Complex Numbers

Question 1

i^2 =

Answer 1

-1

Question 2

All complex numbers are in the form…

Answer 2

A + Bi
A is real
B is imaginary

Question 3

Solve:
X^2 + 49 = 0

Answer 3

X^2 = -49
X = root - 49
X = -7i

Question 4

Z1 = 3 + 2i
Z2 = 4 - 3i

What’s z1 + z2

What’s z1 - z2

Answer 4

7 - i
-1 + 5i

Question 5

5(3i +2i) =

Answer 5

15 + 10i

Question 6

I^3 =

Answer 6

I(i)^2
I(-1) = -i

Split the i’s up into multiple powers of 2,
Replace i squared with a -1

Question 7

What is a complex conjugate symbol

Answer 7

Z*

Question 8

What is the complex conjugate of Z = -2 + 3i

Answer 8

Z* = -2 - 3i

Question 9

Why are complex conjugates useful

Answer 9

When u add them together u get a real number
When u multiply them together u get a real number

Question 10

How to divide complex numbers

Answer 10

Multiply the denominator by its complex conjugate to get a real number
Simplify

Question 11

What is an argand diagram

Answer 11

Y - axis = complex number
X - axis = real number

Question 12

How to represent Z = 3 + 2i on an argand diagram

Answer 12

Much like a vector
3 across and 2 up

Question 13

The complex conjugate is a ________ in the x-axis

Answer 13

Reflection

Question 14

What does the modulus mean

Answer 14

The length of the line

Question 15

How to work out the modulus Z =A + Bi

Answer 15

|z| = root( a ^2 + b^2)

Question 16

What is the argument

Answer 16

The angle of the x axis
In radians

Question 17

How to work out the argument

Answer 17

Draw a sketch
Look at the acute angle of alpha
Tan-1(y/x)
Use the positive values

Question 18

Arg of Z = -3 + 4i

Answer 18

Tan-1(4/3)
= 0.927
-( pi - 0.927) = - 2.215

Question 19

How to generally solve complex numbers equations

Answer 19

Group the real numbers and imaginary parts together
Then equate the real numbers with real numbers
Do the same for the imaginary numbers
Solve both equations simultaneously

Question 20

( a + b )(5 -i) = 10a + b +1 - (2a+1)i

Answer 20

5a - ai + 5b -bi = (10a + b + 1) - (2a +1)i
(5a + 5b) - (a + b)i = (10a +b +1) - (2a + 1)i
5a + 5b = 10a + b + 1
-A - b = -2a -1
Solve simultaneously

Question 21

How to find the square root of -21 - 20i

Answer 21

Let the answer = a + bi
(A + bi)^2 = -21 -20i
A^2 + 2abi - b^2 = -21 -20i
Equate the imaginary numbers with the ir numbers
So the same with the real ones
Solve simultaneously for a and b
Put back into a + bi

Question 22

How to find the roots of a polynomial with no real roots

Answer 22

Use the quadratic formula to find one root
The second root is the complex conjugate

Question 23

Find the quadratic equation that has 5 - 3i as one of its roots

Answer 23

A = 5 - 3i
B = 5 + 3i
Times them together to get your quadratic

Question 24

What did u get in an cubic equation

Answer 24

1 real root
Complex conjugate pair

Question 25

Find all the roots of the equation X^3 -4x^2 + x + 26 = 0

Answer 25

Use factor theorem to find the real root X +2 is a factor Divide the cubic by the factor Find the complex conjugate by using the quadratic formula

Question 26

What would the complex roots of a quadratic equation look like

Answer 26

2 real roots and a complex conjugate pair Or 2 complete conjugate pairs

Question 27

If z = x +iy then the complex conjugate z* = x -iy is a _______ _____ on an argand diagram

Answer 27

Reflection in the x- axis

Question 28

How to find the complex number that represents a reflection of z in the real axis

Answer 28

Find the complex conjugate

Question 29

Z=x + iy What is x in Cartesian form

Answer 29

X = Rcos(theta) R = modulus Theta = argument

Question 30

Z=x + iy What is y in Cartesian form

Answer 30

Y = Rsin(theta)

Question 31

What is the complete Cartesian form of a complex number

Answer 31

Z = r( cos(theta) + isin(theta) )

Question 32

|z1 x z2| = |z1||z2| =

Answer 32

R1 x r2

Question 33

Arg(z1 x z2) = arg(z1) + arg(z2)

Answer 33

= theta1 + theta2

Question 34

What are arg laws similar to

Answer 34

Log laws

Question 35

|z1/z2| = |z1| / |z2| =

Answer 35

R1 / r2

Question 36

Arg|z1/z2| = arg(z1) - arg(z2) =

Answer 36

Theta1 - theta2

Question 37

Circle centre z1 radius r equation

Answer 37

| z - z1|

Question 38

What is the centre and radius of | 4 +5i -z|

Answer 38

|-1( Z - 4 - 5i)| |-1||z - 4 - 5i| |z - (4 + 5i )| Centre 4,5

Question 39

How to find the Cartesian equation

Answer 39

Let z be x + iy Z1 = a + bi |x + iy - (a +bi)| |(x-a) + i(y-b)|=r Square

Question 40

The locus of a point which is equidistant from two fixed points A and B in the complex plane is …

Answer 40

The perpendicular bisector of A and B

Question 41

What’s the general formula for the perpendicular bisector

Answer 41

|z - z1| = |z - z2|

Question 42

What is the equation for a half line

Answer 42

Arg( z - z1)

Question 43

<= means what type of line

Answer 43

Solid

Question 44

Arg(z - z1/ z - z2) = theta

Answer 44

Arg(z - z1) - arg(z - z2) Let Arg(z - z1) = alpha Let Arg(z - z2) = beta Alpha -beta = theta Start at z1 move anti-clockwise to z2 On the major arc if theta is acute and the minor arc if theta is obtuse

Answer 45

Arg(z - z1) - arg(z - z2) Let Arg(z - z1) = alpha Let Arg(z - z2) = beta Alpha -beta = theta Start at z1 move anti-clockwise to z2 On the major arc if theta is acute and the minor arc if theta is obtuse