Properties of Addition of Complex Numbers: Complex Number - Mathematics (Grade 11)

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.

z_1 + z_2 = z_2 + z_1

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.

(z_1 + z_2) + z_3 = z_1 + (z_2 + z_3)

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.

z_1 + 0 = z_1

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 ( $0$ ).

z_1 + (-z_1) = 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.

c(dz_1) = (cd)z_1

Scalar Distributivity

A scalar can be distributed over the addition of scalars.

(c + d)z_1 = cz_1 + dz_1

Complex Distributivity

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

c(z_1 + z_2) = cz_1 + cz_2

Scalar Identity

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

1 z_1 = z_1

Zero Scalar

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

0 z_1 = 0

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).

4z + (-4)z = (4 + (-4))z \quad \text{(Distributive Property)}

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

3z - \frac{1}{2}(2z) = 3z + (-\frac{1}{2})(2z) \quad \text{(Definition of subtraction)}

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