How to find the reciprocal of a complex number?

djeetThis article has been authored by Djeet,the Maths editor at Xamplified,who has over four years of experience in teaching Mathematics.His write-ups at Xamplified can truly be termed ‘Examplified’ as he breaks down complex maths concepts into simple how-to steps.

Table Of Content

Brush Up Basics


Step 1 : Invert the number

If z = a + i b is a complex number, then reciprocal of it is given by

image001

Step 2: Multiply numerator and denominator by conjugate

Multiply numerator and denominator of the inverted number by conjugate of denominator.

image002

Step 3: Simplify and find the reciprocal

Simplify above equation in step (2). Numerator is multiplied by 1 and is already simplified.

Numerator = a + i b

The denominator needs to be simplified.

Denominator = (a + i b) (a – i b) is of the form of (A + B)*(A – B) = A² – B²

Therefore,

(a + i b) (a – i b) = a² – (i b)²
= a² – i² b²
= a² + b²

image003

Therefore,

image004

Hence, it is multiplicative inverse (or reciprocal) of complex number, z.

Example to clear it all


Find the reciprocal of complex number 2 + 2i

Step 1: Let z = 2 + 2i

Then

image005

Step 2: multiply numerator and denominator by conjugate of complex number.

image006

Step 3: Simplify above equation

image007

Observations to give you insight


image0041

Observe the numerator and denominator of multiplicative inverse carefully. Numerator is the conjugate of given complex number while denominator is the square of modulus of same complex number.

image-012

image009

This observation can be used to quickly calculate reciprocal of any complex number.

Tips to make life easier


Always use this result

image010

to establish a flow diagram and perform calculation in mind.

Flow Diagram to find reciprocal of any complex number

Flow Diagram to find reciprocal of any complex number


Let us find reciprocal of complex number 2 + i2 using flow diagram

Reciprocal of complex number 2 + i2 using flow diagram