Properties of Addition of Complex Numbers

Rearranging Complex Expressions without Changing Their Value

Addition and scalar multiplication of complex numbers obey the same familiar algebraic rules as real numbers. Under those rules, expressions can be rearranged and simplified without changing their value.

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

Addition Properties

Addition of complex numbers keeps the familiar properties of real addition, and each check runs on the real part and the imaginary part separately. Splitting a sum into two real sums is what keeps those proofs short.

Commutativity

Changing the order of two complex addends does not change their sum.

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

Multiplying by a real scalar follows the same group of properties, so the same checks apply with one real factor. The scalar touches the real part and the imaginary part in the same way, which is why distributivity and the zero scalar keep working.

Multiplication Associativity

The grouping of scalar multiplication does not affect the result.

Scalar Distributivity

Multiplying a complex number by a sum of scalars is equivalent to multiplying it by each scalar and then adding the results.

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.

Simplifying Expressions with Addition Properties

Distributivity carries a scalar into a sum term by term, and equal coefficients with opposite signs then cancel. Both moves keep the real part and the imaginary part unchanged, so any identity built from them holds for every complex number.

Verifying the Additive Inverse

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

Solution:

Use distributivity to combine the scalar coefficients, then use the fact that multiplying by the zero scalar gives zero.

Apply the distributive property:

Multiplication by the zero scalar gives:

It is proven that $4z + (-4)z = 0$.

Exercise

Using the addition properties, prove that $3z - \frac{1}{2}(2z) = 2z$ for any complex number $z$. Work on the left side only, and name the property you apply at each step, so the argument covers every complex number.

Worked Solution

Write subtraction as addition of the opposite:

Regroup the scalar factors:

Multiply the scalars:

Apply the distributive property:

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