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.z2−1 = 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 × a–iba–ib
⟹ 1z = a–iba2–i2b2 = a–iba2+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(a2b1–a1b2a22+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 = 15–25i
Now,
z1z2 = z1.1z2 = (2 + 3i)( 15–25i)
= (25+65) + i(−45+35) = 85 – 15i