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2. Basic Operations with Complex Numbers

by M. Bourne

Addition and subtraction of complex numbers works in a similar way to that of adding and subtracting surds. This is not surprising, since the imaginary number j is defined as j = √(-1).

Addition of Complex Numbers:

Add real parts, add imaginary parts.

Subtraction of Complex Numbers:

Subtract real parts, subtract imaginary parts.

Examples:

1. (6 + 7j) + (3 − 5j) =

(6 + 3) + (7 − 5)j = 9 + 2j

2. (12 + 6j) − (4 + 5j) =

(12 − 4) + (6 − 5)j = 8 + j

Multiplication of Complex Numbers

Expand brackets as usual, but care with j²!

Examples:

1. 5(2 + 7j) = 10 + 35j

2. (6 − j)(5j) = 30j − 5j² = 5 + 30j

3. (2 − j)(3 + j) = 6 − 3j + 2j − j²

= 6 − (−1) − j

= 7 − j

Let's play with these for a while. You can change any of the numbers in this Livemath example to see what happens in multiplication of complex numbers.

LIVEMath

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4. (5 + 3j)² = 25 + 2(5)(3)j + 9(j²)

= 16 + 30j

5.

6. (3 + 2j)(3 − 2j) = (3)² − (2j)²

= 9 − 4j²

= 9 + 4

= 13

Example 6 is a special case.

3 + 2j is the conjugate of 3 − 2j.

In general:

x + yj is the conjugate of x − yj

and

x − yj is the conjugate of x + yj.

Notice that when we multiply conjugates, our final answer is real only (it does not contain any imaginary terms.

We use the idea of conjugate when dividing complex numbers.

 

Division of Complex Numbers

Earlier, we learned how to rationalise the denominator of an expression like:

We multiplied numerator and denominator by the conjugate of the denominator, 3 + √2:

We did this so that we would be left with no radical (square root) in the denominator.

Dividing with complex numbers is similar.

 

Examples:

1. Express

in the form x + yj.

Solution:

The conjugate of 4 − 2j is 4 + 2j.

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2. Simplify:

Answer:

Let's play with divisions using LiveMath.

LIVEMath

Exercises

1. Express in the form a + bj:

Answer:

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2. Express in the form a + bj.

Answer:

 

Need a break? Play a math game. Well, they all involve math... No, really!

Chapter Contents

• Complex Numbers
• 1. Basic Definitions
• 2. Basic Operations
• 3. Graphical Representation
• 4. Polar Form
• 5. Exponential Form
• 6. Products and Quotients
• 7. Powers and Roots
• 8. AC Circuit Definitions
• 9. Impedance and Phase Angle
• 10. Reactance and Angular Velocity
• 11. AC Circuit Exercises
• 12. Parallel AC Circuits
• Vocabulary
• Fractals

• Comments, Questions?

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