Week 4 Complex Numbers

Question 1

what are natural numbers

Answer 1

counting numbers so positive integers and zero

Question 2

what are rational numbers

Answer 2

numbers that can be represented by a fraction

Question 3

what are real numbers

Answer 3

all rational and irrational numbers

Question 4

what are complex numbers

Answer 4

they are imaginary numbers that involve i

Question 5

what is the value of i

Answer 5

sqrt(-1)

Question 6

what is the general complex number equation and name the two parts

Answer 6

z = x + iy
x is the real part
iy is the imaginary part

Question 7

what are the three possible solutions for a quadratic equation and name the conditions for each

Answer 7

b^2-4ac > 0 => 2 real roots
b^2-4ac = 0 => 1 repeated root
b^2-4ac < 0 => 2 complex roots

Question 8

what happens when a quadratic has 2 complex solutions

Answer 8

the two solutions are a complex conjugate pair

Question 9

what is the complex conjugate pair equation for a general complex number

Answer 9

z* = x - iy

Question 10

how do we obtain the square of the modulus of a complex number

Answer 10

multiplying the complex number by its complex conjugate

I z I^2 = z*z

Question 11

what can we say if two complex numbers are equal

Answer 11

if z1 = z2
then
x1 = x2
y1 = y2

Question 12

what is the general equation for adding or subtracting complex numbers z1 ± z2

Answer 12

z1 ± z2 = (x1 ± x2) + i(y1 ± y2)

Question 13

what can be said if a complex number is equal to 0

Answer 13

x = y = 0

Question 14

what can I z I be thought of as

Answer 14

it can be considered to be the length of a complex number since it is always a real number

Question 15

what is the result of the product of two complex numbers

z1z2

Answer 15

z1z2 = (x1x2 - y1y2) + i(x1y2 + x2y1)

Question 16

what is the procedure for dividing a complex number

Answer 16

multiply both the numerator and denominator by the complex conjugate

Question 17

describe the argand diagram

Answer 17

cartesian diagram where the two axes are Re(z) and Im(z)

Question 18

what are the two components for the polar argand diagram

Answer 18

the two components are theta and I z I

Question 19

what is taking the complex conjugate equivalent to in polar terms

Answer 19

it is equivalent to changing the sign of theta

Question 20

what is the exponential form of a complex number

Answer 20

I z Ie^iθ

Question 21

what is Euler’s formula

Answer 21

e^iθ = cosθ + isinθ

Question 22

what does De Moivres Theorem (DMT) allow us to do

Answer 22

gives us a way of finding powers and roots of complex numbers that has a trigonometric association

Question 23

what is the DMT formula

Answer 23

(cosθ + isinθ)^n = cos(nθ) + isin(nθ)

Question 24

why are complex numbers important in physics

Answer 24

they are useful for expressing oscillating functions