. y is ten, ninety seven against seventy. Now watch the gap, because it never moves.

9:19It is twenty seven every time, the letter falls out of the difference entirely, and so there is no value where they meet.