Complex (could be more)

Question 1

What are the three ways of writing a complex number?

Answer 1

z=x+iy
z=r(cosθ+isinθ)
z=re^(iθ)

Question 2

What do roots of unity add up to?

Answer 2

0

Question 3

What is de moivre’s theorem?

Answer 3

z^n = r^n(cos nθ + i sin nθ)

Question 4

z^n + z^-n

Answer 4

2cos nθ

Question 5

z^n - z^n-1

Answer 5

2sin nθ

Question 6

What is i?

Answer 6

√-1

Question 7

If z = x+iy

z*=?

Answer 7

z = x+iy
z*= x-iy with same modulus but negative argument

Question 8

What are the axis on an argand diagram?

Answer 8

x-axis Real

y-axis Imaginary

Question 9

What is the modulus of an imaginary number?

Answer 9

|z|=√(x²+y²)

Question 10

What is the argument of an imaginary number?

Answer 10

θ=arctan(y/x)

Question 11

What loci does | z - (x+iy) | = r represent?

Answer 11

Circle with centre x+iy and radius r

Question 12

If loci includes signs < > what must the line be?

Answer 12

dashes

Question 13

What loci does | (x+iy) | = | (p+iq) | represent?

Answer 13

Perpendicular bisector between points x+iy and p+iq

Question 14

What loci does arg(z -x -iy) = θ represent?

Answer 14

Half line starting at x+iy making an angle of θ with positive x-axis

Question 15

What is |zw| and arg(zw)?

Answer 15

|z| x |w|

arg z + arg w

Question 16

For roots of unity how do you prove they sum to zero?

Answer 16

Use a geometric series
a = 1
r = w

Question 17

If 2+3i is a root what are the conditions for 2-3i to be a root?

Answer 17

All the constants in equation must be real