Why every rule of arithmetic carries over to algebra

0:00You have just been handed a letter that stands for a number. So the obvious next thing is to start doing algebra with it. And instead, the very next move is to go back to plain arithmetic. Sums with no letters in them anywhere. Numbers you could have done two years ago. That looks like a warm-up, and it is easy to skip. It is not a warm-up. It is the argument.

0:26Because there is a question hiding underneath all of algebra, and almost nobody asks it. You are about to start moving letters around. Rearranging them. Regrouping them. What gives you permission to do that to something when you do not know what it is? Start with something completely ordinary. Twenty three minus ten times two. You know the answer. Twenty three minus twenty, which is three. But watch how it gets written down, because the writing is the point.

1:00It is not treated as a subtraction at all. It is treated as an addition, of two things. The first thing is twenty three. The second thing is minus ten times two, which is minus twenty. And then those two things are added. Twenty three, plus minus twenty. Three. The minus stopped being something you do, and became something the term carries with it. That sounds like a fussy distinction. It is the opposite of fussy.

1:33Here is the difference it makes. If minus is something you do, then it is tied to a position in the line. It happens at that spot, between those two numbers, and it has to stay there. But if the minus belongs to the term, the term can go anywhere and take it along. So every expression becomes one kind of thing. A list of terms, added. Some of the terms happen to be negative, and that is all.

2:02One reading, instead of a different reading for every mixture of plus and minus. And now the useful part, because once everything is an addition, things are allowed. Take eighty three, plus twenty eight, minus thirteen, plus thirty two. As a list of terms: eighty three, twenty eight, minus thirteen, thirty two. Added in that order, they come to one hundred and thirty. Now shuffle the terms. Any order you like.

2:34There are twenty four ways to arrange four things, and every single one of them comes to a hundred and thirty. That is the first rule, and it is called swapping. Terms may be added in any order. But notice what is doing the work, because it is easy to get this wrong. Move the numbers around and leave the signs sitting where they were, and you get ninety two.

2:59Thirty eight out. The rule was never about the numbers. It was about the terms, each carrying its own sign. Here are the same four terms again, and the second rule. You may bundle them however you like before you add. Look at eighty three and minus thirteen. That pair is easy. Seventy. And twenty eight with thirty two. Also easy. Sixty. Seventy and sixty. A hundred and thirty, in two steps instead of four.

3:33That is grouping, and it is worth asking why it is allowed rather than just using it. It is allowed because it is true of numbers. Every bundling of those four terms lands in the same place. Not most of them. Every one, and you can check as many as you have patience for. Which is why nobody has to ask permission before doing it. Now a harder one, because it has a bracket with a minus in front of it.

4:03Sixty eight minus, bracket, eighteen plus thirteen. There are two ways in, and they had better agree. First way: do the bracket. Eighteen plus thirteen is thirty one. So the expression is sixty eight plus minus thirty one. Thirty seven. Second way: get rid of the bracket first, before adding anything. The minus reaches everything inside, so it becomes sixty eight, plus minus eighteen, plus minus thirteen. Now group the easy pair. Sixty eight and minus eighteen make fifty. Fifty, plus minus thirteen.

4:44Thirty seven. The same answer, and it was never in any danger of not being. And here is the version that goes wrong, because almost everybody writes it at least once. Sixty eight minus eighteen plus thirteen. The bracket has been dropped, and only the first thing inside had its sign changed. That comes to sixty three, and the answer is thirty seven. Twenty six out. The minus in front of a bracket reaches every term inside it, not just the nearest one.

5:19And here is the reason this mistake survives, which is genuinely interesting. Sometimes it gives the right answer anyway. It is right exactly when the last term inside is nothing at all, and that happens often enough to keep the habit alive. A method that is occasionally right is worse than one that is always wrong, because nothing ever corrects it. One more rule, and this is the one that does the heavy lifting later.

5:52Suppose you want three lots of, bracket, seven plus five. You can add first. Seven and five make twelve, and three twelves are thirty six. Or you can share the three out. Three sevens are twenty one, three fives are fifteen. Twenty one and fifteen. Thirty six. A multiple of a sum is the sum of the multiples. That is the distributive property, and it is the reason a bracket can ever be opened up.

6:24Again, this is not a fact about seven and five. It is true of every three numbers you could put there. Which brings us to the question this whole detour was for. Swapping. Grouping. Sharing a multiplier out across a bracket. Three rules, and not one of them mentions any particular number. Each one is true of every number there is. Positive, negative, large, small. Take any three numbers at all and check the sharing rule: the two sides differ by nothing.

6:59Not nearly nothing. Nothing, every time, with no exceptions hiding anywhere. So now think about what a letter actually is. It is a number whose identity has been withheld. That is the only thing wrong with it. And a rule that is true of every number does not need to know which number it has got. So it is already true of the letter. Not by a new argument. By the argument you already had.

7:29Which is a licence, and it is worth knowing exactly how wide it is. Because it is not a general permission to move things about once letters appear. Try swapping on a subtraction, as it is written, without turning it into a sum first. Nine take away four is five. Four take away nine is minus five. Those are different, so subtraction does not swap, and nothing may be built on the idea that it does.

8:00The licence is exactly as wide as what was proved about numbers, and not one step wider. That is why the sum of terms reading came first. It converts the whole expression into additions, and addition is the thing all three rules are about. You are not allowed to rearrange because there are letters. You are allowed because the rearrangement was already true. So here is the small sentence all of that was for.

8:32Take the expression a plus three, and put twenty three in place of the letter. Twenty three plus three. The expression takes the value twenty six. Nothing surprising happened. That is precisely the point. The letter did not need special handling, and the arithmetic did not change when it arrived. And from here on, you can do things you could not do before. You can rearrange an expression you cannot evaluate, and be certain the value has not moved.

9:05Because whatever number is hiding behind that letter, the rules were already true of it. Algebra has no rules of its own. It borrows every one of them from arithmetic, and the borrowing is why it works.