Complex Numbers

Question 1

What is the definition of complex numbers?

Answer 1

the set of R^2 together with the (binary) operations +, x, where (a,b) + (c,d) = (a+c, b+d) and
(a,b)(c,d) = (ac-bd, ad+bc)

Question 2

How are two complex numbers equal to each other?

Answer 2

if z=(a,b) and w=(x,y) they are equal if and only if a=x and b=y

Question 3

How are (0,1) and (-1,0) denoted?

Answer 3

(0,1) = I
(-1,0) = I^2

Question 4

What are the real and imaginary parts of the complex number z=(a,b)?

Answer 4

real part = a

imaginary part = b

Question 5

What is (a+ib) + (c+di)?

Answer 5

(a+c) + (b+d)i

Question 6

What is (a+ib)(c+id)?

Answer 6

(ac-bd) + (ad-bc)i

Question 7

If z=a+ib what is the reciprocal of z?

Answer 7

1/(a^2 + b^2) (a-ib)

Question 8

How do you take the square root of the complex number x + iy?

Answer 8

=√(x^2 + y^2 + x )/2 ±i√(x^2 + y^2 −x)/2

Question 9

How does the argument of a complex number vary?

Answer 9

x + 2 pi k where k is an integer

Question 10

What is de Moivre’s theorem?

Answer 10

(cosx + isinx)^n = cosnx + isinnx

Question 11

Define the nth root of unity?

Answer 11

a complex number t is an nth root of unity if z^n=1

Question 12

What is the formula of a circle in the argand plane?

Answer 12

abs(z - (a=ib)) = sqrt(R)