Addition of Complex Numbers

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

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:

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 $z_2$.

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 parts by the same factor.

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

For a zero scalar, both components become zero. The zero vector has no direction.

Negative of a Complex Number

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

For $z\ne0$, 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 $z_2$ ($z_1 + (-z_2)$).

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:

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

1. $2z_1$ (Scalar Multiplication):

   $$\begin{aligned} 2z_1 &= 2(2 + \frac{1}{2}i) \\ &= 2(2) + i(2 \times \frac{1}{2}) \\ &= 4 + i \end{aligned}$$

2. $z_1 + 3z_2$ (Addition and Scalar Multiplication):

   $$\begin{aligned}z_1+3z_2&=(2+\frac12i)+3(-3+\sqrt2i)\\&=(2-9)+i(\frac12+3\sqrt2)\\&=-7+i(\frac12+3\sqrt2)\end{aligned}$$

3. $2z_1 - z_2$ (Subtraction and Scalar Multiplication):

   $$\begin{aligned}2z_1-z_2&=(4+i)-(-3+\sqrt2i)\\&=(4-(-3))+i(1-\sqrt2)\\&=7+i(1-\sqrt2)\end{aligned}$$

Exercise

Calculate the sum and difference, then use the components of the sum as the endpoint coordinates of the diagonal. The subtraction sign applies to the entire number being subtracted, including its imaginary component.

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

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

Solution

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

1. Add the real parts and the imaginary parts separately:

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

2. Subtract the corresponding parts:

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

3. The vectors $z_1$ and $z_2$ form two sides of a parallelogram. Its diagonal ends at $z_3=z_1+z_2=4+i$:

Published: May 1, 2025. Updated: Sep 12, 2026.