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2.07 Complex conjugates and moduli of complex numbers


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$614i
$-15-3i$153i $-15+3i$15+3i
$1-i$1i $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 radicals.  We underwent a process called rationalizing the denominator.

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 rationalize the denominator of the fraction: $\frac{2+3i}{4+5i}$2+3i4+5i


So, to tackle the division we 

  • multiply by a fraction that is equivalent to $1$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 $a^2-b^2$a2b2 (for the distribution $\left(a+bi\right)\left(a-bi\right)$(a+bi)(abi))


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×45i45i
  $=$= $\frac{\left(2+3i\right)\left(4-5i\right)}{\left(4+5i\right)\left(4-5i\right)}$(2+3i)(45i)(4+5i)(45i)
  $=$= $\frac{8+12i-10i-15i^2}{16+20i-20i-25i^2}$8+12i10i15i216+20i20i25i2
  $=$= $\frac{8+2i-15\left(-1\right)}{16-25\left(-1\right)}$8+2i15(1)1625(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.  


Worked examples

Question 1

Rationalize: $\frac{4}{i}$4i

$\frac{4}{i}$4i $=$= $\frac{4}{i}\times\frac{-i}{-i}$4i×ii
  $=$= $\frac{-4i}{-i^2}$4ii2
  $=$= $\frac{-4i}{1}$4i1
  $=$= $-4i$4i


Question 2

Rationalize: $\frac{1-i}{1+i}$1i1+i

$\frac{1-i}{1+i}$1i1+i $=$= $\frac{1-i}{1+i}\times\frac{1-i}{1-i}$1i1+i×1i1i
  $=$= $\frac{\left(1-i\right)\left(1-i\right)}{\left(1+i\right)\left(1-i\right)}$(1i)(1i)(1+i)(1i)
  $=$= $\frac{1-2i+i^2}{1+1}$12i+i21+1
  $=$= $\frac{1-2i-1}{2}$12i12
  $=$= $\frac{-2i}{2}$2i2
  $=$= $-i$i


Question 3

Simplify: $\frac{3}{6+i}+\frac{2}{3-2i}$36+i+232i

$\frac{3}{6+i}+\frac{2}{3-2i}$36+i+232i $=$= $\frac{3}{6+i}\times\frac{6-i}{6-i}+\frac{2}{3-2i}\times\frac{3+2i}{3+2i}$36+i×6i6i+232i×3+2i3+2i
  $=$= $\frac{3\left(6-i\right)}{37}+\frac{2\left(3+2i\right)}{13}$3(6i)37+2(3+2i)13
  $=$= $\frac{18-3i}{37}+\frac{6+4i}{13}$183i37+6+4i13
  $=$= $\frac{456+109i}{481}$456+109i481


Practice questions


Find the value of $\frac{4+6i}{1+i}$4+6i1+i.


Find the value of $\frac{4+7i}{2+i}$4+7i2+i.


Find the value of $\frac{2-3i}{2+3i}$23i2+3i.

Complex moduli

The modulus of a complex number $x+yi$x+yi is the distance of the vector created from the origin to the point $\left(x,y\right)$(x,y). In other words, it is the size of the number.

The modulus is denoted using many different notations in various texts, websites, countries and schools. Here are a list of various notations.

  • the letter $r$r
  • $modz$modz
  • $|z|$|z|
  • $|x+iy|$|x+iy|


 In this diagram, the point $P$P has been plotted.  It corresponds to the complex number $3+4i$3+4i.


We can see the horizontal distance is $3$3, (the $x$x value) and the vertical distance is $4$4, (the $y$y value).  The modulus is the length of the hypotenuse, (which is the length of the vector $P$P)

This distance can be found using the Pythagorean theorem.

$r^2$r2 $=$= $x^2+y^2$x2+y2
$r$r $=$= $\sqrt{x^2+y^2}$x2+y2
  $=$= $\sqrt{3^2+4^2}$32+42
  $=$= $5$5





Worked example

Question 7

If $z=-2+2i$z=2+2i and $w=4-i$w=4i then find

a) $|z|$|z|

$|z|$|z| $=$= $|-2+2i|$|2+2i|
  $=$= $\sqrt{(-2^2)+2^2}$(22)+22
  $=$= $\sqrt{8}$8
  $=$= $2\sqrt{2}$22


b) $|z|^2$|z|2

$|z|^2$|z|2 $=$= $|-2+2i|^2$|2+2i|2
  $=$= $\left(\sqrt{8}\right)^2$(8)2
  $=$= $8$8


c) $|z^2|$|z2|

$|z^2|$|z2| $=$= $\left|\left(-2+2i\right)^2\right|$|(2+2i)2|
$=$= $\left|\left(-2+2i\right)\left(-2+2i\right)\right|$|(2+2i)(2+2i)|
$=$= $\left|-4-4i-4i+4i^2\right|$|44i4i+4i2|
$=$= $\left|-8-8i\right|$|88i|
$=$= $\sqrt{-8^2+-8^2}$82+82
$=$= $\sqrt{64+64}$64+64
$=$= $\sqrt{128}$128
$=$= $8\sqrt{2}$82


d) $|z|-|w|$|z||w|

From part (a), we know that $|z|=2\sqrt{2}$|z|=22.

$|w|$|w| $=$= $\sqrt{4^2+-1^2}$42+12
  $=$= $\sqrt{16+1}$16+1
  $=$= $\sqrt{17}$17
$|z|-|w|$|z||w| $=$= $2\sqrt{2}-\sqrt{17}$2217

which is approximately $1.29$1.29.


e) $|z-w|$|zw|

$|z-w|$|zw| $=$= $\left|\left(-2-2i\right)-\left(4-i\right)\right|$|(22i)(4i)|
$=$= $\left|-2-4-2i+i\right|$|242i+i|
$=$= $\left|-6-i\right|$|6i|
$=$= $\sqrt{(-6)^2+(-1)^2}$(6)2+(1)2
$=$= $\sqrt{37}$37

Practice questions


Find the modulus of $-8+3i$8+3i.


If $z=12-16i$z=1216i and $w=3-4i$w=34i, find:

  1. $\left|z\right|$|z|

  2. $\left|w\right|$|w|

  3. $\left|z+w\right|$|z+w|

  4. $\left|z\right|+\left|w\right|$|z|+|w|



Find the conjugate of a complex number; use the conjugate to find the absolute value (modulus) and quotient of complex numbers.

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