Complex Conjugates
A complex conjugate is defined as a number with equal magnitudes of real and imaginary parts, but opposite in sign.
Let's have a look at some conjugate pairings
| $6+\frac{1}{4}i$6+14i | $6-\frac{1}{4}i$6−14i |
| $-15-3i$−15−3i | $-15+3i$−15+3i |
| $1-i$1−i | $1+i$1+i |
| $26i$26i | $-26i$−26i |
| $7$7 | $7$7 |
There is a special symbol we use to denote the conjugate of $z$z. It is called zed bar, and looks like this 
.
We have dealt with a similar concept before when we were studying surds. We underwent a process called rationalising the denominator, you can refresh here if you need to.
Just like this process, we use complex conjugates to help us define a process for division of complex numbers.
Instead of a formal division process, we use conjugates to turn the operation of division into an operation of multiplication.
Let's look at this question as an example. $\frac{2+3i}{4+5i}$2+3i4+5i
So, to tackle the division we
- multiply by a fraction that is equivalent to 1
- use the conjugate of the denominator to construct this fraction
- multiply the binomials on the numerator and denominators and simplify
- the denominator will always simplify to $a^2-b^2$a2−b2 (for the expansion $\left(a+bi\right)\left(a-bi\right)$(a+bi)(a−bi))
Let's finish off this question $\frac{2+3i}{4+5i}$2+3i4+5i
| $\frac{2+3i}{4+5i}$2+3i4+5i | $=$= | $\frac{2+3i}{4+5i}\times\frac{4-5i}{4-5i}$2+3i4+5i×4−5i4−5i |
| $=$= | $\frac{\left(2+3i\right)\left(4-5i\right)}{\left(4+5i\right)\left(4-5i\right)}$(2+3i)(4−5i)(4+5i)(4−5i) | |
| $=$= | $\frac{8+12i-10i-15i^2}{16+20i-20i-25i^2}$8+12i−10i−15i216+20i−20i−25i2 | |
| $=$= | $\frac{8+2i-15\left(-1\right)}{16-25\left(-1\right)}$8+2i−15(−1)16−25(−1) | |
| $=$= | $\frac{8+2i+15}{16+25}$8+2i+1516+25 | |
| $=$= | $\frac{23+2i}{41}$23+2i41 | |
| $=$= | $\frac{23}{41}+\frac{2i}{41}$2341+2i41 |
This last step may or may not be necessary. Sometimes it's easier to do further work when the real and imaginary parts are separate.
Here are some more examples
Example 1
| $\frac{4}{i}$4i | $=$= | $\frac{4}{i}\times\frac{-i}{-i}$4i×−i−i |
| $=$= | $\frac{-4i}{-i^2}$−4i−i2 | |
| $=$= | $\frac{-4i}{1}$−4i1 | |
| $=$= | $-4i$−4i |
Example 2
| $\frac{1-i}{1+i}$1−i1+i | $=$= | $\frac{1-i}{1+i}\times\frac{1-i}{1-i}$1−i1+i×1−i1−i |
| $=$= | $\frac{\left(1-i\right)\left(1-i\right)}{\left(1+i\right)\left(1-i\right)}$(1−i)(1−i)(1+i)(1−i) | |
| $=$= | $\frac{1-2i+i^2}{1+1}$1−2i+i21+1 | |
| $=$= | $\frac{1-2i-1}{2}$1−2i−12 | |
| $=$= | $\frac{-2i}{2}$−2i2 | |
| $=$= | $-i$−i |
Example 3
| $\frac{3}{6+i}+\frac{2}{3-2i}$36+i+23−2i | $=$= | $\frac{3}{6+i}\times\frac{6-i}{6-i}+\frac{2}{3-2i}\times\frac{3+2i}{3+2i}$36+i×6−i6−i+23−2i×3+2i3+2i |
| $=$= | $\frac{3\left(6-i\right)}{37}+\frac{2\left(3+2i\right)}{13}$3(6−i)37+2(3+2i)13 | |
| $=$= | $\frac{18-3i}{37}+\frac{6+4i}{13}$18−3i37+6+4i13 | |
| $=$= | $\frac{456+109i}{481}$456+109i481 |
Try this for yourself before you check out the solutions.
Verify which of the following hold true, check with $z=1+i$z=1+i and $w=2−2i$w=2−2i.
Then prove for $z=a+bi$z=a+bi and $w=c+di$w=c+di
- $\overline{z-w}=\overline{z}-\overline{w}$z−w=z−w
- $\overline{zw}=\overline{z}\times\overline{w}$zw=z×w
- $\overline{z^n}=\left(\overline{z}\right)^n$zn=(z)n use (b) to help here
- $\overline{\left[\frac{z}{w}\right]}=\frac{\overline{z}}{\overline{w}}$[zw]=zw
- What is $z+\overline{z}$z+z ? Generalise
-
What is $z-\overline{z}$z−z ? Generalise
(go here for the solution)
More Worked Examples
QUESTION 1
Find the value of $\frac{4+6i}{1+i}$4+6i1+i.
QUESTION 2
Find the value of $\frac{4+7i}{2+i}$4+7i2+i.
QUESTION 3
Find the value of $\frac{2-3i}{2+3i}$2−3i2+3i.