Addition and Multiplication of Complex Numbers

On The Set of Complex Numbers page we defined the set of complex numbers, $\mathbb{C}$, as follows:

(1)

\begin{align} \quad \mathbb{C} = \{ a + bi : a, b \in \mathbb{R}, \: \mathrm{and}, \: i = \sqrt{-1} \} \end{align}

We will now define two operations on $\mathbb{C}$: addition and multiplication.

Definition: If $z = a + bi, w = c + di \in \mathbb{C}$ then the operation of Addition denoted $+$ between $z$ and $w$ is defined to be $z + w = (a + c) + (b + d)i$ and $z + w$ is called the Sum of $z$ with $w$ .

In other words, if $z, w \in \mathbb{C}$ then the sum of $z$ with $w$ is the new complex number $z + w$ whose real part is the sum of the real parts of $z$ and $w$, and whose imaginary part is the sum of the imaginary parts of $z$ and $w$.

For example:

(2)

\begin{align} \quad (2 + 4i) + (3 - 2i) = (2 + 3) + (4 - 2)i = 5 + 2i \end{align}

Graphically, we can add complex numbers similarly to how we graphically add vectors in $\mathbb{R}^2$. If we represent two complex numbers $z$ and $w$ as position vectors in the complex plane, then their sum $z + w$ will be the position vector whose initial point is the origin and whose terminal point is $a + c$ units in the direction of the real axis and $b + d$ units in the direction of the imaginary axis:

It's not hard to see that complex number addition is commutative ($z + w = w + z$) and associative ($z + (w + v) = (z + w) + v$) for all $z, w, v \in \mathbb{C}$) by the commutativity and associativity of the real numbers.

Definition: If $z = a + bi, w = c + di \in \mathbb{C}$ then the operation of Multiplication denoted $\cdot$ between $z$ and $w$ is defined to be $z \cdot w = (ac - bd) + (ad + bc)i$ and $z \cdot w$ is called the Product of $z$ with $w$.

The "$\cdot$" is often dropped with regards to multiplication and the product of $z$ with $w$ can simply be written as $zw$.

The product of two complex numbers can easily be calculated by expanding and simplifying the product of two complex numbers as the product of two binomials. For example:

(3)

\begin{align} \quad (1 + 2i)(3 + 4i) = 3 + 4i + 6i + 8i^2 = 3 + 10i + 8i^2 = (3 - 8) + 10i = -5 + 10i \end{align}

Notice that if $z = a + bi \in \mathbb{C}$ and $k \in \mathbb{R}$ then:

(4)

\begin{align} \quad kz = k(a + bi) = ka + kbi \end{align}

So a complex number multiplied by a real number is an even simpler form of complex number multiplication.

Once again, it's not too hard to verify that complex number multiplication is both commutative and associative.