Properties of Addition of Complex Numbers

Basic Operation Properties

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).

Commutativity

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

z_1 + z_2 = z_2 + z_1

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

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)

Additive Identity (Zero 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

Additive Inverse (Opposite)

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

Associativity of Scalar Multiplication

The grouping of scalar multiplication does not affect the result.

c(dz_1) = (cd)z_1

Distributivity of Scalar over Scalar Addition

A scalar can be distributed over the addition of scalars.

(c + d)z_1 = cz_1 + dz_1

Distributivity of Scalar over Complex Addition

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

c(z_1 + z_2) = cz_1 + cz_2

Scalar Multiplicative Identity

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

1 z_1 = z_1

Multiplication by Zero Scalar

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

0 z_1 = 0

Using the Operation Properties

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

Example of Using Properties

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)}

= (0)z \quad \text{(Scalar addition)}

= 0 \quad \text{(Multiplication by Zero Scalar 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)}

= 3z + ((-\frac{1}{2}) \times 2)z \quad \text{(Associativity of Scalar Multiplication)}

= 3z + (-1)z \quad \text{(Scalar multiplication)}

= (3 + (-1))z \quad \text{(Distributive Property)}

= (2)z \quad \text{(Scalar addition)}

= 2z

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

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