2.7.2 Euler’s Form of the complex number

The following identity is known as Euler’s formula e Euler formula gives the polar form z r e i Note When performing multiplication or finding powers or roots of complex numbers, Euler form can also be used. Re Im O - i - cis π cis π cis Re Im O cis − Fig. . Fig.

. Example . Find the modulus and principal argument of the following complex numbers. (i) + i (ii) (iii) − (iv) − i (i) + i Modulus = x α = tan Since the complex number + i lies in the first quadrant, has the principal value θ = .

Therefore, the modulus and principal argument of + i are and p respectively. (ii) Modulus = and α = tan Since the complex number i lies in the second quadrant has the principal value θ = . Therefore the modulus and principal argument of i are and p respectively. (iii) − r = and .

Since the complex number − i lies in the third quadrant, has the principal value, θ = = . Therefore, the modulus and principal argument of − i are and respectively. (iv) − i r = and . Since the complex number lies in the fourth quadrant, has the principal value, θ = Fig.