Solving Equations Involving z, bar z and |z|

Transcript

0:00Introduction

Hello everyone! So, one small puzzle before we start. Take the equation z square equals z bar. z bar, the z with a bar on top, is the conjugate of z. There is a z square in it, so it looks like a quadratic. Two answers, right? Hmm. What if I say it has four answers? Four? How? Today we learn to solve equations that mix z, z bar and mod z, the z between two vertical bars. And these equations come up often in JEE papers. By the end, you will find all four answers yourself, okay?

0:38Three standard methods

So, an equation mixes z, z bar and mod z, all together. There are three standard methods, okay? But first, you tell me: what is the safest first move? Yes, break z into its parts! Put z equals x plus iota y, with x and y real. Then z bar is x minus iota y, and mod z is root of x square plus y square. Now everything is in x and y. With me? Then compare the real parts on both sides, and the imaginary parts on both sides: two real equations. This method always works. But be warned, it can be long, with squares and roots everywhere. Second method: the polar form, z equals r e to the power iota theta, where r is mod z, and theta, the Greek letter, is the argument. It is best when powers of z and mod z appear. Simple reason: in this form, mod z is just r. And the third method, the smart one: take modulus, or take conjugate, of both sides. Very often this gives mod z at once. Keep this one in mind; it will crack our puzzle.

1:51Topper's trick: modulus first

So, here is the topper's trick. Look for this form: z square equals a real number times z bar. The word real in brackets stands for that real number, and the dot means times. Or the form mod z equals something; the three dots stand for that something, okay? Step one: take modulus of both sides first, and that gives you mod z; the double arrow means gives. Don't jump to x plus iota y. Modulus first, that's it! Step two: put that value of mod z back into the equation. Now the equation is much easier to finish. Clear? Now, let's see when to use what.

2:29Which method? Read the equation first

Alright, now a small game. Before solving, a topper first reads the equation, and then picks the method. Look at these three. One: mod z minus z equals one plus two iota. Two: z to the power four, minus mod z to the power four, equals four iota z square. Three: z square equals z bar. Which method would you pick for each? First one: only z and mod z, no powers. So method one, z equals x plus iota y, is short and safe here. Second one: z to the power four, and mod z to the power four. Powers of both! So, polar form, yes! This one is a JEE Advanced question, and it is solved on this page for you. And the third, our puzzle: z square equals z bar. It is z square equals a real number, here just one, times z bar. Exactly the topper's form, so modulus first. Now let's solve two of these, step by step.

3:34Example 1: solve |z| - z = 1 + 2i

Right, on to Example one, the first equation from our game. Solve: mod z minus z equals one plus two iota. We picked method one, remember? Pause the video, and try it. Done? Okay, let's check. Put z equals x plus iota y. Mod z becomes root of x square plus y square. So: root of x square plus y square, minus x, minus iota y, equals one plus two iota. Compare the imaginary parts first. On the left it is minus y, on the right it is two. So minus y equals two, and y equals minus two. Easy, right? Now the real parts. With y equal to minus two, y square is four. So root of x square plus four, minus x, equals one. Move the x across: root of x square plus four equals one plus x. Square both sides. x square plus four equals one plus two x plus x square. The x squares cancel, so two x equals three, and x equals three by two. So x is three by two, and we are done, right? Hmm, wait. We squared both sides. Is x equals three by two surely right? Not until we check! The left side is a square root, so it is never negative; so one plus x must not be negative either. Here one plus x is five by two, greater than zero; the tick mark means the check passes. So z equals three by two minus two iota. Never skip this check after squaring, okay?

5:13Example 2: all z with z^2 = z bar, four solutions

Now, our puzzle from the start. Find all z with z square equals z bar. This one is harder, so go slowly. Pause and try it. Hint: use the topper's trick, modulus first. Done? Okay, let's check. Take modulus of both sides. Mod of z square is mod z, whole square. And mod z bar equals mod z: the mirror image of z is at the same distance from the origin, isn't it? So, mod z square equals mod z. Now the easy step, right? Divide both sides by mod z, and we get mod z equals one. Correct? No, no, careful! What if mod z is zero? Then you just divided by zero. So don't divide. Take mod z common instead: mod z, times, mod z minus one, equals zero. So mod z is zero, or mod z is one. Two cases! Divide, and you lose a solution. This is the JEE trap here, got it? Case one, mod z is zero: then z itself is zero. That is one solution. Case two, mod z is one. We know z z bar equals mod z square, which is one here. So z bar is one by z. Put this back into the equation: z square equals one by z. Multiply by z: z cube equals one. Its roots are one, omega and omega square. Omega, the Greek letter, is a cube root of unity. Here, omega is minus half plus root three by two iota. And omega square is minus half minus root three by two iota. So, all together: zero, one, and minus half plus or minus root three by two iota. Four solutions! There you go, puzzle solved. Because of that z bar, this was never an ordinary quadratic in z. And modulus first finished it in a few lines. Simple as that!

7:11Quick revision: which method for which equation

So, quick revision. Equations mixing z, z bar and mod z: put z equals x plus iota y, compare real and imaginary parts, and check after squaring. Powers of z and mod z together: polar form. And for z square equals a real number times z bar, or mod z equals something: modulus first, and don't divide by mod z, because mod z may be zero. Now go and try the practice problems on this page. You can do it. All the best!