Reading the plane: real axis, imaginary axis, and a mirror image for the conjugate

0:00A complex number is two real numbers written in a fixed order. A point of the coordinate plane is two real numbers written in a fixed order. That is the whole observation, and everything here comes out of it. So attach each number to the point carrying the same two reals, and the plane acquires a complex number at every point. Read this way, the plane has two names: the complex plane, and the Argand plane.

0:31It is not a new plane. It is the same one, with a number attached to each point. The picture that results is not an illustration of the algebra. It is the algebra, drawn. Which is a strong claim, so we are going to test it rather than admire it. Two definitions that arrived as formulas will turn into things you can see. The size of a number becomes a distance. The conjugate becomes a reflection.

1:02But neither of those is free, and finding out what they cost is the point. First, the pairing itself has to be honest. Take forty-nine numbers, every combination of two parts from minus three to three, and the forty-nine points with the same coordinates. Two things must hold. No two different numbers may land on the same point. And every point must be able to say which number is there, and be right.

1:33Both hold, with nothing left over. Zero collisions, zero points lost on the round trip. That matters because it is easy to write down a pairing that fails. Throw the second part away and send every number to the point with its first coordinate. Now two hundred and ninety-four pairs of different numbers collide, and not one of the forty-nine points can say which number is there. That is not a correspondence. It is a projection, and information has gone.

2:08That difference is what lets the picture carry meaning rather than merely suggest it. Six numbers, placed. Two plus four i goes to the point two across, four up. Call it A. Minus two plus three i goes to minus two across, three up. That is B. Plain i goes to nought across, one up: C. And two, with no second part at all, goes to two across and nought up: D.

2:38Minus five minus two i lands down and to the left, at E. One minus two i lands down and to the right, at F. Notice the last two of those six. C has an empty first part and sits on the vertical axis. D has an empty second part and sits on the horizontal one. Those two are not decoration. They are the cases the axes are named for, and we come back to both.

3:10Now the first of the two claims. The size of a number is the distance of its point from the origin. Before proving that, notice what it needs. Along an axis, distance is easy: it is just a difference of two reals, and no formula is involved. Off the axes it is not, because neither coordinate matches, and that needs a rule. So the rule gets handed in rather than assumed. Three of them.

3:41The first is the one this plane actually uses: square the two gaps and add. The second measures along the grid lines, as though you had to walk street corners. The third takes only the longer of the two gaps and ignores the shorter. On the point three across and four up, those three give twenty-five, forty-nine and sixteen. They are genuinely three different rules. But on the point three across and nought up, all three give nine. Along an axis they agree, because one gap is zero and there is nothing to disagree about.

4:21With the first rule in hand, here is the claim. The squared distance of a point from the origin is the sum of the squares of its two coordinates. The squared size of a number is the sum of the squares of its two parts. Across all forty-nine numbers, those two quantities never once differ. But agreeing at forty-nine places is a demonstration, not a reason, and the reason is better.

4:50Under the pairing, the coordinates of the point are the parts of the number. Not equal to them, not matching them: the same two reals. So the two expressions are not two formulas that happen to give the same answer. They are one expression, with different names written on the letters. That is what the correspondence buys you. Once the two pairs of reals are the same pair, any formula built out of them says the same thing on both sides.

5:24So how much of that came from the picture, and how much from the rule? Take the rule away. Keep the same pairing, the same axes, the same numbers, and measure distance by street corners instead. Almost everything survives. The pairing still runs both ways. The two axes hold exactly the numbers they held before. The conjugate is still a mirror image. One thing breaks, and only one. The distance from the origin no longer matches the size.

5:59And look at where it breaks. Of the forty-nine numbers, thirteen have an empty part, and at every one of those thirteen the two still agree. At the other thirty-six, both parts filled, the two disagree every time. The third rule breaks in exactly the same place, at the same thirty-six. So size being distance is not a fact about pictures. It is a fact about this plane and this rule, doing real work at every point off both axes.

