Addition of Complex Numbers

Addition of Two Complex Numbers

How do you add two complex numbers?

Suppose we have two complex numbers:

z_1 = x_1 + iy_1

z_2 = x_2 + iy_2

To add them ( $z_1 + z_2$ ), simply add the real parts together and the imaginary parts together.

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

Addition Example

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:

z_1 + z_2 = (2 + 1) + i(3 + (-1)) = 3 + i(2) = 3 + 2i

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.

Besides addition, other operations work similarly:

Scalar Multiplication

Multiplying a complex number $z = x + iy$ by a real number (scalar) $c$ is straightforward. Just multiply $c$ into 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 $c = -1$ .

-z = -(x + iy) = -x + i(-y) = -x - iy

Geometrically, $-z$ is a vector with the same length as $z$ but pointing in the opposite direction (180 degrees 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 $z_2$ ( $z_1 + (-z_2)$ ).

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)

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 $z_1$ .

Example of Combined Operations

Suppose we have:

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

z_2 = -3 + \sqrt{2}i

Let's calculate some operations:

$2z_1$ (Scalar Multiplication):

$2z_1 = 2(2 + \frac{1}{2}i) = 2(2) + i(2 \times \frac{1}{2}) = 4 + i$
$z_1 + 3z_2$ (Addition and Scalar Multiplication):

$z_1 + 3z_2 = (2 + \frac{1}{2}i) + 3(-3 + \sqrt{2}i)$
$= (2 + \frac{1}{2}i) + (3(-3) + i(3\sqrt{2}))$
$= (2 + \frac{1}{2}i) + (-9 + 3\sqrt{2}i)$
$= (2 - 9) + i(\frac{1}{2} + 3\sqrt{2})$
$= -7 + i(\frac{1}{2} + 3\sqrt{2})$
$2z_1 - z_2$ (Subtraction and Scalar Multiplication):

$2z_1 - z_2 = (4 + i) - (-3 + \sqrt{2}i)$
$= (4 - (-3)) + i(1 - \sqrt{2})$
$= (4 + 3) + i(1 - \sqrt{2})$
$= 7 + i(1 - \sqrt{2})$

Exercise

If $z_1 = 1 + 2i$ and $z_2 = 3 - i$ . Determine:

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

Answer Key

$z_1 + z_2 = (1+3) + i(2+(-1)) = 4 + i$
$z_1 - z_2 = (1-3) + i(2-(-1)) = -2 + i(3) = -2 + 3i$
Visualization of $z_1$ , $z_2$ , and $z_3 = z_1 + z_2$ on the complex plane using the parallelogram rule:

Addition of Complex Numbers
Visualization of $z_1$ , $z_2$ , and $z_3 = z_1 + z_2$ on the complex plane using the parallelogram rule.

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