Complex Numbers/argand diagrams

Question 1

i

Answer 1

√-1
Written as b i
Or a + b i as a complex number where a is real

Question 2

√-36

Answer 2

√36 x √-1

6i

Question 3

√-7

Answer 3

√7 x √-1

√7 i

Question 4

i
i^2
i^3
i^4

Answer 4

i
-1
-i
1

Question 5

(2+3i)(3-2i)

Answer 5

Expand as normal collecting like terms 
6 - 4i + 9i -6i^2
6 + 5i - 6(-1)
6 + 5i + 6
12 + 5i

Question 6

Complex conjugate

Answer 6

The same as the original complex number but with the second sign reversed

Question 7

What does the complex conjugate do

Answer 7

Multiplying or dividing it by the complex number gives you a real number

Question 8

Symbol for the conjugate of z

Answer 8

z*

Question 9

How to use the complex conjugate

Answer 9

Typically it is used the same way as rationalising a denominator

Question 10

What do you know if one root is a complex number

Answer 10

Another root must be the complex conjugate

Question 11

What to remember if a root is complex

Answer 11

The bracket is (x-(a + bi))

x - a - bi

Question 12

Roots if a cubic touches the x axis 3/2/1 time(s)

Answer 12

3 times : 3 real roots
2 times: 3 real roots - 2 of them the same
1 time : 1 real root - 2 complex (conjugates)

Question 13

Roots if a quartic touches the x axis

4/3/2/1/0 time(s)

Answer 13

4 times: 4 real roots
3 times: 4 real roots - 2 identical
2 times: 2 real roots - 2 complex (conjugates)
1 time: 2 identical real - 2 complex (conjugates)
0 times: 4 complex (2 conjugate pairs)

Question 14

Argand Diagrams

Answer 14

Used for plotting complex numbers
x-axis is real, y is imaginary
Plot like coordinates and draw a line to the origin

Question 15

Argand diagram solutions trick

Answer 15

The real (x)-axis is a line of symmetry for solutions to a polynomial

Question 16

|z|

Answer 16

Use pythagoras on the real part and the coefficient of i to find the modulus of z

Question 17

arg(z)

Answer 17

Argument of z - anti-clockwise angle in radians from the real axis, give as negative value if more than pi

Question 18

arg(z) when x and y values are positive

Answer 18

tan-1(y/x)
Quadrant 1

USE POSITIVE VALUES OF Y AND X

Question 19

arg(z) when x is positive and y is negative

Answer 19

-(tan-1(y/x))
Quadrant 4

USE POSITIVE VALUES OF Y AND X

Question 20

arg(z) when x is negative and y is positive

Answer 20

π - tan-1(y/x)
Quadrant 2

USE POSITIVE VALUES OF Y AND X

Question 21

arg(z) when x and y are both negative

Answer 21

-π + tan-1(y/x))
Quadrant 3

USE POSITIVE VALUES OF Y AND X

Question 22

Modulus argument form

Answer 22

If r = |z| and o = arg(x)

z = r(cos(o) + i sin(o))

MUST BE PLUS SIN AND COS

Question 23

How to find the point from the modulus argument form

Answer 23

x of z = r cos theta

y of z = r sin theta

Question 24

Modulus of multiplied complex numbers

Answer 24

The combined modulus is each individual modulus multiplied together

Question 25

Argument of multiplied complex numbers

Answer 25

The combined argument is the sum of the individual arguments

Question 26

Modulus of divided complex numbers

Answer 26

The combined modulus is the division of the individual moduluses

Question 27

Argument of divided complex numbers

Answer 27

The combined argument is the same as subtracting the individual arguments

Question 28

How to go from modulus argument form to complex number

Answer 28

Solve by putting the modulus argument form in your calculator

Question 29

What to know for negatives arguments

Answer 29

``` Cos(-θ) = Cos(θ) Sin(-θ) = -Sin(θ) ``` IF YOU CHANGE ONE ARG THE OTHER MUST CHANGE WITH IT

Question 30

How to get an argument into the principal argument form

Answer 30

The argument is the smallest difference (positive or negative) between the worked value and an even coefficient of pi

Question 31

Principal argument form

Answer 31

-π < arg(x) < π

Question 32

How to solve for the complex solutions when not given any

Answer 32

Complete the square

Question 33

How to adjust arg depending on what direction you have it measured from

Answer 33

If anti-clockwise from up: add half pi If anti-clockwise from east: subtract pi If anti-clockwise from down: subtract half pi

Question 34

How to go from two multiplied mod arg to x+ yi

Answer 34

Multiply mods Add args Carry out that For divide: divide mods and subtract args

Question 35

How to solve from one complex root

Answer 35

Another is the complex conjugate Do (x- a - bi)(x - a + bi) Divide the equation by the answer (by inspection) and solve normally