Complex Numbers Flashcards

For how many integers n with 1 ≤ n ≤ 100 is iⁿ + i^(−n) = 0?

Answer: 50

iⁿ + i^(−n) = 0 exactly when n is odd: for odd n, iⁿ = i or −i and i^(−n) is its opposite, giving 50 values.

If z satisfies z + |z| = 2 + i, what is |z|?

Answer: 5/4

Writing z = x + yi gives y = 1 and x + √(x²+1) = 2; solving yields x = 3/4 and |z| = 2 − 3/4 = 5/4.

What is the product of all primitive 6th roots of unity?

Answer: 1

The primitive 6th roots are e^(iπ/3) and e^(5iπ/3); their product is e^(2πi) = 1.

In the complex plane, what is the area of the triangle with vertices 0, z, and iz where z = 3 + 4i?

Answer: 25/2

Multiplication by i rotates z by 90°, so the triangle is a right isosceles triangle with legs |z| = 5; area = ½·5·5 = 25/2.

If |z − 1| = |z + 1|, which best describes the locus of z in the complex plane?

Answer: The imaginary axis

z equidistant from 1 and −1 lies on the perpendicular bisector of the segment [−1, 1], which is the imaginary axis.

What is the value of (1 + i√3)^6?

Answer: 64

Writing 1 + i√3 = 2e^(iπ/3), we get (2e^(iπ/3))^6 = 64·e^(2πi) = 64.

Let z₁ and z₂ be complex numbers with |z₁| = 2 and |z₂| = 3. What is the maximum possible value of |z₁ + z₂|?

Answer: 5

By the triangle inequality |z₁ + z₂| ≤ |z₁| + |z₂| = 5, with equality when z₁ and z₂ point in the same direction.