Imaginary Numbers

Question 1

What is the modulus of z= x + iy?

Answer 1

The modulus is the length from the origin to z, defined as |z| = sqrt(x^2 + y^2).

Question 2

What is the difference between the argument (arg(z)) and the principal value of the argument (Arg(z))?

Answer 2

arg(z) is the angle θ from the origin to z. Arg(z) is this value, θ, between -π<θ<π. Or, Arg(z) is the unique value of arg(z).

Question 3

What is the polar form of a complex number?

Answer 3

A complex number in form z = rcis(θ) is described as polar.

Question 4

What is an Argand diagram?

Answer 4

An Argand diagram is a visual representation of a complex number.

Question 5

How to you divide two complex numbers?

Answer 5

Multiply the question by the complex conjugate of the denominator.

Question 6

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

Answer 6

x - iy.

Question 7

What do you do to the value of inverse tan(y/x) to find arg(z) if you know z lies within the second or third quadrants of an Argand diagram?

Answer 7

Add π.

Question 8

What is the value of (cis θ1)(cis θ2)?

Answer 8

(cis θ1)(cis θ2) = cis(θ1 + θ2)

Question 9

What is the value of 1/cis(θ)?

Answer 9

1/cis(θ) = cis(-θ)

Question 10

|mod(z1) mod(z2)| = ?

arg(z1z2) = ?

Answer 10

|mod(z1)mod(z2)| = |mod(z1)||mod(z2)|.

arg(z1z2) = arg(z1) + arg(z2).

Question 11

What is de Moivres theorom for an integral index?

Answer 11

(cis θ)^n = cis (nθ).

Question 12

What is de Moivres theorom for a fractional index?

Answer 12

cis((p/q)θ) is one of the values of (cis θ)^(p/q).

Question 13

With z = cis(θ), what is a trigonometric identity in regards to cos(θ)?

(optional help: cosθ has a plus)

Answer 13

cos(θ) = 1/2(z + 1/z).

Question 14

With z = cis(θ), what is a trigonometric identity in regards to sin(θ)?

(optional help: sinθ has a minus)

Answer 14

sin(θ) = 1/2i(z - 1/z)

Question 15

Consider the trigonometric identity of cos(θ) with z = cis(θ). What difference does having cos(mθ) make to the identity?

Answer 15

In the identity of cos(mθ), all references of z in the identity are to the power of m.

cos(θ) = 1/2(z + 1/z)
cos(mθ) = 1/2(z^m + 1/z^m)

Question 16

How do you undergo synthetic division?

Answer 16

Divide the first value of the numerator by the first value of the denominator. This value is the first value of the result. Then, multiply this value by the denominator. Subtract this value from the numerator until you run out of values. the remaining values in the numerator are the remainder, and the values extracted is the completed division.

Question 17

What is the term “the zeros of the polynomial” synonymous with?

Answer 17

The roots of the equation.

Question 18

What term is given to the value n, where n is the highest power in a polynomial?

Answer 18

It is said that the polynomial is of degree n.

Question 19

What does the Maclaurin series of some function f (i.e f(x) = sin(x)) describe?

Answer 19

The Maclaurin series of f describes an infinite expansion of values that some functions satisfy.

For f(x) = sin(x), the Maclaurin expansion is sin(x) = x - (x^3)/3! + (x^5)/5! + …

Question 20

Define sinh(x) (pronounced shine x), and cosh(x).

Tip: involves e^x.

Answer 20

sinh(x) = 1/2(e^x - e^-x).

cosh(x) = 1/2(e^x + e^-x)

Question 21

What is the value of log(z)?

Answer 21

log(z) = ln(|z|) + iarg(z)

Question 22

How would you describe the principal value of the logarithm function?

Answer 22

Log(z) = ln(|z|) + iArg(z)