Complex Numbers

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Complex Analysis › Complex Numbers

Questions 1 - 10

1

What is the magnitude of the following complex number?

$\sqrt{29}$

$\sqrt{14}$

None of these

Explanation

The magnitude of a complex number is defined as

$\vert a+bi\vert=\sqrt{a^2+b^2}.$

So the modulus of is

$\vert5+2i\vert=\sqrt{5^2+2^2}=\sqrt{25+4}=\sqrt{29}$ .

2

Evaluate:

$\left| 6 - 3 i \right|$

$3 \sqrt{5}$

$- 3 \sqrt{5}$

$\frac{3 \sqrt{5}}{2}$

Explanation

The general formula to figure out the modulus is

$\left|a+bi\right|=\sqrt{a^2+b^2}$

We apply this to get

$\left| 6 - 3 i \right| = \sqrt{ 36 + 9 }$

$= \sqrt{ 45 }$

$= 3 \sqrt{5}$

3

Evaluate:

$\left| ( 8 + 5 i ) ( -10 + 3 i ) \right|$

$\sqrt{9701}$

$- \sqrt{9701}$

$\frac{\sqrt{9701}}{2}$

Explanation

The general formula to figure out the modulus is

$\left|a+bi\right|=\sqrt{a^2+b^2}$

We apply this to get

$\left| ( 8 + 5 i ) ( -10 + 3 i ) \right| = \sqrt{ 64 + 25 }\cdot \sqrt{ 100 + 9 }$

$= \sqrt{89} \cdot \sqrt{109}$

$= \sqrt{9701}$

4

What is the magnitude of the following complex number?

$\sqrt{10}$

$\sqrt{8}$

None of these

Explanation

The magnitude of a complex number is defined as

$\vert a+bi\vert=\sqrt{a^2+b^2}.$

So the modulus of is

$\vert3-i\vert=\sqrt{3^2+(-1)^2}=\sqrt{9+1}=\sqrt{10}$ .

5

Evaluate:

$\left| 9 - 9 i \right|$

$9 \sqrt{2}$

$- 9 \sqrt{2}$

$\frac{9 \sqrt{2}}{2}$

Explanation

The general formula to figure out the modulus is

$\left|a+bi\right|=\sqrt{a^2+b^2}$

We apply this to get

$\left| 9 - 9 i \right| = \sqrt{ 81 + 81 }$

$= \sqrt{ 162 }$

$= 9 \sqrt{2}$

6

What is the magnitude of the following complex number?

$2\sqrt2$

None of these

Explanation

The magnitude of a complex number is defined as

$\vert a+bi\vert=\sqrt{a^2+b^2}.$

Because the complex number has no imaginary part, we can write it in the form . Then the modulus of is

$\vert8+0i\vert=\sqrt{8^2+0^2}=\sqrt{64+0}=\sqrt{64}=8$ .

7

Evaluate $(\sqrt{3} - i)^6$

-64

64

64i

-64i

None of the other answers

Explanation

Converting from rectangular to polar coordinates gives us

$r = \sqrt{x^2+y^2} = \sqrt{3+1} = 2$

$\theta = tan^{-1}(y/x) = tan^{-1}\left(-\frac{1}{\sqrt{3}}\right) = -\frac{\pi}{6}$

So

$(\sqrt{3}-i)^6 = (2e^{-\frac{\pi}{6}i})^6 = 64e^{-\pi i} = -64$

8

Evaluate: $\left| 1 - 3 i \right|$

$\sqrt{10}$

$- \sqrt{10}$

$\frac{\sqrt{10}}{2}$

Explanation

The general formula to figure out the modulus is

$\left|a+bi\right|=\sqrt{a^2+b^2}$

We apply this to get

$\left| 1 - 3 i \right| = \sqrt{ 1 + 9 }$

$= \sqrt{ 10 }$

9

What is the argument of the following complex number?

$\frac{\pi}{4}$

None of these

$\frac{\pi}{2}$

$\frac{-3\pi}{4}$

Explanation

Note that the complex number lies in the first quadrant of the complex plane.

For a complex number , the argument of is defined as the real number $\theta$ such that

$\tan(\theta)=\frac{b}{a}$ ,

where $-\pi<\theta\leq\pi$ is in radians.

Then the argument of is

$\tan(\theta)=\frac{1}{1}=1$

$\theta=\frac{-3\pi}{4},\frac{\pi}{4}$ .

The angle $\frac{-3\pi}{4}$ lies in the third quadrant of the complex plane, but the angle $\frac{\pi}{4}$ lies in the first quadrant, as does . So $\theta=\frac{\pi}{4}$ .

10

What is the value of $\vert e^{i\theta}\vert$ , where $-\pi<\theta\leq\pi$ is in radians?

Not enough information is given.

Explanation

The magnitude of a complex number $z=re^{i\theta}$ is defined as
$\vert z\vert=r$ .

If $z=e^{i\theta}$ , then , so $\vert e^{i\theta}\vert$ =1.

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