(-3+5i)/(-9+4i)

2 min read Jun 16, 2024

Dividing Complex Numbers: A Step-by-Step Guide

Dividing complex numbers can seem intimidating, but it's actually a straightforward process using a clever trick. Let's break down how to divide (-3 + 5i) / (-9 + 4i).

1. Multiply by the Conjugate

The key to dividing complex numbers is multiplying both the numerator and denominator by the conjugate of the denominator. The conjugate of a complex number is formed by simply changing the sign of the imaginary part.

In this case, the conjugate of (-9 + 4i) is (-9 - 4i).

2. Simplify the Expression

Now, we multiply:

( -3 + 5i) / (-9 + 4i) * (-9 - 4i) / (-9 - 4i)

This might look complex, but remember: we're essentially multiplying by 1, so we don't change the value of the expression.

Numerator:

We expand using the FOIL method (First, Outer, Inner, Last):
- (-3)(-9) + (-3)(-4i) + (5i)(-9) + (5i)(-4i)
- 27 + 12i - 45i - 20i²
Since i² = -1, we substitute to simplify:
- 27 + 12i - 45i + 20 = 47 - 33i

Denominator:

Again, expand using FOIL:
- (-9)(-9) + (-9)(-4i) + (4i)(-9) + (4i)(-4i)
- 81 + 36i - 36i - 16i²
Substitute i² = -1:
- 81 + 16 = 97

3. Final Result

We now have:

(47 - 33i) / 97

Finally, express the result in standard complex number form (a + bi):

(47/97) - (33/97)i

Conclusion

By multiplying by the conjugate of the denominator and simplifying, we successfully divided the complex numbers and arrived at our final answer: (47/97) - (33/97)i.