Complex Numbers: Definition and Equality

Transcript

0:00Introduction

Hello everyone! Let me start with a small puzzle. You know the rule: one equation can find one unknown. Two unknowns need two equations, right? But today I will give you one single equation, with two unknowns, x and y, and we will find both of them. How is that possible? The secret is in what it means for two complex numbers to be equal. This idea comes up often in JEE questions on complex numbers. So, let's begin.

0:32What is a complex number?

So, what is a complex number? Here, i is iota, the number whose square is minus one. Combine it with real numbers: z equals x plus iota y, where x and y are real numbers. See the symbol like a rounded E? It means belongs to. And the R with a double line, called double struck R, is the set of all real numbers. Okay? Now, x is called the real part of z. On screen it is written R E of z, in brackets, short for real part. So the real part of z equals x. And y is called the imaginary part, written I M of z. Look carefully: the imaginary part is y, just y, not iota y. With me? All such numbers together make the set of complex numbers, the double struck C you see on screen. Clear?

1:24Trap: the imaginary part is a real number

Let's test this right away. Take z equals three minus four iota. Its real part is three, easy. Now tell me, what is its imaginary part? Many students say minus four iota. No, no, wait! It is just minus four, the real number sitting next to iota. The imaginary part is always a real number. Got it?

1:49Purely real and purely imaginary

Now, two special cases. What happens if the imaginary part, y, is zero? Correct! Then z is just x, an ordinary real number. We call it purely real. So every real number is also a complex number. That is why double struck R is a subset of double struck C; the sideways U symbol means subset. Okay? And if the real part x is zero, but y is not zero, z equals iota y: purely imaginary. The equals sign with a slash through it means not equal to. Now a tricky one. What about zero itself? Purely real, or purely imaginary? Zero is zero plus zero iota. Its imaginary part is zero, so it is purely real. And its real part is also zero, so many books call it purely imaginary too. Both! So in JEE, purely imaginary usually means the real part of z is zero. But always check whether the question allows z equals zero. We will meet this trap again very soon. Keep it in mind.

2:56Equality of complex numbers

Now the main idea of today. Take two complex numbers. When will you call them equal? Yes, exactly! When their real parts are equal, and their imaginary parts are equal. On screen: x one plus iota y one equals x two plus iota y two. The small one and two written below are subscripts, labels for the first and second number. The double headed arrow means if and only if. So x one equals x two, and y one equals y two. Clear? And look what this gives us. One complex equation is the same as two real equations. That is the answer to our puzzle! Two real equations hide inside one, so two unknowns can be found. This is the most used idea in the whole chapter. Got it?

3:44Trap: complex numbers have no order

Now one more trap. Equal, we understand. But what about greater than? Is two plus three iota greater than one plus iota? Two is bigger than one, three is bigger than one, so yes, right? No, no, careful! This statement has no meaning at all. Complex numbers have no order. Let me show you why, with iota itself. Is iota greater than zero, or less than zero? Suppose iota is greater than zero. Two positive numbers multiply to a positive number, so iota times iota is greater than zero; the dot here means times. But iota times iota is minus one. Minus one greater than zero? Absurd! Now suppose iota is less than zero. Then minus iota is greater than zero. So minus iota times minus iota is greater than zero. But that product is again minus one. Absurd again! Neither works. Okay? So, inequalities in complex number questions are always between real quantities. Like mod z, with the two vertical bars, the real part, the imaginary part, and arg z. You will meet mod z and arg z very soon. With me?

4:56Example 1: find real x and y by equating parts

Now, Example one. Remember our puzzle? Find real x and y, if three x minus seven, in brackets, plus two iota y, equals minus five y, plus, five plus x, in brackets, times iota. One equation, two unknowns. Pause the video and try it. Done? Okay, let's check. Compare the real parts first. On the left, three x minus seven. On the right, minus five y. So three x minus seven equals minus five y. Now the imaginary parts. On the left, it is two y, not two iota y; remember the trap! On the right, five plus x. So two y equals five plus x, which means x equals two y minus five. The double arrow pointing right means implies, okay? Now put this x into the first equation. Three times, two y minus five, minus seven, equals minus five y. That is six y minus twenty two equals minus five y, so eleven y equals twenty two. With me? So y equals two, and x equals two times two, minus five, that is minus one. One equation, and we found both unknowns. Puzzle solved! Quick check, always. Three times minus one, minus seven, is minus ten, and minus five times two is also minus ten. Two times two is four, and five plus minus one is four. Perfect!

6:25Example 2: when is it purely real or purely imaginary

Now a slightly harder one. For what real x is this number, x square minus five x plus six, plus iota times, x square minus three x plus two, first, purely real, and second, purely imaginary? Pause the video and try it. Careful with part two! Done? Okay, let's check. First, factorise both parts. The real part, x square minus five x plus six, is x minus two, times x minus three. Right? And the imaginary part, x square minus three x plus two, is x minus one, times x minus two. Part one: purely real means the imaginary part is zero. So x equals one, or x equals two. Really simple. Part two: purely imaginary means the real part is zero, and the imaginary part is not zero. The real part is zero at x equals two, or three. So both are answers, right? Oh no, careful! At x equals two, the imaginary part is also zero. So x must be in the set two, three, shown in curly brackets, but x cannot be one or two. Only x equals three is left. Look, at x equals two, the whole number becomes zero. Remember the zero trap? That is exactly why we check whether zero is allowed. So, part one: x equals one or two. Part two: x equals three. Clear? So, quick revision. z equals x plus iota y; the real part x and the imaginary part y are both real numbers.

7:59Quick revision: real part, imaginary part, equality

Two complex numbers are equal when their real parts are equal and their imaginary parts are equal. No order on complex numbers; and purely imaginary means the real part is zero, but check whether zero is allowed. Now go and try the practice problems on this page. You will find them really simple. All the best!