audio read-through Arithmetic with Complex Numbers

Whenever you get a new mathematical object (like complex numbers), you need to develop tools to work with the new object.

Arithmetic with complex numbers was introduced in a prior lesson:   Arithmetic with Complex Numbers in the Algebra II materials. Quickly read through this earlier section and do plenty of the online exercises to make sure that you understand all the concepts there.

For your convenience, this current lesson gives a quick summary of important concepts from the earlier lesson, and also discusses two new topics: the complex conjugate, and division of complex numbers.

Arithmetic With Complex Numbers: Adding, Subtracting, Multiplying, Dividing, Complex Conjugate, More

Let $\,a\,,$ $\,b\,,$ $\,c\,,$ and $\ d\ $ be real numbers, and let $\,i:=\sqrt{-1}\,.$

Addition and Subtraction

$$ \begin{align} &(a+bi)\, + \, (c+di)\cr &\qquad =\ (a+c) + (b+d)i \end{align} $$

$$ \begin{align} &(a+bi)\, - \, (c+di)\cr &\qquad =\ (a-c) + (b-d)i \end{align} $$

To add complex numbers:

The same idea holds for differences.

You can also just think of combining like terms—the real number terms and the $\,i\,$ terms.

Multiplication

$$ \begin{align} &(a+bi)(c+di)\cr &\qquad =\ (ac-bd) + (ad+bc)i \end{align} $$

You certainly don't want to memorize this formula. Just use FOIL to multiply things out!

$$ \begin{align} &\cssId{s22}{(a+bi)(c+di)}\cr &\ \cssId{s23}{= \overbrace{\ ac\ \strut}^{\text{F}} + \overbrace{\ adi\ \strut}^{\text{O}} + \overbrace{\ bci\ \strut}^{\text{I}} + \overbrace{\ bdi^2\ \strut}^{\text{L}}}\cr &\qquad \cssId{s24}{\text{(using FOIL)}}\cr\cr &\ \cssId{s25}{= (ac + bdi^2) + (adi + bci)}\cr &\qquad \cssId{s26}{\text{(re-group)}}\cr\cr &\ \cssId{s27}{= (ac - bd) + (ad + bc)i}\cr &\qquad \cssId{s28}{(i^2 = -1 \text{ and factoring})} \end{align} $$

Multiplication by a Real Number

A special case of multiplication (above) is multiplication by a real number.

For all real numbers $\,k\ $:

$$\cssId{s32}{k(a + bi) = ka + kbi}$$

Notice that both the real and imaginary parts get multiplied by $\,k\,.$

The Complex Conjugate

By definition, the complex conjugate of $\,a$ $+\,bi\,$ is $\,a$ $-\,bi\,$.

(Hover over the yellow with your cursor.) Note that the real part stays the same; you take the opposite of the imaginary part.

The complex conjugate of $\,a + bi\,$ is notated by $\overline{a + bi}\,,$ so:

$$ \cssId{s40}{\overline{a + bi} = a - bi} $$

When a complex number is multiplied by its conjugate, a nonnegative real number results. Letting $\,z = a + bi\,$:

$$ \begin{align} \cssId{s43}{z\overline{z}}\ &\cssId{s44}{= (a+bi)(a-bi)}\cr\cr &\cssId{s45}{= a^2 - abi + abi - b^2i^2}\cr\cr &\cssId{s46}{= a^2 + b^2} \end{align} $$

Division

For division, we need the additional requirement that $\,c\,$ and $\,d\,$ are not both zero.

$$ \begin{align} &\cssId{s49}{\frac{a+bi}{c+di}} \cr &\qquad \cssId{s50}{=\ \left(\frac{ac + bd}{c^2 + d^2}\right) + \left(\frac{bc - ad}{c^2 + d^2}\right)i} \end{align} $$

Again, you certainly don't want to memorize this formula. Just multiply by $\,1\,$ in the form of the complex conjugate of the denominator over itself, and then simplify:

$$ \begin{align} &\cssId{s53}{\frac{a+bi}{c+di}}\cr &\ \cssId{s54}{= \frac{a+bi}{c+di}\cdot\overbrace{\frac{c-di}{c-di}}^{= 1}}\cr &\quad \cssId{s55}{\text{(multiply by $\,1\,$)}}\cr\cr &\ \cssId{s56}{= \frac{(a+bi)(c-di)}{(c+di)(c-di)}}\cr &\quad \cssId{s57}{\text{(multiply across)}}\cr\cr &\cssId{s58}{= \frac{(ac+ bd) + (bc-ad)i}{c^2 + d^2}}\cr &\quad \cssId{s59}{\text{(multiply out numerator and denominator)}}\cr\cr &\ \cssId{s60}{= \left(\frac{ac + bd}{c^2 + d^2}\right) + \left(\frac{bc - ad}{c^2 + d^2}\right)i} \end{align} $$

Naming Conventions for Complex Numbers

Two common variable names for complex numbers are $\,z\,$ and $\,w\,.$

Examples

Let $z = 2 - 3i\,$ and $\,w = -5 + 7i\,.$

Then:

$\displaystyle \begin{align} \cssId{s66}{z + w}\ &\cssId{s67}{= (2-3i) + (-5 + 7i)}\cr &\cssId{s68}{=\ (2-5) + (-3 + 7)i}\cr &\cssId{s69}{=\ -3 + 4i}\cr\cr \cssId{s70}{z - w}\ &\cssId{s71}{= (2-3i) - (-5 + 7i)}\cr &\cssId{s72}{=\ (2+5) + (-3 - 7)i}\cr &\cssId{s73}{=\ 7 - 10i}\cr\cr \cssId{s74}{zw}\ &\cssId{s75}{= (2 - 3i)(-5 + 7i)}\cr &\cssId{s76}{=\ -10 + 14i + 15i -21i^2}\cr &\cssId{s77}{=\ (-10 + 21) + 29i}\cr &\cssId{s78}{=\ 11 + 29i}\cr\cr\cr \cssId{s79}{\frac{z}{w}}\ &\cssId{s80}{= \frac{2-3i}{-5 + 7i}\cdot\frac{-5-7i}{-5-7i}}\cr\cr &\cssId{s81}{=\ \frac{(2-3i)(-5-7i)}{25 + 49}}\cr\cr &\cssId{s82}{= \frac{(-10-21) + (15-14)i}{74}}\cr\cr &\cssId{s83}{=\ \frac{-31 + i}{74}}\cr\cr &\cssId{s84}{=\ -\frac{31}{74} + \frac{1}{74}i}\cr\cr\cr \cssId{s85}{\overline{z}}\ &\cssId{s86}{= \overline{2-3i}}\cr &\cssId{s87}{=\ 2 + 3i}\cr\cr \cssId{s88}{\overline{w}}\ &\cssId{s89}{= \overline{-5 + 7i}}\cr &\cssId{s90}{=\ -5 - 7i}\cr\cr \cssId{s91}{z\overline{z}}\ &\cssId{s92}{= (2-3i)(2+3i)}\cr &\cssId{s93}{=\ 4 + 6i - 6i - 9i^2}\cr &\cssId{s94}{=\ 4 + 9}\cr &\cssId{s95}{=\ 13}\cr\cr \cssId{s96}{-4z + \overline{2w}}\ &\cssId{s97}{= -4(2 - 3i) + \overline{2(-5 + 7i)}}\cr &\cssId{s98}{=\ -8 + 12i + (\overline{-10 + 14i})}\cr &\cssId{s99}{=\ -8 + 12i + (- 10 - 14i)}\cr &\cssId{s100}{=\ -18 - 2i} \end{align} $

Concept Practice