Size and reflection: two quantities that turn algebra into geometry

0:00Every complex number carries two extra quantities, and they usually arrive as definitions to memorise. One is its modulus. The other is its conjugate. One measures how big the number is. The other reflects it, turning over the sign of just one of the two parts. Size and reflection, filed under separate headings. They are one mechanism, with one job. Multiply any complex number by its own conjugate, and every trace of the symbol disappears. What is left is an ordinary real number.

0:40Not just any real number. It is precisely the modulus, squared. That identity is why a complex denominator can always be made real, and why the formula for a multiplicative inverse looks as it does. So let us earn it, rather than accept it. Start with the size. For a number with parts a and b: square both, add, take the square root. That non-negative root is the modulus. For three plus i, nine plus one, so ten under the root. For two minus five i, twenty-nine.

1:19The minus sign in the second changes nothing, because the part is squared first. Now a decision that runs through this video. We are never going to take that root. The root of ten is not something you can finish writing down. So the modulus stays characterised, not computed: the non-negative thing whose square is that sum. Claims about size are then checked on that sum, which is safe only if two different non-negative quantities never share a square.

1:53None here do. Drop the word non-negative and four pairs appear at once. So the restriction carries the argument: comparing sums and comparing sizes say one thing. Now the reflection. The conjugate keeps the first part and turns over the sign of the second. Three plus i becomes three minus i. The commonest slip here is confusing that with the additive inverse, which turns over both parts. They are different maps. Across forty-nine sample numbers they disagree on forty-two.

2:33They agree on exactly the seven where the first part is empty. It has two obvious properties. Do it twice and you are back where you started. And it never changes the size, since squaring minus b matches squaring b. Neither is special. Write down six maps: leave both parts alone, flip the second, flip the first, flip both, swap them, double the second. Five of the six undo themselves, and the same five leave the size alone. Only the doubling fails.

3:09So neither property picks out the conjugate. Something else has to. Here is the test that does. Multiply each number by its own flipped version. Flip only the second part and the symbol vanishes every time. Zero failures out of forty-nine. Here is the surprise. Flip only the first part, and it vanishes every time as well. Two of the five give a real answer for every number. So realness is not what makes the conjugate right.

3:44Look at what those two answers are. Three plus i, times minus three plus i, is minus ten. Flipping the first part gives the size with its sign turned over, negative at forty-eight of them. Only one map lands on a squared plus b squared itself, and it lands there every time. That is the one that can be a size, because a size may not be negative. Non-negativity is the test, not realness.

4:16Why does it work? Open the brackets. A plus b i, times a minus b i, gives four products, not two. Top left, a times a, which is a squared, staying in the first slot. Top right, a times minus b, and bottom left, b times a, both landing in the second slot. Those two are one product written twice with opposite signs. Counted rather than assumed: both cells in the second slot have a partner cancelling them, and neither in the first has any.

4:52So the second slot adds up to nothing, and the symbol is gone. That leaves the corner. B times minus b is minus b squared, times whatever the symbol squares to. Since that is minus one, the minus flips to a plus. For three plus i, the second slot adds to nothing and the first to ten. And the corner is the only cell the relation touches. Make the symbol square to plus one, and the corner turns from plus one into minus one. Every other cell sits unchanged.

5:31That is worth chasing, because it separates two things taught together. Make the relation a dial. Nine settings, the symbol squaring to everything from minus four to plus four. At all nine, the two middle cells still cancel. Every time. So the answer coming out real never depended on the relation, only on a times b matching b times a. So what does the relation buy? Two things. First, whether it can be negative. From minus four up to zero it never is. From plus one upward it is, at nineteen of the forty-nine and rising.

6:13Second, and this is the one that matters, whether it is the sum of the two squares. At eight of the nine it misses at forty-two numbers. At one it misses nowhere. That setting is the symbol squaring to minus one. One of nine, so the minus is not decoration. Now cash it in. A number times its conjugate is the sum of the two squares, an ordinary real. Divide through by it, and you have a number times something equalling one.

6:48So the multiplicative inverse is the conjugate over the squared modulus. Nothing more. That is the same object as the formula you meet before the conjugate appears: the first part over a squared plus b squared, and minus the second part over the same. Written out independently, the two agree at all forty-eight non-zero numbers here, and each really does multiply back to one. That is a fact about this multiplication rule: move the symbol so it squares to plus one, and the two part company at forty-two numbers.

7:28Take two minus three i. Its conjugate is two plus three i, its squared modulus thirteen. So its inverse is two thirteenths plus three thirteenths i. The a squared plus b squared underneath was never arbitrary. It is a number multiplied by its own reflection. The same identity does the thing you will use most: it clears a complex denominator. Multiply top and bottom by the conjugate of the bottom. That is legitimate for the dullest reason: the same quantity sits above and below, so you multiplied by one.

