## Example 1.1.9 (b): Algebra with complex numbers |

Simplify |

Here the problem is to simplify the denominator *(3i-4)*. The trick
is to multiply top and bottom by 1, expressed as *(-3i-4)/(-3i-4)*.
The expression *(-3i-4)* is what will later be called the
*conjugate* of *(3i-4)*. That trick uses the general fact that

(a + ib) (a - ib) = a^{2}- iab + iab - i^{2}b^{2}= a^{2}+ b^{2}

where *(a + ib)* and *(a - ib)* are again called
conjugate (as defined later). Anyway, using this trick the original expression
simplifies considerably: