An equation is a claim of equality, not an instruction to compute

0:00Here is a bar hanging from a ring, with something on each end. The ring is marked four. Each end is marked two. And the bar hangs dead level. Here is a second one. The ring says seven, one end says four, the other says three. This bar tips. It goes down on the four side. Two different things are being said here. The ring tells you the whole weight hanging underneath. Four is two and two; seven is four and three.

0:31The bar tells you something else entirely: whether the two sides agree. So look at what that second bar does. Its ring is correct. Four and three really is seven. And it still tips, because four is not three. Now notice something — a change of habit, not a new fact. A level bar does not tell you to do anything. It is not saying: add these up and write the answer over there.

1:00It reports: these two sides agree. That is the whole message. For years the equals sign has been an instruction — work out the left, put the answer on the right. From here it means what the bar means: these two things are the same size. Let us make that pay. This one hangs from a ring marked sixteen. On one string: a leaf, a small bud, another leaf. On the other: a single flower.

1:31You are told one number and one number only. A leaf weighs three. That looks nowhere near enough. The bar hangs level, so the leaf, bud and leaf together weigh exactly what the flower weighs. And the ring says everything together weighs sixteen. So the flower is half of sixteen. The flower is eight. The other side is eight too, and two of that eight is leaves. So the bud is two.

1:59Nobody handed you the bud. You read it off two true sentences. Now a harder one. Notice what is missing. Someone holds a bar level in their hand. No ring. No total anywhere. Three slices of bread hang on one side. Two eggs hang on the other. One slice of bread weighs two. That is everything you are given. Three slices at two each is six. The bar is level, so the two eggs together also weigh six.

2:32Two eggs, six. So one egg is three. And I never needed a total. The level bar on its own was enough. That is reporting doing the work computing used to do. Let me write that down instead of saying it. The egg is the thing I do not know, so I will give it a name. Call it e. Not because e is mysterious. Because I chose it, and it is short for egg.

3:00A letter is a name somebody picked, not a box you are meant to guess. Now the sentence. Two plus two plus two, on this side. And e plus e on that side. Two plus two plus two equals e plus e. Or, tidied up: two e equals six. Look at what that line is. It is not an instruction to compute anything. It is a claim. It says: whatever e turns out to be, twice it is six.

3:31This looks like a different subject. It is not. Matchsticks, laid out in a strip of triangles. Position one is a single triangle. Count the sticks: three. Position two adds a second triangle — but they share a side, so it does not cost three more. Count it. Five. Position three: seven. Position four: nine. The sharing is doing real work here. If those triangles were drawn apart, position four would need twelve sticks instead of nine.

4:06Three, five, seven, nine. Every single one of them is odd, and hold on to that. Now I could keep counting, one position at a time, forever. Instead, let me write those counts down in an awkward way. On purpose. Three is two times one, plus one. Five is two times two, plus one. Seven is two times three, plus one. I have not tidied those up, because tidying hides the only thing worth seeing.

4:38The one at the end never moves. The two at the front never moves. Only the middle number changes, and it is just the position. So call the position n. The number of sticks is two n plus one. One short line, and it covers every position at once. Here is where it gets interesting. So far it runs one way: you hand it a position, it hands back a stick count.

5:07Position ten? Two times ten, plus one. Twenty one sticks. Easy. But now somebody asks the question the other way round. I have ninety nine sticks. Which position uses exactly that many? You could grind through it: three, five, seven, nine, and keep going. Or you could write one line. Two n plus one equals ninety nine. That line has a name — the word this video has been walking towards.

5:40It is an equation. An equation is a written line claiming that two expressions name the same number. Not an instruction. A claim. It has two halves, and they get names. Everything before the equals sign is the left hand side. Two n plus one. Everything after it is the right hand side. Ninety nine. Compare that with two n plus one on its own, with no equals sign anywhere.

6:10That is only an expression. Hand it a position and it names a number. It does not claim anything. Put the equals sign and the ninety nine after it and it claims something: true for exactly one position, false for every other. So what does it mean to solve it? It means finding which value of n makes the two sides agree. Evaluating: here is n, go and find the number. Solving: here is the number, go and find n.

6:43In this case the answer is forty nine. Check it the way you check a level bar: put forty nine in and see if the sides match. Two times forty nine is ninety eight, plus one is ninety nine. Right hand side, ninety nine. They agree. Try forty eight and the left hand side comes out ninety seven. Ninety seven is not ninety nine, so forty eight is out. One position works. Every other position fails.

7:15Now a question that sounds the same and is not. Can any position in this sequence use exactly two hundred sticks? Every count in this sequence is odd. Three, five, seven, nine, and on. Two hundred is even. So no. No position reaches it. But there is a sloppy way to say that. The sloppy way is: two n plus one equals two hundred has no answer. That is not true. It has a perfectly good answer.

7:48Two n would be a hundred and ninety nine, so n is half of that. Ninety nine and a half. Put it back in and you really do get two hundred. The arithmetic is fine. What fails is that there is no position ninety nine and a half. Positions are whole. So it is the strip that rules two hundred out, not the algebra. Ask for two hundred and one and everything is fine: that is position one hundred.

8:19Two habits to break before we finish. The first: the answer does not live on the right. Here is one written the other way round. Twenty equals y minus three. Twenty three works, because twenty three minus three is twenty. Swap the sides over and it is the same claim. The second habit: letters do not have to sit on one side only. Two z plus four equals five z minus fourteen.

8:48Letters on both sides, and it is still just a claim about a number. Six works. Two sixes plus four is sixteen. Five sixes minus fourteen is also sixteen. The bar hangs level. One last thing, and it is the point of all of it. When we wrote two n plus one equals ninety nine, we had no method for solving it. So why write it? Because the moment you can state what must be true about a number, you have changed the order of the work.

9:23Before, you had to know how to find the answer before you could begin. Now you write down what you know first, and go looking for the answer afterwards. The level bar said the same thing without a single symbol. It never told you to do anything. It told you what was true. Everything that comes next is a method for hunting that number down. But the sentence comes first, and the sentence is the hard part.