Question

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

`1/(1 + "i")`

Answer 1

Let z = `1/(1 + "i")`

= `(1 - "i")/((1 + "i")(1 - "i"))`

= `(1 - "i")/(1 - "i"^2)`

= `(1 - "i")/(1 - (-1))`   ...[∵ i² = – 1]

= `(1 - "i")/2`

∴ z = `1/2 - 1/2"i"`

∴ a = `1/2`, b = `(-1)/2`

∴ | z | = r

= `sqrt("a"^2 + "b"^2)`

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

= `sqrt(1/4 + 1/4)`

= `1/sqrt(2)`

Here `(1/2, (-1)/2)` lies in 4^th quadrant

θ = amp (z)

= `2pi + tan^-1("b"/"a")`

= `2pi + tan^-1(((-1)/2)/(1/2))`

= 2π + tan^–1(–1)

= 2π – tan^–1(1)

= `2pi - pi/4`

= `(7pi)/4`

∴ θ = 315° = `(7pi)/4`

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

= `1/sqrt(2)(cos 315^circ +  "i"  sin315^circ)`

= `1/sqrt(2)[cos((7pi)/4) + "i"  sin((7pi)/4)]`

The exponential form of z = re^iθ

= `1/sqrt(2)"e"^((7pi)/4"i"`.