Properties of Addition of Complex Numbers

Operation Basics

Addition and scalar multiplication operations on complex numbers have interesting properties, similar to those of real numbers. These properties help us in performing calculations.

Let $z_1$, $z_2$, and $z_3$ be any complex numbers, and let $c$ and $d$ be any scalars (real numbers).

Addition Properties

Commutativity

The order of addition does not matter; the result remains the same.

Example: $(2+i) + (1-3i) = (1-3i) + (2+i) = 3-2i$

Addition Associativity

When adding three complex numbers, the grouping of the addition does not affect the result.

Identity Element

There exists a complex number $0 = 0 + 0i$ (zero) such that when added to any complex number $z_1$, the result is $z_1$ itself.

Inverse Element

Every complex number $z_1 = x + iy$ has an additive inverse (opposite), denoted by $-z_1 = -x - iy$, such that their sum is the zero element (0).

Example:

If $z_1 = 5-2i$, then $-z_1 = -5+2i$.

Then $(5-2i) + (-5+2i) = (5-5) + i(-2+2) = 0 + 0i = 0$.

Scalar Multiplication

Multiplication Associativity

The grouping of scalar multiplication does not affect the result.

Scalar Distributivity

A scalar can be distributed over the addition of scalars.

Complex Distributivity

A scalar can be distributed over the addition of complex numbers.

Scalar Identity

Multiplying a complex number by the scalar 1 does not change the complex number.

Zero Scalar

Multiplying a complex number by the scalar 0 results in the complex number zero.

Property Applications

These properties can be used to simplify or prove expressions involving complex numbers.

Application Example

Show that for any complex number $z$, $4z + (-4)z = 0$ holds.

Solution:

We can use the distributivity of scalar over scalar addition (property f) and the multiplication by zero scalar property (property i).

Thus, it is proven that $4z + (-4)z = 0$.

Exercise

Using the properties above, prove that $3z - \frac{1}{2}(2z) = 2z$ for any complex number $z$.

Answer Key

Thus, it is proven that $3z - \frac{1}{2}(2z) = 2z$.