Question

Express the following complex numbers in polar form and exponential form:

−1

Answer 1

Let z = −1 = −1 + 0.i

This is of the form a + bi, where a = −1, b = 0

∴ r = `sqrt("a"^2 + "b"^2)`

= `sqrt((-1)^2 + 0^2)` = 1

Also, cos θ = `"a"/"r" = (-1)/1` = −1

and sin θ = `"b"/"r" = 0/1` = 0

∴ θ = π ...[∵ cos π = −1 and sin π = 0]

∴ the polar form of z = r(cos θ  + i sin θ)

= 1(cos π + i sin π)

and the exponential form of z = re^iθ

= 1·e^iπ

= e^πi