6:33Now the axes, and they are easier. A point sits on the horizontal axis when its up-coordinate is nought. Under the pairing, that coordinate is the second part of the number. So the numbers on the horizontal axis are exactly those with no second part. Exactly, in both directions: across all forty-nine, there is no number that sits there without an empty second part, and none with an empty second part that sits anywhere else.

7:07Seven of the forty-nine qualify, and D is the one among our six. D is the number two, with nothing attached to i. It is an ordinary real number, and it has landed on the line where ordinary real numbers have always lived. Which is why that line gets called the real axis. The other axis is the mirror of that argument. A point sits on the vertical axis when its across-coordinate is nought, and that coordinate is the first part of the number.

7:43So the vertical axis holds exactly the numbers with no first part. Again seven of the forty-nine, and again exactly, in both directions. C is ours: plain i, at nought across and one up. That line is called the imaginary axis, and the name misleads a little, so be careful. The coordinate measured up that axis is an ordinary real number. One, in C's case. What makes the axis imaginary is the part it carries, not the quantity marked along it.

8:21One number is on both axes at once, and there is exactly one: nought, sitting at the origin, with both parts empty. The second claim. The conjugate is a reflection in the horizontal axis. Saying one point is the mirror image of another is a statement about a picture, so settle it in the picture's own terms, not by pointing at a minus sign. Three conditions. The line joining the two points must be vertical. Both must stand the same distance from the axis. And they must be on opposite sides of it.

9:02Test all three, on the point of a number and the point of its conjugate, across all forty-nine. Every one passes. Not one failure. Take A, at two across and four up. Its conjugate is two minus four i, which sits two across and four down. Same first coordinate, so the join is vertical. Sixteen for the squared height in each case, so the distances match. And they are on opposite sides.

9:34Ask the same three questions about the vertical axis instead, and forty-eight of the forty-nine fail. The reflection is in the horizontal axis and nowhere else. A reflection always leaves something where it is, and here that something is worth naming. Of the forty-nine, the mirror moves forty-two and leaves seven exactly where they were. Those seven are precisely the numbers with an empty second part. There is no other number the reflection fixes, and no number with an empty second part that it moves.

10:13So the points the mirror leaves alone are exactly the points of the horizontal axis. Which is the same seven the previous scene counted. D is one of them. The conjugate of the number two is the number two, and its point does not move. That is worth holding onto, because it says the same thing twice over: a number is its own conjugate exactly when it is real, and exactly when it sits on the real axis.

10:45The algebra and the picture are not agreeing here. They are the same statement. One question is still open. Why that pairing? Write down six, and put all six through the same five tests. The one we used. One that reads the second part upside down. One that reads the parts the other way round. One that adds them together for the first coordinate. One that is the right map with the wrong return trip. And the one that throws the second part away.

11:21Test them on collisions, on the round trip, on distance against size, on the axes, and on the mirror. No single test turns down more than two of the six. Distance catches the shear and the projection, and lets the swap straight through, because a sum of two squares does not care which order the squares come in. The axis test catches the swap and the projection, and lets the shear through.

11:49The mirror catches the swap and the shear, and passes the projection, because when everything collapses onto the axis every point is trivially its own reflection. And the one with the wrong return trip is caught by nothing but the round trip, because going forwards it is the correct map. Two of the six pass every test there is. So the picture does not pin the pairing down completely, and knowing how far it does is the difference between a proof and a habit.

12:24Finally, put the six back on the axes and read off something that was there all along. Each of those six points has a distance from the origin, and we can get it without measuring. Square the parts and add. Twenty for A, thirteen for B, one for C, four for D, twenty-nine for E, five for F. So C is exactly one away, and D exactly two. A is the square root of twenty, which is two root five, because two root five squared is four times five.

12:59Change the rule to street corners and those six become thirty-six, twenty-five, one, four, forty-nine and nine. Only two of the six are unchanged, and they are C and D, the two that sit on an axis. A number is two reals in order, a point is two reals in order, and the pairing lets a formula about one be read as a fact about the other. The size became a distance, and it cost one rule about the plane. The conjugate became a mirror, and it cost nothing at all.