Week 4

Question 1

The inverse of function f can be written as f^-1.
What is f^-1(f^-1)?

Answer 1

f

Question 2

If f and g both have inverses, what can (g.f)^-1 be written as?

Answer 2

(g.f)^-1 = (f^-1).(g^-1)
Note the change in the order in which the functions are applied

Question 3

What are the steps of proof by mathematical induction?

Answer 3

Step 1 - Base Case - Prove that the first possible case is true.
Step 2 - Induction Assumption - Assume that n=k and that the statement is true for n=k.
Step 3 - Induction Step - Prove that the statement is true for n=k+1.

Question 4

What are the steps of proof by strong mathematical induction?

Answer 4

Step 1 - Base Case - Prove that the first possible case is true.
Step 2 - Induction Assumption - Assume that n=k AND the statement is true for all values of n from 0 to k.
Step 3 - Induction Step - Prove that the statement is true for n=k+1.

Question 5

What is a rational number?

Answer 5

A number that can be expressed as a fraction of two integers. (a/b) a is an integer and b is a non-zero integer.

Question 6

What is the symbol for rational numbers?

Answer 6

Q

Question 7

What is a real number?

Answer 7

A number that is not imaginary or complex.

Question 8

What is the general form of a complex number?

Answer 8

a+bi (a is the real part and bi is the imaginary part)

Question 9

If a+bi is a complex number, what is it’s conjugate?

Answer 9

a-bi

Question 10

What is a+bi expressed in polar form?

Answer 10

a=rcos(theta)
b=rsin(theta)
a+bi = rcos(theta)+i*rsin(theta)=rcis(theta)

Question 11

What is De Moivre’s Theorem? and when can it be applied?

Answer 11

De Moivre’s Theorem explains when a complex number is raised to a power n.
z = complex number = rcis(theta) = re^i(theta)
z^n = (re^i(theta))^n = (r^n)(e^intheta)