Manipulating Complex Numbers

Now that we understand how to work with the powers of $i$, we can learn the various arithmetic operations involving complex numbers.

Addition and subtraction

When adding or subtracting two complex numbers, we simply add/subtract their respective real and imaginary components: $$ \begin{equation}\begin{aligned} z_1&=a+bi\\ z_2&=c+di\\ z_1+z_2&=(a+bi)+(c+di)\\ &=(a+c)+(bi+di)\\ &=(a+c)+(b+d)i\\ z_1-z_2&=(a+bi)-(c+di)\\ &=(a-c)+(bi-di)\\ &=(a-c)+(b-d)i\\ \end{aligned}\end{equation} $$

Examples

• Find the sum of $z_1=3+2i$ and $z_2=4-5i$.

$$ \begin{equation}\begin{aligned} z_1+z_2&=(3+2i)+(4-5i)\\ &=(3+4)+(2+-5)i\\ &=7-3i\\ \end{aligned}\end{equation} $$

• Find the difference between $z_3=7+2i$ and $z_4=2+i$.

$$ \begin{equation}\begin{aligned} z_3-z_4&=(7+2i)-(2+i)\\ &=(7-2)+(2-1)i\\ &=5+i\\ \end{aligned}\end{equation} $$

Multiplication

When multiplying two complex numbers, the result is achieved in the same way as multiplying two binomials - we multiply each algebraic term in the first bracket by each term in the second bracket: $$ \begin{equation}\begin{aligned} (a+bi)(c+di)&=(a)(c)+(a)(di)+(bi)(c)+(bi)(di)\\ &=ac+adi+bci+bdi^2\\ &=ac+adi+bci+bd(-1)\\ &=(ac-bd)+(ad+bc)i\\ \end{aligned}\end{equation} $$ $$ \begin{equation}\begin{aligned} (3+2i)(4-5i)&=(3)(4)+(3)(-5i) +(2i)(4)+(2i)(-5i)\\ &=12-15i+8i-10i^2\\ &=12-15i+8i-10(-1)\\ &=(12+10)+(-15+8)i\\ &=22-7i\\ \end{aligned}\end{equation} $$

Conjugate of a complex number

Given the complex number $z=a+bi$, the conjugate is $$ \begin{equation}\begin{aligned} \bar{z}=a-bi\\ \end{aligned}\end{equation} $$

Thus for $3+4i$, the conjugate is $3-4i$ and for $5i$, the conjugate is $-5i$. We simply negate the imaginary component of the complex number.

Challenge

Write the conjugates of the following complex numbers:

• $4-i$
• $-7i$
• $9i-3$

Realization of complex numbers

We do not divide complex numbers. Instead, we realize the divisor. Realization of a complex number is where we convert it to a real number. That is, we get rid of the imaginary component. The easiest way to realize a complex number is to multiply it by its conjugate.

Examples

• Simplify the quotient $\frac{1+2i}{4i}$

$$ \begin{equation}\begin{aligned} \frac{1+2i}{4i}&=\frac{1+2i}{4i}\times \frac{-4i}{-4i}\\ &=\frac{(1+2i)\times -4i}{4i\times -4i}\\ &=\frac{-4i-8i^2}{16}\\ &=\frac{8-4i}{16}\\ &=\frac{2-i}{4}\\ \end{aligned}\end{equation} $$

• Simplify the ratio $\frac{1-3i}{2-i}$

$$ \begin{equation}\begin{aligned} \frac{1-3i}{2-i}&=\frac{1-3i}{2-i}\times \frac{2+i}{2+i}\\ &=\frac{(1-3i)(2+i)}{2^2-i^2}\\ &=\frac{2+i-6i-3i^2}{2-(-1)}\\ &=\frac{2+3+i-6i}{2-(-1)}\\ &=\frac{5-5i}{3}\\ \end{aligned}\end{equation} $$

Note how for each example above, the denominator moves from being a complex number to being a real number.

Polar form of a complex number

$$ \begin{equation}\begin{aligned} z=r(\cos{\theta}+\sin{\theta})\\ \end{aligned}\end{equation} $$

Exponential form of a complex number

$$ \begin{equation}\begin{aligned} z=re^{i\theta}\\ \end{aligned}\end{equation} $$