Complex Numbers - Complex Analysis

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Question

Evaluate: $\left| 9 - 7 i \right|$

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Answer

The general formula to figure out the modulus is

.

We apply this notion to get.

$\left| 9 - 7 i \right| = $\sqrt{ 81 + 49 }$$

$= $\sqrt{130}$$

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Question

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

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Answer

The general formula to figure out the modulus is

We apply this to get

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

$= $\sqrt{ 10 }$$

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Question

Evaluate:

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

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Answer

The general formula to figure out the modulus is

We apply this to get

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

$= $\sqrt{ 162 }$$

$= 9 $\sqrt{2}$$

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Question

Evaluate:

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

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Answer

The general formula to figure out the modulus is

We apply this to get

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

$= $\sqrt{ 45 }$$

$= 3 $\sqrt{5}$$

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Question

Evaluate:

$\left| ( 9 + 2 i ) ( -9 + 4 i ) \right|$

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Answer

The general formula to figure out the modulus is

We apply this to get

$\left| ( 9 + 2 i ) ( -9 + 4 i ) \right| = $\sqrt{ 81 + 4 }$\cdot $\sqrt{ 81 + 16 }$$

$= $\sqrt{85}$ \cdot $\sqrt{97}$$

$= $\sqrt{8245}$$

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Question

Evaluate:

$\left| ( 7 + 6 i ) ( -8 + 6 i ) \right|$

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Answer

The general formula to figure out the modulus is

We apply this to get

$\left| ( 7 + 6 i ) ( -8 + 6 i ) \right| = $\sqrt{ 49 + 36 }$\cdot $\sqrt{ 64 + 36 }$$

$= $\sqrt{85}$ \cdot 10$

$= 10 $\sqrt{85}$$

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Question

Evaluate:

$\left| ( 9 + 7 i ) ( -5 + 6 i ) \right|$

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Answer

The general formula to figure out the modulus is

We apply this to get

$\left| ( 9 + 7 i ) ( -5 + 6 i ) \right| = $\sqrt{ 81 + 49 }$\cdot $\sqrt{ 25 + 36 }$$

$= $\sqrt{130}$ \cdot $\sqrt{61}$$

$= $\sqrt{7930}$$

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Question

Evaluate:

$\left| ( 5 + 6 i ) ( -8 + 5 i ) \right|$

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Answer

The general formula to figure out the modulus is

We apply this to get

$\left| ( 5 + 6 i ) ( -8 + 5 i ) \right| = $\sqrt{ 25 + 36 }$\cdot $\sqrt{ 64 + 25 }$$

$= $\sqrt{61}$ \cdot $\sqrt{89}$$

$= $\sqrt{5429}$$

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Question

Evaluate:

$\left| ( 10 + 3 i ) ( -5 + 4 i ) \right|$

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Answer

The general formula to figure out the modulus is

We apply this to get

$\left| ( 10 + 3 i ) ( -5 + 4 i ) \right| = $\sqrt{ 100 + 9 }$\cdot $\sqrt{ 25 + 16 }$$

$= $\sqrt{109}$ \cdot $\sqrt{41}$$

$= $\sqrt{4469}$$

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Question

Evaluate:

$\left| ( 9 + i ) ( -9 + 6 i ) \right|$

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Answer

The general formula to figure out the modulus is

We apply this to get

$\left| ( 9 + i ) ( -9 + 6 i ) \right| = $\sqrt{ 81 + 1 }$\cdot $\sqrt{ 81 + 36 }$$

$= $\sqrt{82}$ \cdot 3 $\sqrt{13}$$

$= 3 $\sqrt{1066}$$

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Question

Evaluate:

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

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Answer

The general formula to figure out the modulus is

We apply this to get

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

$= $\sqrt{89}$ \cdot 5 $\sqrt{2}$$

$=5 $\sqrt{178}$$

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Question

Evaluate:

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

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Answer

The general formula to figure out the modulus is

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}$$

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Question

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

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Answer

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

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Question

Compute $(-16)^$\frac{1}{4}$$

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Answer

Converting from Rectangular to Polar Coordinates

$(-16)^$\frac{1}{4}$ = $(16e^{i\pi + 2\pi k}$)^$\frac{1}{4}$ = $2e^{i$\frac{\pi}${4}$ + $\frac{\pi}{2}$k}$

Evaluating for

we get that

$\pm$\sqrt{2}$(1+i) $\text{ and }$ \pm$\sqrt{2}$(1-i)$

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Question

What is the magnitude of the following complex number?

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Answer

The magnitude of a complex number is defined as

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

So the modulus of is

.

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Question

What is the magnitude of the following complex number?

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Answer

The magnitude of a complex number is defined as

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

So the modulus of is

.

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Question

What is the magnitude of the following complex number?

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Answer

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

.

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Question

What is the argument of the following complex number?

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Answer

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}$$ .

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Question

The 5th roots of unity are the five unique solutions to which equation?

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Answer

A 5th root of unity is a complex number such that . Manipulating this equation yields .

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Question

What is the argument of the following complex number in radians, rounded to the nearest hundredth?

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Answer

Note that the complex number lies in the fourth 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{-3}{2}$$

$\theta=-0.98,2.16$ .

The angle lies in the second quadrant of the complex plane, but the angle lies in the fourth quadrant, as does . So $\theta=-0.98$ .

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