A letter-number stands for any number, not one number

0:00Here is a sentence with no algebra in it at all. Shabnam is three years older than Aftab. That is a relationship between two ages, and it never changes. Whatever age Aftab happens to be, Shabnam is that, and three more. So if somebody tells you how old Aftab is, you can answer for Shabnam. Aftab is four. Then Shabnam is seven. Aftab is ten. Then Shabnam is thirteen. Aftab is twenty three. Then Shabnam is twenty six.

0:35Three questions, three sums, three answers. And every one of them is correct. So let us put those in a table, because that is what a table is for. One column for Aftab's age, one column for Shabnam's. And notice what is actually written in the second column. Not seven. Four plus three. Not thirteen. Ten plus three. Not twenty six. Twenty three plus three. Because the sum is more honest than the answer. It shows where the answer came from.

1:12Now here is the next row, and it is the important one. A question mark. Some age nobody has told you yet. And the table has just run out of usefulness. Not because it is wrong. Every row in it is right. But it can only ever be finished for the rows somebody sat down and worked out. There are always more ages, and the table can never have them all.

1:40So here is the move that the whole of this subject is built on. Take the words Aftab's age, and rub them out. And in the space they leave, write a single letter. Call it a. Everything else in that row stays exactly as it was. Still plus three. The last row of the table reads: a, and beside it, a plus three. Now that row is doing something none of the others can.

2:08It is not an answer. It is the rule that made all the answers. And the letter is worth being careful about, because it is easy to misread. It is not a new kind of number. Arithmetic has all the numbers it needs. It is a new kind of name. A name that deliberately refuses to say which number it is. And that refusal is the entire point. Because it never commits, it covers everything.

2:38One line, holding every case at once, including the cases nobody has worked out. A letter used this way has a name of its own: a letter-number. And what you build with it is an algebraic expression. Which brings us to the mistake almost everybody makes here. The letter looks like a puzzle. It looks like something you are supposed to find. So let us be blunt: nothing here is asking what a is.

3:09There is no answer to that question, and there was never meant to be one. The letter is not hiding a number. It is holding a place. Watch what happens if you fill that place in. Put four where the a is, and the expression reads four plus three. Seven. Put twenty three where the a is, and it reads twenty three plus three. Twenty six. Every single age gives an answer, every answer is different, and all of them are right.

3:42Now let us run the same relationship the other way. Suppose you are told Shabnam's age and asked for Aftab's. Same sentence. Nothing about the world has changed. Shabnam is three years older, so Aftab is three years younger. Call Shabnam's age s. Then Aftab's age is s minus three. So one sentence has given you two expressions, and it is worth seeing that they are not rivals. They are one relationship, read from two different ends.

4:16But they are not the same expression, and here is the proof. Feed twenty into each. The first gives twenty three. The second gives seventeen. Let us do the whole move again on something that is not an age. Here are matchsticks, arranged into the letter L. One L takes two matchsticks. Two Ls take four. Three Ls take six. And you could keep going, but you can already feel the table coming.

4:47How many for five Ls? Five times two. Ten. How many for seven? Seven times two. Fourteen. How many for forty five Ls? Nobody is going to lay out forty five Ls to find out. Forty five times two. Ninety. So call the number of Ls n, and the number of matchsticks is two times n. Same move. A multiplication this time instead of an addition. Now a situation where one letter is not enough.

5:21You are at a market. An apple costs thirty five cents. A kilogram of flour costs sixty cents. You buy ten apples and five kilograms of flour. What do you pay? Ten apples at thirty five is three hundred and fifty. Five kilograms at sixty is three hundred. Six hundred and fifty cents altogether. Now do it again for eight apples and nine kilograms. Two hundred and eighty, and five hundred and forty.

5:54Eight hundred and twenty. And already you would rather have the rule than keep doing this. So call the apples c, and the kilograms of flour j. And the bill is c times thirty five, plus j times sixty. Two letters, because two different things can vary. And here is how you know you really needed both. Hold the apples still at ten, and change only the flour. The bill changes anyway. So the total is not decided by the apples alone.

6:28But suppose the rule were: buy one kilogram of flour with every apple. Then the flour is no longer free to vary. It follows the apples, and one letter does the whole job. So the number of letters is not the number of things in the sentence. It is the number of things that can move on their own. One warning, and it takes ten seconds. People say a letter stands for any number, and that is nearly right.

6:59What it refuses to do is name one particular number. What it is allowed to be is still decided by the situation. Watch: put minus five into a plus three, and the arithmetic answers cheerfully. Minus two. The expression has no idea it is talking about somebody's age. Nobody is minus five years old, and nobody has half a matchstick. That restriction is not in the letter. It never was. It is in the sentence that introduced the letter.

7:33So the letter goes as wide as the situation allows, and not one step wider. Now let us aim this at something you have known for years, to see what it buys. The perimeter of a square. All four sides the same length. Call that length q. Then the perimeter is four times q. Nothing new has been claimed about squares. That is the same fact you already had. But look what comes with it for free. A triangle with three equal sides: three times q.

8:08A regular pentagon: five times q. A regular hexagon: six times q. Which is one rule wearing three different numbers. An expression that captures a general relationship like this has a name. It is called a formula. And if the side of a square is seven, the perimeter is four times seven. Twenty eight. Straight back down to a number. So look at what has actually happened here. No new numbers were invented. Every answer in this video was ordinary arithmetic.

8:45What was invented is a way of writing a relationship down before you know the numbers. And it works both directions. Up, from worked cases to a rule that covers all of them. And down, by putting a number back in place of the letter, whenever you want an answer. A worked case tells you about one situation. A formula tells you about every situation. So when you meet a letter in mathematics, do not ask what it is equal to.

9:17Ask what it counts, and ask what it is allowed to be. It is a name that refuses to say which number it is. And that refusal is the most useful thing about it.