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Division of Complex Numbers

Here you will learn what is the division of complex numbers with examples.

Let’s begin –

Division of Complex Numbers

The division of a complex number z1 by a non-zero complex number z2 is defined as the multiplication of z1 by the multiplicative inverse of z2 and is denoted by z1z2.

Thus, z1z2 = z1.z21 = z1.(1z2)

How to Find Muliplicative Inverse :

Let z = a + ib be a non-zero complex number. Then,

1z = 1a+ib

Multiply numerator and denominator by conjugate of denominator,

1z = 1a+ib × aibaib

1z = aiba2i2b2 = aiba2+b2

1z = aa2+b2 + i(b)a2+b2

Clearly, 1z is equal to the multiplicatve inverse of z.

How to Divide Two Complex Numbers :

Let z1 = a1+ib1 and z2 = a2+ib2. Then

z1z2 = (a1+ib1){a2a22+b22 + i(b2)a22+b22}

[ z = a + ib 1z = aa2+b2 + i(b)a2+b2

By definition of multiplication,

z1z2 = (a1a2+b1b2a22+b22) + i(a2b1a1b2a22+b22)

Example : If z1 = 2 + 3i and z2 = 1 + 2i, then find z1z2.

Solution : We have z1 = 2 + 3i and z2 = 1 + 2i,

1z2 = 11+2i = 1525i

Now, 

z1z2 = z1.1z2 = (2 + 3i)( 1525i)

= (25+65) + i(45+35) = 8515i

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