8:11But the shape changes, because the bottom is now a number times its own conjugate. And that is real. Take five plus root two i, over one minus root two i. The denominator becomes one plus two, which is three. The numerator becomes three plus six root two i. Divide through by three, and the answer is one plus two root two i. Two routes exist, not obviously the same: clear the denominator, or multiply by the inverse.

8:45They never once disagree. Under a rule that forgets which number was written first, they disagree a hundred and forty-four times. Compounded, it is the same move. Twelve plus five i, over four plus three i, clears to sixty-three twenty-fifths minus sixteen twenty-fifths i. With the identity in hand, five results about products and quotients get listed almost as an afterthought. The size of a product is the product of the sizes; the size of a quotient, the quotient of the sizes, provided the divisor is not nothing.

9:27The reflection of a product is the product of the reflections, and the reflections of a sum and of a quotient split the same way. They almost always arrive as a list, with a remark that they can be derived, and no derivation. Two of the five compare ordinary values and three compare two-slot numbers, which is what size and reflection mean here. Do they hold? Across two hundred and fifty-six pairs, not one fails.

9:59But a failure count alone is a poor check. The two quotient results speak for only two hundred and forty of those pairs; at the other sixteen the divisor is nothing. Test one properly. Three plus i times two minus five i is eleven minus thirteen i. Its squared modulus is two hundred and ninety. And ten times twenty-nine is also two hundred and ninety. One pair is a demonstration, not a proof.

10:34So break the multiplication rule, and see which of the five fall over. Take a rule that forgets which number was written first. The size of a product now fails a hundred and eighty-four times. The size of a quotient fails seventy-two. The two reflection results about products and quotients report zero failures, and that is not agreement. Their coverage has collapsed to ninety-six pairs, because the denominator no longer comes out real and the manoeuvre refuses to run.

11:10Those zeros are silence, not success. Two other broken rules do the same kind of damage. So put all six rules together. Which of the five covers every pair under every rule and never fails? Exactly one. The reflection of a sum. Not luck. It is the only one whose statement never mentions multiplication. Break that rule as hard as you like: a result built purely out of addition cannot notice.

11:42Which is the general lesson. A result survives a broken rule exactly when its argument never used that rule. Two of those five carried a condition. The divisor must not be nothing. That looks like small print. It is not. Hand each of the five a divisor that is nothing, and watch which refuse to answer. Exactly two refuse, and they are exactly the two about quotients. The condition belongs to the statements themselves.

12:16The condition says the divisor must not be nothing. What the method needs is that its squared modulus is not nothing, since that is what you divide by. Two different requirements, and everything rests on them being one. They are, for a reason about ordinary reals. A sum of two squares is zero only when both parts are, because neither square can be negative, so neither can cancel the other.

12:46So the size vanishes for exactly one number, and that number is nothing itself. That is what makes the inverse formula safe. Without it, an ordinary-looking number could have zero underneath. That guarantee is worth taking seriously, so watch it fail. Keep everything else, and let the symbol square to plus one. Take one plus i, whose conjugate is one minus i. Neither is nothing. Multiply them under the new relation and the answer is nothing.

13:22Two things that are not nothing, multiplying to nothing. And now one plus i can have no inverse. Settled by an argument, not by hunting for an inverse and failing. Suppose something did multiply it to one. Push that inverse through, and the one minus i on the other side is carried to nothing. But it is not nothing. The argument needs permission to regroup a triple product, which this system gives.

13:53Run the same argument where the symbol squares to minus one and it refuses, because there the product is not nothing. Ten of the forty-nine behave that way once the symbol squares to plus one. Where it squares to minus one, none do. That is what the minus really buys. Not elegance, but the guarantee that every number except nothing can be divided by. Finally, what it is worth, because recognising it turns long problems short.

14:28Ask for the imaginary part of the reciprocal of two minus i times its own conjugate. Without the identity that is an expansion, a clearing and a division. With it, the product is just the squared modulus, five. The reciprocal is one fifth, and its imaginary part is nothing. One line. And the best of them. Multiply out four brackets: one plus two i, three minus i, two plus two i, one minus four i.

15:02That product is eighty plus twenty i, whose squared modulus is six thousand eight hundred. Now the other way. One sum of squares from each bracket, multiplied together. Also six thousand eight hundred, and the brackets were never opened. Not that the shortcut is free: under a slot-by-slot rule those brackets give thirty-six. Size and reflection, one mechanism. Multiply a number by its own reflection, the symbol cancels itself, and the size is left behind, squared.