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An introduction to complex numbers This topic is part of the TCS FREE high school mathematics 'How-to Library', and introduces the concept of complex numbers. (See the index page for a list of all available topics in the library.)Â To make best use of this topic, you need to download the Maths Helper Plus software. Click here for instructions. ### Theory:

This topic introduces complex numbers, and demonstrates how complex number calculations can be done using complex calculator data sets in Maths Helper Plus.

Complex numbers are an extension of the real number system which give meaning to expressions such as:

Â x² + 5 = 0 where: The complex number system depends upon the definition of the symbol ' i ' such that i 2 = -1. ' i ' is called the 'imaginary' number.

A complex number is made up of two real numbers: the 'real' part and the 'imaginary' part. The imaginary part of a complex number is multiplied by ' i '.

A complex number can be written as the sum of its real and imaginary parts, eg: 3 + 4i, or as an ordered pair: (3,4), ie: If the ordered pair is plotted as a point on the x-y plane, the x axis being the real axis, and the y axis being the imaginary axis, then the complex number can also be expressed in polar form, ie. as a magnitude (distance of the point from the origin) and an angle.

The magnitude of a complex number is equivalent to its absolute value, while the angle is called the argument ( written as 'arg') of the complex number.Â

The magnitude and argument are written as: 'magnitude cis argument'

eg. complex number z = 1 + Ö3i or (1,Ö3) has magnitude: |z| = 2 and arg z = 60° , written as: 2cis60. See diagram below: ### Method:

Maths Helper Plus can perform many complex number operations. It has a feature called the 'complex calculator' that can demonstrate basic complex number operations, as well as evaluate many kinds of complex number expressions.

#### Step 1Â  Start with an empty Maths Helper Plus document

If you have just launched the software then you already have an empty document, otherwise hold down ‘Ctrl’ while you briefly press the ‘N’ key.

#### Step 2Â  Create a complex calculator

1. Press the F3 key to activate the 'input box' for typing (see below):

2. Type: com into the input box:

Â Â Â Â Â Â Â Â Â Â Â Â Â 3. Press Enter to complete the entry

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The 'options' box for the complex calculator will display immediately, like this:

Â This is the 'Program' tab, where you can type complex number expressions to be evaluated.

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#### Step 3Â  Enter the expressions to evaluate

1. Click on the large edit box.

2. Type the expressions to be evaluated.

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Here is a selection of complex number operations that the complex calculator can understand:

NOTE: 'sqrt' means 'square root'

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 Â 3+4i Â sqrt(-5) Â i² Â 2cis60° Â (2+3i)[(-1+2i)+(5-4i)] Â [(1-2i)/(2-i)]² - [(2+3i)/(1-2i)]³ Â 4cis(pi/12) / 2cis(-5pi/12) Â

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3. Click the 'Run' button to evaluate the expressions you entered.

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The text view displays the results. To see the results, you will probably need to close the complex calculator options box by clicking the OK button.

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Here are the results for the example expressions above:

Output

3+4i = 3 + 4i

sqrt(-5) = 2.236067977i

i² = -1

2cis60° = 1 + 1.732050808i

(2+3i)[(-1+2i)+(5-4i)] = 14 + 8i

[(1-2i)/(2-i)]² - [(2+3i)/(1-2i)]³ = -3.912 - 0.904i

4cis(pi/12) / 2cis(-5pi/12) = 2i

#### How to make changes to your complex number expressions

To display the complex calculator options box at any time, double click on the text view to the left of the results display, OR if there are no results displayed, to the left of the words 'Complex Calculator'.

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 This picture (right) shows you where to double click... Â If there are results displayed, double click in the area we have shaded blue. Otherwise, double click in the area we have shaded pink, then click the 'Program' tab at the top of the options box when it is displayed. Â Â Â Â

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#### How to use complex number variables

The complex calculator includes five variables: 'A', 'B', 'C', 'D' and 'X' that you can use in any expressions.

The example at the beginning of this article will demonstrate how they can be used.

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Recall the following equation:Â  Â x² + 5 = 0 where: are the two solutions.

The solutions can be typed into the complex calculator like this: X = -sqrt(-5) and X = sqrt(-5).

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We want to substitute these two complex numbers into the equation Â x² + 5 = 0 to show that they satisfy it. We will also add some comment lines. These begin with a double quote: " . Comment lines make the output more meaningful. A double quote on a line by itself makes a blank line in the output. The completed steps are shown below:

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 Â "Set 'X' to negative root: Â X = -sqrt(-5) Â " Â "Evaluate equation: Â X² + 5 Â " Â "Set 'X' to positive root: Â X = sqrt(-5) Â " Â "Evaluate equation: Â X² + 5 Â

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After clicking the 'Run' button, the text view displays this output:

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Output

Set 'X' to negative root:

X = -sqrt(-5)

Â  = -2.236067977i

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Evaluate equation:

X² + 5 = 0

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Set 'X' to positive root:

X = sqrt(-5)

Â  = 2.236067977i

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Evaluate equation:

X² + 5 = 0

By using variable 'X', we have shown that the equation x² + 5 = 0 is indeed satisfied by the two complex roots.

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