Multiplying, then inverting: how division becomes possible

0:00Here is a rule you have probably been handed. a plus i b, times c plus i d, is a c minus b d, plus i times a d plus b c. Look at it and ask an honest question: how would you know if it were wrong? A rule you were handed is a rule you cannot check. So put six of them up together, all six taking four ordinary numbers and returning two.

0:28One multiplies slot by slot. One forgets the minus sign. One has a minus in both slots. One swaps the answers over, one throws the last term away — and one is the rule you were handed. Now run the usual tests. Does the order matter? Does the grouping of three? Is there something that changes nothing? Five of the six do not notice the order. Four of the six do not notice the grouping. Four of the six have something that changes nothing.

1:03And four of the six pass all three at once. So the three tests everybody reaches for cannot tell you which rule is right. Something else has to. Here is what does. Stop treating i as a special object, and treat it as a letter. You are multiplying a plus i b by c plus i d. That is two brackets, and you have been opening brackets since you were twelve.

1:32Every term in the first meets every term in the second. a meets c. a meets i d. i b meets c. And i b meets i d. Four terms. Not two. That is worth stopping on, because the commonest wrong answer here — multiply the first parts, multiply the second parts — has two terms in it. It has lost half the multiplication. Now count how many times the symbol appears in each of the four.

2:03None in the first. One in the second. One in the third. And two in the fourth. Nought, one, one, two. Nothing has been substituted yet. No new rule has been used. Two brackets were opened, and that is all. Now, and only now, the one thing you know about this symbol. Its square is minus one. Look back at the four terms and ask which of them that touches.

2:32Exactly one. The first three carry the symbol nought times or once, and a rule about the symbol squared has nothing to say about any of them. The fourth carries it twice, and that one becomes b d, times minus one. Minus b d — with no symbol left in it at all. So it stops being part of the second slot and joins the first. Gather what is left. The first slot has a c, and now minus b d.

3:04The two middle terms are the only ones carrying a single symbol, and together they make a d plus b c. That is the rule. All of it. One bracket expansion, one substitution, and collect. And set all six candidates against that expansion, across four hundred pairs of numbers: exactly one agrees everywhere, and it is this one. Trace the minus sign, because it is the part that looks arbitrary and is not.

3:37It is not in the question anywhere. Both brackets are additions. It arrives at exactly one moment: when i times i is replaced by minus one. That minus is the entire fingerprint the substitution leaves on the answer. Which means you never have to remember it. If you can expand two brackets, you can rebuild this rule in about ten seconds, and the minus will be sitting where it has to be.

4:08Now the wrong rule, and why it is dangerous rather than merely wrong. Multiplying slot by slot — first parts together, second parts together — is wrong at most of those four hundred pairs. But it is right at forty three of them. Forty three. So if you check one example, and it happens to land in those forty three, you will walk away certain of a rule that is false.

4:36One example can knock a rule down. It can never stand one up. You will usually meet a list of properties at this point, given without proof. Multiplying two of these gives another one. The order does not matter. The grouping does not matter. There is a number that changes nothing. Every number except one has something that multiplies it to that number. And multiplication spreads over addition. A list given without proof is worth checking.

5:10Across four hundred pairs, the order matters in none of them. Across two hundred and sixteen triples, the grouping matters in none of them. Spreading over a sum holds in all two hundred and sixteen, from the left and from the right. And every one of the four hundred products comes back as two slots again. That last one is not free, and here is the reason. Before the substitution, two hundred and eighty nine of those four hundred products really do climb up to the symbol squared.

5:45They leave the two-slot shape. The substitution is what brings them back, every time, and that is what the first property is actually claiming. Which number leaves everything alone when you multiply by it? Do not answer. Search. Forty nine candidates were tried here, and the test was strict: it has to leave every number unchanged, and from both sides. Exactly one survived. One, plus i nothing. The ordinary one, sitting in this system the way every ordinary number sits in it, with an empty second slot.

6:27And it did not have to come out that way. Having something that changes nothing is a property of the rule, not of the notation. Four of those six candidate rules had one, and two of them had none at all. The searching is the point. Written down, one plus i nothing is a guess that happens to be right. Found, it is a fact about this system. Now the harder one. Given a number, what multiplies it to one?

6:59Take two plus i three, and search again — three hundred and twenty four candidates this time. Exactly one works: two thirteenths, minus i three thirteenths. Look at what came out. The two and the three are still there, the sign of the second has flipped, and both sit over thirteen. Thirteen is two squared plus three squared. So the general shape is: a over a squared plus b squared, and minus b over the same thing.

7:31But do not take that on trust either — multiply it back. The first slot of the product comes to a squared plus b squared, over a squared plus b squared. That is one. The second slot comes to minus a b plus a b. That is nothing. One plus i nothing. The identity, exactly. Across nineteen different numbers, that check fails zero times. One line of algebra, and the formula is no longer something you hope is right.

