Complex Numbers

Question 1

What is Euler’s equation for complex numbers?

Answer 1

Question 2

What is a complex (aka Argand) plane?

Answer 2

It’s just like the normal Cartesian plane, but Re(z) is the horizontal axis and Im(z) is the vertical axis.

Question 3

Complete the following rule for complex conjugates.

Answer 3

Question 4

Complete the following rule for complex conjugates.

Answer 4

Question 5

What is the modulus of a complex number?

Answer 5

It’s the magnitude of the vector from the origin to point z on a complex plane.

Question 6

Complete the following property of a modulus:

Answer 6

Question 7

Complete the following property of a modulus:

Answer 7

Question 8

Complete the following property of a modulus:

Answer 8

Question 9

Complete the following property of a modulus:

Answer 9

Question 10

What is the argument, arg z, of a complex number?

Answer 10

The argument of a complex number is the angle between its vector and the positive real axis.

Question 11

How do you represent a complex number in polar form?

Answer 11

Question 12

How do you write the complex conjugate of z in polar form?

Answer 12

Question 13

According to De Moivre’s Theorem,

Answer 13

Question 14

How many solutions are there for the following equation?

Answer 14

The n nth roots of c are the solutions.

Question 15

What is the modulus for all roots of the following equation?

Answer 15

All of the roots will have modulus,

Question 16

If you are asked to simplify something with a z in the denominator, what is 1 good stragety?

Answer 16

• Multiply the denominator to the other side of the equation and solve for z.
• In the last moment change z to the form required.

Question 17

If you are asked to isolate the real (or imaginary) part of an expression of complex numbers, what is a good strategy?

Answer 17

1. Simplify the denominator.
2. Multiply by the complex conjugate of the denominator to get rid of the fraction.
3. Group the real parts together and the imaginary parts together.

Question 18

How many complex solutions must an odd polynomial have? (How many complex solutions are there to x^n=1 if n is odd)?

Answer 18

Odd polynomials may not have any complex roots. However, if there exists any complex root, then they must always come in pairs.

Question 19

If you know that w is a non-real solution to z^3=1, how can you best use this to simplify expressions of w?

Answer 19

1. Recall that both w^3=1 and w*^3=1.
2. Replace any 1s in the expression with w^3.
3. Factor and rearrange, somehow taking advantage of the fact that w^3-1=0.

Question 20

If you’re given distinct roots of a complex equation/function, how do you know the other roots?

Answer 20

For an even function, the roots always come in conjugate pairs.