>Calculations>Addition and Subtraction of Complex Numbers

Addition and Subtraction of Complex Numbers

Addition and subtraction of complex numbers is carried out element-wise, with the real part and the imaginary part calculated separately. The set of complex numbers is closed under addition and subtraction. This means that you get a new complex number when you add and subtract complex numbers.

Formula

Addition and Subtraction of Complex Numbers

Let $z = a + b i$ and $w = c + d i$ be complex numbers, then

z + w = (a + c) + (b + d) i

and

z - w = (a - c) + (b - d) i .

Example 1

Find $z + w$ and $z - w$ for the complex numbers $z = 4 - 3 i$ and $w = 2 + i$

Addition is carried out element-wise, with the real part and the imaginary part worked out separately:

\begin{array}{l} z + w & = (4 - 3 i) + (2 + i) \\ = (4 + 2) + (- 3 + 1) i \\ = 6 - 2 i . \end{array}

z + w = (4 - 3 i) + (2 + i) = (4 + 2) + (- 3 + 1) i = 6 - 2 i .

Equivalently, for subtraction:

\begin{array}{l} z - w & = (4 - 3 i) - (2 + i) \\ = (4 - 2) + (- 3 - 1) i \\ = 2 - 4 i . \end{array}

z - w = (4 - 3 i) - (2 + i) = (4 - 2) + (- 3 - 1) i = 2 - 4 i .

If you draw complex numbers in the complex plane, addition and subtraction of complex numbers can be thought of in the same way as addition and subtraction of vectors:

Geometric visualization of addition of complex numbers.

(a) Addition

Geometric visualization of subtraction of complex numbers.

(b) Subtraction

As with the real numbers, the complex numbers are commutative and associative under addition and subtraction:

Rule

Commutative and Associative Properties

For all complex numbers $z_{1}$ , $z_{2}$ and $z_{3}$ , both the commutative property

z_{1} + z_{2} = z_{2} + z_{1}

and the associative property

z_{1} + (z_{2} + z_{3}) = (z_{1} + z_{2}) + z_{3}

holds.

The commutative and the associative property state that you can freely change the order of numbers and parentheses as long as the calculation only consists of addition and subtraction.

Think About This

Addition and subtraction are important operations. For instance, you can find the distance between the midpoints of two complex numbers by using addition and subtraction.

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