. . 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.
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. . Fig.
. Re Im O r = q Re Im O r = q Re Im O r = Re Im O r = - - Complex Numbers Therefore, the modulus and principal argument of − i are and . In all the four cases, modulus are equal, but the arguments are depending on the quadrant in which the complex number lies. Example .
Represent the complex number (i) −− i (ii) + i in polar form. (i) Let −− i = r (cos sin ) We have r = x α = tan Since the complex number −− i lies in the third quadrant, it has the principal value, θ = = Therefore, −− i = −− i = , k Î . Note Depending upon the various values of k , we get various alternative polar forms. (ii) + i r = z θ = tan