2 Complex numbers
2.6 Division of complex numbers
The second of the conjugate–modulus properties enables us to find reciprocals of complex numbers and to divide one complex number by another, as shown in the next example. As for real numbers, we cannot find a reciprocal of zero, nor divide any complex number by zero.
Example 2
- (a) Find the reciprocal of 2 − 5i.
- (b) Find the quotient
Solution
- (a) We want to find the complex number which represents
- We multiply the numerator and denominator by 2 + 5i, the complex conjugate of 2 − 5i, to give
- (b) We multiply the numerator and denominator by 1 −2i, the complex conjugate of 1 + 2i, to give
The method used in Example 2, of multiplying the numerator and denominator by the complex conjugate of the denominator, enables us to find the reciprocal of any non-zero complex number z, and the quotient z_{1}/z_{2} of any two complex numbers z_{1} and z_{2}, where z_{2} ≠ 0. We can obtain general formulas as follows.
For the reciprocal, we have
If z = x + iy, so and |z|^{2}= x^{2} + y^{2}, we obtain
For the quotient z_{1}/z_{2}, we have
If z_{1} = x_{1} + iy_{1} and z_{2} = x_{2} + iy_{2}, this can be rewritten as
These formulas may be used in theoretical work, but for calculations of reciprocals and quotients it is simplest to use the method of Example 2.
Exercise 13
Find the reciprocal of each of the following complex numbers.
- (a) 3 − i
- (b) −1 + 2i
Solution
In each case we multiply both the numerator and the denominator by the complex conjugate of the denominator.
- (a)
- (b)
Exercise 14
Evaluate each of the following quotients.
- (a)
- (b)
Solution
In each case we multiply the numerator and denominator by the complex conjugate of the denominator.
- (a)
- (b)