Addition of Complex Numbers

Adding the Real and Imaginary Parts Separately

Start with two complex numbers:

z_1 = x_1 + iy_1

To form $z_1 + z_2$ , add the real parts to each other and do the same with the imaginary parts.

z_1 + z_2 = (x_1 + x_2) + i(y_1 + y_2)

Adding Two Specific Complex Numbers

Let $z_1 = 2 + 3i$ and $z_2 = 1 - i$ .

The real part of $z_1$ is $2$ , the real part of $z_2$ is $1$ .
The imaginary part of $z_1$ is $3$ , the imaginary part of $z_2$ is $-1$ .

Then their sum is:

\begin{aligned} z_1 + z_2 &= (2 + 1) + i(3 + (-1)) \\ &= 3 + i(2) \\ &= 3 + 2i \end{aligned}

Adding on the Complex Plane

Using the parallelogram rule, the addition of complex numbers can be viewed geometrically on the complex plane. If we represent $z_1$ and $z_2$ as vectors (arrows) from the origin $(0, 0)$ , then their sum, $z_1 + z_2$ , is the diagonal vector of the parallelogram formed by $z_1$ and $z_2$ .

Geometric Addition of Complex Numbers

Visualization of the sum

z_1 = 2+3i

and

z_2 = 1-i

using the parallelogram rule.

Scaling Negating and Subtracting

The same separation of real and imaginary parts also applies to scalar multiplication, negation, and subtraction. Multiplying by a real number scales both parts by that number, negation flips both signs, and subtraction combines negation with addition. Since each rule acts on the two parts independently, you never have to mix them.

Scalar Multiplication

When a complex number $z = x + iy$ is multiplied by a real scalar $c$ , the scalar multiplies both the real and imaginary parts.

cz = c(x + iy) = cx + i(cy)

Geometrically, this scales the vector $z$ by a factor of $c$ . If $c$ is negative, the vector's direction is reversed.

Negative of a Complex Number

The negative of $z = x + iy$ is $-z$ . This is the same as scalar multiplication by $c = -1$ .

\begin{aligned} -z &= -(x + iy) \\ &= -x + i(-y) \\ &= -x - iy \end{aligned}

Geometrically, $-z$ is a vector with the same length as $z$ but pointing in the opposite direction ( $180^\circ$ rotation).

Subtraction of Two Complex Numbers

Subtracting $z_2$ from $z_1$ ( $z_1 - z_2$ ) is the same as adding $z_1$ to the negative of $z_2$ ( $z_1 + (-z_2)$ ).

\begin{aligned} &z_1 - z_2 = z_1 + (-z_2) \\ &= (x_1 + (-x_2)) + i(y_1 + (-y_2)) \\ &= (x_1 - x_2) + i(y_1 - y_2) \end{aligned}

So, subtract the real parts and subtract the imaginary parts.

Geometrically, $z_1 - z_2$ is the vector from the tip of $z_2$ to the tip of $z_1$ .

Combining Addition Subtraction and Scaling

Use the following two numbers:

z_1 = 2 + \frac{1}{2}i

Calculate three expressions by handling the real and imaginary parts separately:

$2z_1$ (Scalar Multiplication):

$\begin{aligned} 2z_1 &= 2(2 + \frac{1}{2}i) \\ &= 2(2) + i(2 \times \frac{1}{2}) \\ &= 4 + i \end{aligned}$
$z_1 + 3z_2$ (Addition and Scalar Multiplication):
$z_1 + 3z_2 = (2 + \frac{1}{2}i) + 3(-3 + \sqrt{2}i)$
$2z_1 - z_2$ (Subtraction and Scalar Multiplication):
$2z_1 - z_2 = (4 + i) - (-3 + \sqrt{2}i)$

Exercise

This set practises addition and subtraction in both forms, then asks you to place the results in the complex plane. Compute each result first, and only then draw the three vectors.

Given $z_1=1+2i$ and $z_2=3-i$ , determine:

$z_1 + z_2$
$z_1 - z_2$
For $z_3=z_1+z_2$ , draw $z_1$ , $z_2$ , and $z_3$ on the complex plane.

Solution

Add and subtract the corresponding parts, then read the sum as a diagonal on the complex plane.

Add the real parts and the imaginary parts separately:

$\begin{aligned} z_1+z_2 &=(1+3)+i(2+(-1)) \\ &=4+i \end{aligned}$
Subtract the corresponding parts:

$\begin{aligned} z_1-z_2 &=(1-3)+i(2-(-1)) \\ &=-2+3i \end{aligned}$
The vectors $z_1$ and $z_2$ form two sides of a parallelogram. Its diagonal ends at $z_3=z_1+z_2=4+i$ :

Addition of Complex Numbers
The vectors $z_1$ and $z_2$ form the sides of the parallelogram. $z_3=z_1+z_2$ is its diagonal.