Using Complex Numbers

Question 1

i^2=

Answer 1

-1

Question 2

x^2 = -9

Answer 2

±3i

Question 3

What does Re(z) mean?

Answer 3

Real part of the complex number

Question 4

What does Im(z) mean?

Answer 4

Imaginary part of the complex number (contains i)

Question 5

What is a complex number, what form does it take?

Answer 5

Contains real and imaginary parts usually in the form x+yi where x and y are real (denoted using letter z).

Question 6

How do you add complex numbers?

(3+4i) + (2-8i) = ???

Answer 6

Add imaginary parts and add complex parts

(3+2) + (4-8)i = 5-4i

Question 7

How do you subtract complex numbers?

(6-9i) - (1+6i) = ???

Answer 7

Subtract imaginary parts and subtract complex parts

(6-1) + (-9-6)i = 5-15i

Question 8

How do you multiply complex numbers?

(7+2i)(3-4i) = ???

Answer 8

Use foil to multiply brackets in normal way but remember i^2 is -1

21 - 28i + 6i - 8i^2
21 - 22i + 8
29-22i

Question 9

What does it mean if two complex numbers are equal?

Answer 9

Imaginary parts are equal and real parts are equal

Question 10

How would u find a and b and find the two complex numbers?

z1 = (3-a) + (2b-4)i 
z2 = (7b-4) + (3a-2)i

Answer 10

By equating Re(z) and Im(z)

Re(z) 3-a = 7b-4
Im(z) 2b-1 = 3a-2

Solve these simultaneously to find a and b. Sub a and b back into originals to find the two complex numbers.

(Answer : a = 0 and b = 1 so z1= 3-2i and z2= 3-2i)

Question 11

How do you find the square root of a complex number?

e.g find the square roots of 21-20i

Answer 11

1) let (a+bi)^2 = 21-20i
2) expand LHS
3) equate imaginary and real parts
4) solve simultaneously by substitution method
5) REMEMBER A AND B ARE REAL
6) put values for a and b back in (a+bi)

(Answer 5-2i and -5+2i)

Question 12

What is the complex conjugate of z = x+yi?

Answer 12

z* = x- yi

Question 13

If the coefficients of a quadratic equation are real and the roots are complex what can you deduce about the roots?

Answer 13

The roots will be complex conjugates pairs so z and z*

Question 14

What can you deduce about the sum and product of two conjugate pairs?

Answer 14

They will be real

Question 15

How do you divide complex numbers?

e.g (2+7i) / (3-5i)

Answer 15

zz* = real

so multiply top and bottom by complex conjugate of denominator so here multiply by 3+5i

(answer -29/34 + 31/34i )

Question 16

How do you represent complex numbers geometrically on an Argand diagram?

Answer 16

x axis is real and y axis is imaginary

Question 17

What is the modulus of z= x+yi, |z|?

Answer 17

The distance from the origin- given as √(x^2+y^2)

Think of it as finding the distance between two coordinates where you use pythagorus theorem.

Question 18

What does |z|^2 equal?

Answer 18

zz*

Question 19

What is the argument?

Answer 19

The angle the complex number makes on the Argand diagram with the x axis - angle ranges from -π to π measured anticlockwise in radians

Question 20

How would you find the argument of (-5-5i)?

Answer 20

1) arctan(y/x) ignoring any negative signs
2) work out what quadrant you will be in
3) this is quadrant 3 so do π-ans and make it negative as you are going in the clockwise direction

(answer -3π/4)

Question 21

What is the modulus argument form of a complex number?

Answer 21

r(cosϴ+isinϴ)

Sometimes written as [r,ϴ] or r cis ϴ

Question 22

How would you derive the modulus argument form?

Answer 22

Draw a triangle:
hypotenuse = r 
ϴ = angle 
adjacent = x
opposite = y
generate equations for sinϴ and cosϴ and rearrange to get x and y - stick these values into x+yi

Question 23

In modulus argument form what values can r take?

Answer 23

r>0

Question 24

How do you multiply in modulus argument form?

Answer 24

Multiply the moduli and add the arguments

Question 25

How do you divide in modulus argument form?

Answer 25

Divide the moduli and subtract the arguments

Question 26

What does |z-a| = r represent in loci?

Answer 26

A circle of points |z| where ‘a’ is the centre and ‘r’ is the radius.

Question 27

How would you represent this |Z-6+4i| = 3

Answer 27

A circle where 6-4i (swap signs) is the centre and has a radius of 3

Question 28

What does arg(z-a) = ϴ represent in loci?

Answer 28

Start at point ‘a’ and move anticlockwise through the angle ϴ. The points z lie on this angle line

Question 29

What does -π/4 < arg(z-3+4i) < π/4 look like?

Answer 29

The area of point in-between -π/4 and π/4 from the point 3-4i

Question 30

What does |z-a| = |z-b| represent in loci?

Answer 30

The perpendicular bisector of a and b