Polar Form of Complex Numbers and De Moivre flashcards

Question 1

In the complex plane, what does the horizontal axis represent?

Accepted Answer

The real part of the number (a in a + bi); it is labeled the real axis.

Question 2

How do you plot a + bi in the complex plane?

Accepted Answer

Move a units horizontally and b units vertically from the origin.

Question 3

What does |z| measure for a complex number z?

Accepted Answer

The distance from the origin to the point representing z (its magnitude).

Question 4

For z = x + yi, what is the formula for |z|?

Accepted Answer

|z| = sqrt(x^2 + y^2)

Question 5

What is the modulus of a complex number?

Accepted Answer

Its absolute value: the distance r from the origin to the point (x, y).

Question 6

What does the symbol theta (θ) represent in the polar form of a complex number?

Accepted Answer

The argument: the angle of direction of the point from the origin.

Question 7

What is the polar form of a complex number with modulus r and argument θ?

Accepted Answer

z = r(cos θ + i sin θ)

Question 8

What does the abbreviation r cis θ stand for?

Accepted Answer

r(cos θ + i sin θ)

Question 9

Which conversion formula gives the real part x of z from its polar form?

Accepted Answer

x = r cos θ

Question 10

Which conversion formula gives the imaginary part y of z from its polar form?

Accepted Answer

y = r sin θ

Question 11

How is r found from x and y when writing x + yi in polar form?

Accepted Answer

r = sqrt(x^2 + y^2)

Question 12

For z1 = r1(cos θ1 + i sin θ1) and z2 = r2(cos θ2 + i sin θ2), what is z1 z2?

Accepted Answer

z1 z2 = r1 r2 [cos(θ1 + θ2) + i sin(θ1 + θ2)]

Question 13

In words, how do you multiply two complex numbers in polar form?

Accepted Answer

Multiply the moduli and add the angles.

Question 14

For z1 = r1(cos θ1 + i sin θ1) and z2 = r2(cos θ2 + i sin θ2), what is z1/z2?

Accepted Answer

z1/z2 = (r1/r2)[cos(θ1 − θ2) + i sin(θ1 − θ2)], with z2 ≠ 0

Question 15

In words, how do you divide two complex numbers in polar form?

Accepted Answer

Divide the moduli and subtract the angles.

Question 16

In words, how does De Moivre's Theorem find z^n?

Accepted Answer

Raise the modulus to the nth power and multiply the argument by n.

Question 17

What is the nth Root Theorem formula for z^(1/n), where z = r(cos θ + i sin θ)?

Accepted Answer

z^(1/n) = r^(1/n) [cos(θ/n + 2kπ/n) + i sin(θ/n + 2kπ/n)]

Question 18

In the nth Root Theorem, which values of k are used?

Accepted Answer

k = 0, 1, 2, 3, ..., n − 1

Question 19

Find |z| for z = 3 − 4i.

Accepted Answer

5

Question 20

Find |z| for z = sqrt(5) − i.

Accepted Answer

sqrt(6)

Question 21

Write −4 + 4i in polar form.

Accepted Answer

4 sqrt(2) cis(3π/4)

Question 22

Convert z = 12(cos(π/6) + i sin(π/6)) to rectangular form.

Accepted Answer

6 sqrt(3) + 6i

Question 23

Find (1 + i)^5 using De Moivre's Theorem.

Accepted Answer

−4 − 4i

Polar Form of Complex Numbers and De Moivre flashcards

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Polar Form of Complex Numbers and De Moivre flashcards

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