Complex numbers

Question 1

What is i equal to?

Answer 1

sqrt(1)

Question 2

What is i² equal to

Answer 2

-1

Question 3

What is the form of a complex number?

Answer 3

a + bi where a and b are real numbers.

Question 4

What is the imaginary part of a + bi

Answer 4

b

Question 5

What is the real part of a + bi

Answer 5

a

Question 6

Real part and imaginary part

Answer 6

Question 7

Answer 7

Question 8

State the quadratic formula

Answer 8

Question 9

Answer 9

Question 10

What is a complex conjugate and how do you denote it?

Answer 10

It is when the imaginary part of a complex number is multiplied by -1. and is denoted with a line above it

Question 11

What is a complex number multiplied by its complex conjucate?

Answer 11

The sum of the real part squared and the imaginary part squared

It returns a real number

Question 12

Answer 12

Question 13

What is meant by “the modulus of complex number, z”. How do you calculate it?

Answer 13

The magnitude of it.

Question 14

What does the following equal to?

Answer 14

Question 15

When does a complex number equal its conjugate?

Answer 15

If and only if there is only a real component

Question 16

Is the following true for all cases?

Answer 16

Yes. It is all true.

Question 17

What is the diagram where complex numbers are represented on a plane?

Answer 17

Argand diagram

Question 18

What is arg(z)

Answer 18

The angle that the complex number, z makes to the real axis. It is the argument of z

Question 19

True or false? A complex number z has an arbitirary number of arg(z) values

Answer 19

True. Because you can just keep adding 2π or subtract 2π and it will be the same angle

Question 20

What is the difference between arg(z) and Arg(z)

Answer 20

Arg(z) is the prinicipal argument and is bounded by [-π, π)

Question 21

Answer 21

Question 22

What is the polar form of a complex number?

Answer 22

Question 23

Answer 23

Question 24

How would you multiply two complex numbers in polar form?

Answer 24

Multiply the modulus’s and add the angles/arguments

Question 25

How would you divide two complex numbers in polar form?

Answer 25

Divide the modulus's and in the cis, do the first angle minus the second angle.

Question 26

Answer 26

Question 27

True or false, all real numbers are complex numbers.

Answer 27

True, this is because they can all be represented with the a+b`i`

Question 28

How do you represent a complex number in exponential form?

Answer 28

Question 29

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Question 30

Answer 30

Question 31

Answer 31

This is known as the most beautiful equation

Question 32

What happens when a complex number is multiplied by `i`?

Answer 32

An angle of π/2 is added to the complex number's arg(z).

Question 33

What is De Moivre's Theorum?

Answer 33

Increasing a complex number by an index, n. The modulus will be raised to the nth power and the angle will be multiplied by n.

Question 34

# True or false? De Moivre's theorum works False . Only works when a complex number is raised by all numbers

Answer 34

False . Only works when a complex number is raised by a **natural number**

Question 35

Answer 35

Question 36

Answer 36

z=1,`i`,-1,-`i`

Question 37

Answer 37

Question 38

Answer 38

a) is a polynomial equation as it = 0 b) is neither as it raises z to a negative number c) is a polynomial expression as have an equal sign

Question 39

What does the fundamental theorum of algebra state?

Answer 39

That given a polynomial equation with complex or real coefficients, there is always a solution in the complex numbers. ## Footnote Remember that real numbers is a subset of complex numbers (real numbers inside complex numbers)

Question 40

Answer 40

(z+1) is a factor as subbing z=-1 makes P(z)=0. (z-2) is not a factor as subbing z=2 makes P(z)≠0.

Question 41

# Describe the following polynomial equation P(z)=(z-a)Q(z)

Answer 41

1. (z-a) is a factor of P(z) 2. Q(z) is the quotient of P(z) 3. Q(z) is one degree less than P(z)

Question 42

# Factorise the following: x²+y²

Answer 42

(x-y)(x+y) | This is just something that you should remember

Question 43

# Factorise the following: x²+2xy+y²

Answer 43

(x+y)(x+y) | This is just something that you should remember

Question 44

# Factorise the following: x²-2xy+y²

Answer 44

(x-y)(x-y) | This is just something that you should remember

Question 45

Answer 45

Question 46

Answer 46

Question 47

Answer 47

Question 48

# True or false? A polynomial consisting of real coefficients will have solutions that are real numbers

Answer 48

False. It will have solutions that can either be real or complex/imaginary **pairs**.

Question 49

# True or false? Similar to polynomials with real roots, polynomials with complex roots can have only one solution

Answer 49

False. Polynomials with complex solutions must occur in **conjugate pairs**

Question 50

Answer 50