Introduction to Complex Numbers

Question 1

Define the conjugate of z

Answer 1

If z = a + ib, z* = a - ib

Question 2

Give the fundamental theorem of algebra

Answer 2

p(z) = a0 + a1z + a2z^2 + … + anz^n can be rewritten as p(z) = an(z-c1)(z-c2)…(z-cn) for n complex numbers c

Question 3

Define the modulus of z

Answer 3

If z = x + iy,
|z| = sqrt(x^2+^2)

Question 4

Define the argument of z

Answer 4

If z = x + iy,
arg(z) = arctan(y/x), though by convention, 0 ≤ arg(z) < 2π

Question 5

Define an isometry

Answer 5

A translation where distance is preserved