This 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
Step 2: Multiply numerator and denominator by conjugate
Multiply numerator and denominator of the inverted number by conjugate of denominator.
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
(a + i b) (a – i b) = a² – (i b)²
= a² – i² b²
= a² + b²
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
Step 2: multiply numerator and denominator by conjugate of complex number.
Step 3: Simplify above equation
Observations to give you insight
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.
This observation can be used to quickly calculate reciprocal of any complex number.
Tips to make life easier
Always use this result
to establish a flow diagram and perform calculation in mind.
Flow Diagram to find reciprocal of any complex number
Let us find reciprocal of complex number 2 + i2 using flow diagram