8:06There is a quiet question underneath that, and it is the most important thing here. A fraction needs its bottom not to be nothing. So when is a squared plus b squared nothing? Do not go hunting for examples. Argue. a is an ordinary real number, so it is below nothing, exactly nothing, or above nothing. Three cases, and there is no fourth. The same for b, so nine between them.

8:36Squaring kills the sign. Whatever a was, a squared is nothing or above it, and the same for b squared. So the bottom is a sum of two things, neither of which is ever below nothing. A sum like that is nothing only when both pieces are. Of the nine cases, exactly one gives nothing, and it is the case where a is nothing and b is nothing. Which is the number nought itself — the one number excluded.

9:06That is not fine print. It is the whole condition, and it covers every number at once. There is a way this is usually done by hand, and it looks like a different method. It is not. You want one over two plus i three. Multiply above and below by two minus i three — the same number with the sign of its second slot flipped. Below, that product comes to thirteen.

9:36Thirteen and nothing. An ordinary real number, with an empty second slot. And now you can divide, because dividing by an ordinary number is something you have always been able to do. Why does the bottom come out clean? The second slot of that product is a times minus b, plus b times a. Those cancel before anything is substituted. The symbol never enters it. Across all nineteen numbers, that product leaves something in the second slot exactly zero times — and it stays zero for every one of the nine things the symbol could have squared to.

10:19So it is not a trick that happens to work. It is the same computation, written in a different order. Now a place where the wording you will usually meet is narrower than the mathematics. The condition beside that formula is often written as: a is not nothing, and b is not nothing. Read literally, that refuses any number with an empty slot. Of the nineteen numbers in this sweep, four have an empty slot — two with nothing in the first, two with nothing in the second.

10:55And all four of them have an inverse. Take minus i. Its first part is nothing and its second is minus one, so the bottom is nought plus one, which is one. The formula returns nought plus i one. Just i. Multiply back: minus i, times i, is minus i squared, which is one. It works. The condition the mathematics needs is the one the argument gave you: the number is not nought. Not both parts non-zero, not either part non-zero. Just not nought.

11:33If a stated condition and an argument disagree, believe the argument. Which brings us to dividing, and to the good news. There is nothing to define. z one divided by z two means z one, multiplied by the inverse of z two. No second rule and no long-division procedure. Dividing is multiplying, by something you already know how to build. So take six plus three i, divided by two minus i.

12:08First the inverse of two minus i. The bottom is four plus one, which is five. So the inverse is two fifths, plus i one fifth. Now multiply. Six plus three i, times two plus i, all over five. The top: twelve, plus six i, plus six i, plus three i squared. The three i squared is minus three, so the first slot is twelve minus three, which is nine. The second is six plus six, which is twelve.

12:42Nine plus twelve i, over five. Nine fifths, plus twelve fifths i. One inverse, one multiplication, and the answer is back in the shape it started in. A few more, quickly. Five i, times minus three fifths i. The i times i gives minus one, which cancels the minus already there. Three, plus i nothing. Three times seven plus seven i, plus i times the same bracket. That is twenty one plus twenty one i, plus seven i, plus seven i squared — and the last one is minus seven.

13:24Fourteen, plus twenty eight i. One minus i, to the fourth power. Square it first. One minus two i plus i squared, which is nought minus two i. Square that. Minus four, plus i nothing. Write the whole fourth power out in one go instead, and it climbs to five powers of the symbol before anything is substituted — and comes back to the same minus four. The inverse of four minus three i: sixteen plus nine is twenty five, so four twenty fifths plus three twenty fifths i.

14:02The inverse of root five plus three i: five plus nine is fourteen, so root five over fourteen, minus three fourteenths i. Notice that first slot. It is not a fraction of whole numbers at all, and nothing in the argument ever needed it to be. One last thing, and it is the reason any of this was worth doing. Suppose the symbol had squared to something else. Try nine candidates, from minus four up to four.

14:36In each, ask: is there a number that is not nought which you cannot divide by? At four of the nine, the answer is no. Every number except nought has an inverse, and those four are exactly the ones where the square is below nothing. At the other five, there is a number that kills. If the symbol squares to nothing, the symbol times itself is nothing — two things that are not nought, multiplying to nought.

15:06If it squares to one, one plus the symbol times one minus the symbol does it. If it squares to four, two plus the symbol does. And the awkward pair, two and three, need a number carrying a root two or a root three — which is why this has to be argued over the whole line and not over fractions. So the choice was never only about solving one equation. A negative square is what makes the bottom of that fraction a sum of two squares.

15:39And a sum of two squares is what makes this a system you can divide in.