Complex Numbers

Question 1

What is a complex number?

Answer 1

A complex numbers has a real part and an imagined part.
Ex. 3+4i

3 is the real part
4i is the imagined part

Question 2

What symbol is used to represent √-1 ?

Answer 2

i is used to represent imaginary numbers
√-1 = i
√-4 =4i
√-25=5i

Question 3

What capital letter is used to represent the set of complex numbers?

Answer 3

C represents the set of complex numbers.

Question 4

How do you add or subtract complex numbers?

Example
3+2i)+(4-3i

Answer 4

You add the real parts and the imaginary parts separately.
3+4+2i-3i=
7-i

Question 5

How do you remove brackets from complex numbers?

Example
3(2-4i)

Answer 5

You multiply each term inside the bracket by the number outside the bracket.
3(2-4i)=
6-12i

Question 6

How do you multiply two complex numbers?

Example
2+3i)(-4+5i

Answer 6

You split the first bracket and multiply the second bracket by both parts.

2(-4+5i)+3i(-4+5i)=

Question 7

How do you divide by a complex number?
or
How do you express 2-3i in the form a+bi?
———-
4+2i

Answer 7

Multiply above and below the line by the conjugate of the bottom line.

(4+2i)(4-2i)

Question 8

What is the conjugate of z if z=a+bi?

Answer 8

The conjugate of a complex number you change the sign of the imaginary part.

The conjugate of a+bi is a-bi.

Question 9

What do you always get when you multiply a complex number by its conjugate?

Answer 9

The result is always a real number.

Question 10

What is an Argand diagram used to represent?

Answer 10

An Argand diagram is used to represent complex numbers on a graph.
The horizontal axis (-) is called the real axis (Re).
The vertical axis (I) is called the imaginary axis (Im).

Question 11

What happens to a point (a+bi) on a argand diagram when you multiply it by its conjugate (a-bi)?

Answer 11

It rotates anticlockwise.

Question 12

What is the length (or modulus)) of a complex number?

Ia+biI

Answer 12

The length (or modulus) of a complex numbers is its distance from the origin (0+0i).

Question 13

How do you find the length (or modulus) of a complex number?
If z= a+bi
IzI or Ia+biI =?

Answer 13

IzI = I a+bi I =

√ a squared + b squared

Question 14

How do you solve a quadratic equation of the form

ax squared +bx+c = 0?

Answer 14

1. Use factors
2. If it cannot be factorised use the formula

x=-b+ or - √bsquared-4ac
——————————-
2a

Question 15

If you are given the roots of a quadratic equation, how do you form the equation?

Answer 15

x squared - x(sum of roots) + the product of the roots =0

Question 16

What can you tell about the roots of a quadratic equation if the roots work out to be a conjugate pair?

Answer 16

All its roots are conjugate pairs.

Question 17

How do you express 3+4i in the form a+bi?
——-
5

Answer 17

3 + 4i
– —
5 5