A First Course in Linear Algebra, 2017a

332 Complex Numbers
Example 6.14: Polar to Standard Form
Let z = 2e 2πi/3 . Write z in the standard form
z = a + bi
Solution. Let z = 2e 2πi/3 be the polar form of a complex number. Recall that e iθ = cosθ + isinθ. Therefore
using standard values of sin and cos we get:
z = 2e i2π/3 = 2(cos(2π/3)+isin(2π/3))
(
= 2 − 1 √ )
3
2 + i 2
= −1 + √ 3i
which is the standard form of this complex number.
♠
You can always verify your answer by converting it back to polar form and ensuring you reach the
original answer.
Exercises
Exercise 6.2.1 Let z = 3+3i be a complex number written in standard form. Convert z to polar form, and
write it in the form z = re iθ .
Exercise 6.2.2 Let z = 2i be a complex number written in standard form. Convert z to polar form, and
write it in the form z = re iθ .
Exercise 6.2.3 Let z = 4e 2π 3 i be a complex number written in polar form. Convert z to standard form, and
write it in the form z = a + bi.
Exercise 6.2.4 Let z = −1e π 6 i be a complex number written in polar form. Convert z to standard form,
and write it in the form z = a + bi.
Exercise 6.2.5 If z and w are two complex numbers and the polar form of z involves the angle θ while the
polar form of w involves the angle φ, show that in the polar form for zw the angle involved is θ + φ.