Algebraic Operations on Complex Numbers

Transcript

0:00Introduction

Hello everyone! One small challenge first. Take one plus iota, and raise it to the power two thousand and twenty six. Would you really multiply one plus iota by itself, over two thousand times? Keep that thought. By the end of this video, you will do it in a few lines. Today: adding, subtracting, multiplying and dividing complex numbers. You will use these steps all through complex numbers in JEE, okay? So, let's begin.

0:31Add and subtract: like terms together

So, one rule runs this whole video. The i on screen is iota, the imaginary unit. Treat it like an ordinary letter in algebra. Only one extra step: whenever iota square appears, the i with a small two on top, replace it by minus one. Clear? Now take z one equals a plus iota b, and z two equals c plus iota d, with a, b, c and d real. The small one and two below z are just labels. Now you tell me: what is z one plus z two? Yes, exactly! Real with real: a plus c. Imaginary with imaginary: b plus d, with the iota outside the bracket. Like terms together. Okay? Subtraction works the same way: a minus c, plus iota times b minus d. With me?

1:23Multiplication: expand, then use i squared = -1

Now multiplication: a plus iota b, times c plus iota d. Multiply every term by every term, like in algebra. We get a c, plus iota a d, plus iota b c, plus iota square b d. So the real part is a c plus b d, right? No, no, careful! Iota square is minus one, so the last term is minus b d. Real part: a c minus b d. Imaginary part: a d plus b c. This minus sign is a very common slip. Don't fall for it, okay?

1:56Division: multiply top and bottom by c-id

Now division: a plus iota b over c plus iota d, with the fraction bar between them. The equals sign with a slash through it means not equal: z two must not be zero. We want real plus iota times real. So how do we remove iota from the bottom? Very good! Multiply the top and the bottom by c minus iota d. It is the bottom number, with the sign of its iota part flipped. Look at the bottom now. The middle terms cancel, leaving c square minus iota square d square. Iota square is minus one, so this is c square plus d square. It is a real number! So the fraction becomes a plus iota b, times c minus iota d, over c square plus d square. Expand the top and split it. Real part: a c plus b d, over c square plus d square. Imaginary part: b c minus a d, over the same. Just remember the move. Got it? And the reciprocal: one over a plus iota b equals a minus iota b, over a square plus b square. Same move, nothing new.

3:07Laws and useful products

Now some good news. All the usual laws of algebra hold in this double struck C, the set of complex numbers. Identities like a plus b whole square equals a square plus two a b plus b square stay true too. So, what is one plus iota, whole square? One, plus two iota, plus iota square. The one and the iota square cancel, so it is just two iota! Similarly, one minus iota, whole square, is minus two iota. Multiply them: one plus iota, times one minus iota, is just two. Divide them: one plus iota over one minus iota is iota, and the flipped fraction is minus iota. Keep these five results at your fingertips. Clear?

3:55Topper's trick: high powers through the square

Now a trick that toppers use. Find one plus iota, to the power ten. Would you multiply ten brackets, one by one? No way! Go through the square. One plus iota, whole square, is two iota, and the double arrow means so. So one plus iota to the power two k is two iota to the power k, that is, two to the power k, times iota to the power k. Okay? For the power ten, k is five. Two to the power five is thirty two. And as the bracket says, iota to the power four is one, so iota to the power five is just iota. Thirty two iota, in one line! We need this trick in Example two.

4:38Example 1: write products and quotients as a + ib

Alright, Example one. Write each of these in the form a plus iota b; the small roman numerals number the parts. Three plus two iota, times four minus five iota. Two minus three iota, whole square. One plus two iota, whole cube. And five plus iota, over two minus three iota. Pause the video, and try all four. Done? Okay, let's check. Part one: twelve, minus fifteen iota, plus eight iota, minus ten iota square. Minus ten iota square is plus ten. So twenty two minus seven iota. Clear? Part two: two minus three iota, whole square. Two square is four, and three iota, whole square, is minus nine. So the answer is just minus five, right? Oh no, careful! You missed the middle term: minus two, times two, times three iota, that is minus twelve iota. So four, minus twelve iota, plus nine iota square: minus five minus twelve iota. Part three: expand the cube of one plus two iota, like a plus b whole cube. One, plus three times two iota, plus three times two iota whole square, plus two iota whole cube. The first two terms give one plus six iota. Two iota, whole square, is minus four, so the third term is minus twelve. Two iota, whole cube, is minus eight iota. Total: minus eleven minus two iota. With me? Part four: multiply top and bottom by two plus three iota, the bottom with its iota sign flipped. The bottom becomes four plus nine, thirteen. The top: ten, plus fifteen iota, plus two iota, plus three iota square. Three iota square is minus three, so the top is seven plus seventeen iota. Answer: seven by thirteen, plus seventeen by thirteen iota. Perfect!

6:40Example 2: (1 + i) to the 2026 plus (1 - i) to the 2026

Now, our challenge from the start, a little bigger. Find one plus iota to the power two thousand and twenty six, plus one minus iota to the same power. Pause the video and try it. Hint: go through the square. Done? Okay, let's check. Two thousand and twenty six is two times one thousand and thirteen. So square one plus iota first, inside the big brackets, then raise it to the power one thousand and thirteen. That gives two iota, to the power one thousand and thirteen. So what is iota to the power one thousand and thirteen? One thousand and thirteen is four times two hundred and fifty three, plus one; the dot means times. Iota to the power four is one, so only the remainder matters. Iota to the power one thousand and thirteen is just iota! So the first term is two to the power one thousand and thirteen, times iota. The second term looks just the same, so it gives the same answer, right? No, no, wait! One minus iota, whole square, is minus two iota. And minus two iota, raised to an odd power, keeps its minus sign. Iota to the power one thousand and thirteen is iota again. So the second term is minus two to the power one thousand and thirteen, times iota. Now add. The two terms are equal and opposite. Zero! That giant power, finished in a few lines. That's the square trick!

8:12Quick revision: multiply, divide, powers of 1 + i

So, quick revision. Add, subtract and multiply like algebra, with iota square equal to minus one. To divide, multiply the top and the bottom by c minus iota d, okay? For high powers of one plus or minus iota, written with a plus sign over a minus sign, go through the square: it is plus or minus two iota. Now go and try the practice problems on this page. All the best!