Question 3. Sketch the Argand diagram of each of the complex...

Question 3. Sketch the Argand diagram of each of the complex numbers $z = x + iy$ in parts (a) - (c) below, and from this determine their modulus, $r = |z|$, and argument $ heta$ as an angle between $-pi$ and $pi$ radians. Then write each number in the polar form $z = re^{i heta}$. (a) $z = 1 + i$ (b) $z = -1 + isqrt{3}$ (c) $z = -3$ (d) Find the complex solutions of the equation $z^2 + z + 1 = 0$ and write them in both Cartesian ($x + iy$) and polar ($re^{i heta}$) form.

Transcript

00:03 Now here in this quote, we look at a few complex numbers, right? so basically we asked to sketch the arg and diagram, right, for each of the complex numbers.

00:14 So let's look at this, look at them one by one, right? so the first one, you can look at the, let me just draw this.

00:21 So this is basically a complex plan, right? and then we're going to look at this is orangey, and suppose this is one and this is two and this is three, right? and i'll come similar here, one, two, three, right? so let me first look at the a, right, a is one.

00:39 So the real part is one.

00:42 So the image is also one, so the number will be here, right? so obviously you see that this is the radius and this will be the angle of theta, right? so i can see, of course, is 45 degrees and the distance is square of two, right? so this one is going to be given by a square root of two times one, in 45 degrees that's actually given by pi over or four.

01:05 So how about the next one? well, the next one, let's look at the number.

01:09 You have a real part, minus one.

01:12 Real part is here, right? minus one.

01:15 And the margin part actually positive.

01:18 So it's slightly larger close to two.

01:21 So it would be here, right? so that would be for the second number.

01:25 And it's very clear that the radius actually is one plus, it's basically two, right? so the radius is two.

01:31 And so you'll find this one to begin by two times, explanation amount of i times the angle.

01:38 And you can see that this angle is 1 to 20 degrees...

Question

Please give Ace some feedback