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This topic is part of the TCS FREE high school mathematics 'How-to Library', and shows you how to add,  subtract, multiply and divide complex numbers. (See the index page for a list of all available topics in the library.) To make best use of this topic, you need to download the Maths Helper Plus software. Click here for instructions.

Theory:

This topic shows you how to add,  subtract, multiply and divide complex numbers.

1 Adding complex numbers

If 'A' and 'B' are two complex numbers, so that:

A = a + bi      and     B = c + di

Then we can sum A and B like this:

A + B  =  a + bi  + c + di

           =  (a + c) + (b + d)i

For example, if A = 1 + 2i, and B = 3 - 4i, then:

             A + B  =  (1 + 3) + (2 - 4)i

                         =  4 - 2i

2 Subtracting complex numbers

If 'A' and 'B' are two complex numbers, so that:

A = a + bi      and     B = c + di

Then we can subtract B from A like this:

A - B  =  a + bi  - ( c + di)

           =  (a - c) + (b - d)i

For example, if A = 1 + 2i, and B = 3 - 4i, then:

             A - B  =  (1 - 3) + (2 - -4)i

                         =  -2 + 6i

3 Multiplying complex numbers

If 'A' and 'B' are two complex numbers, so that:

A = a + bi      and     B = c + di

Then we can multiply A and B like this:

A × B  =  ( a + bi ) × ( c + di )

           =  a × c  +  a ×di   +   bi × c  +  bi × di

           =  ac  +  adi   +   bci  +  bdi^²

  But  i^² =  -1

 So  A × B   =   ( ac  -  bd ) + ( ad  +  bc )i

  For example, if A = 1 + 2i, and B = 3 - 4i, then:

             A × B  =  (1×3 - 2×(-4)) + (1×(-4) + 2×3)i

                         =  11 + 2i

4  Dividing complex numbers

If 'A' and 'B' are two complex numbers, so that:

A = a + bi      and     B = c + di

Then we can divide A by B like this:

For example, if   A = 1 + 2i, and B = 3 - 4i, then:

It should be noted that the denominator terms: ( c² + d² ), are the square of the magnitude of complex number 'B'.  ( c² + d² ) = |B|², which leads to the more compact version of the division formula:

where |B|² = ( c² + d² )

The 'Method' section below shows you how Maths Helper Plus can display all of the working steps for adding, subtracting, multiplying or dividing complex numbers.

Method:

In the following steps, we will use the examples from the 'Theory' section above. To solve your own complex number problems, insert your own numbers.

Step 1  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 2  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:

             

3. Press Enter to complete the entry

 

The 'options' box for the complex calculator will display immediately. 

Click the 'Complex Calculations' tab (see below) at the top of the options box to display the calculation options:

 

Click to select the operations that you want to see working steps for.

 

Step 3  Enter the complex numbers A and B

Click the 'Complex Editor' tab at the top of the options box. This displays the 'Complex Editor' tab where you enter the complex numbers A and B to create the working steps. See below:

 

Click on the 'A' edit box and type complex number A like this:

real part , imaginary part

( So, to enter A = 1 + 2i, type:  1,2 )

 

The edit box displays in yellow to show that the values have been changed.

 

Similarly, click on the 'B' edit box and type complex number B.

( So, to enter B = 3 - 4i, type:  3,-4 )

 

Now, click the 'Apply' button at the bottom of the edit box. The text view will display the complex number calculations. 

 

IMPORTANT: Each time you change either 'A' or 'B', remember to click the 'Apply' button to update the calculations.

 

 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.

 

Here are the results displayed by Maths Helper Plus for the 'A' and 'B' values entered above:

 

Complex calculations

Complex Sum:

  Let A = a + bi, and B = c + di, then

  A + B = (a + c) + (b + d)i

        = (1 + 3)

          + (2 + -4)i

        = 4 + -2i

|A + B| = 4.47214, arg(A+B) = -26.5651°

 

Complex Difference:

  Let A = a + bi, and B = c + di, then

  A - B = (a - c) + (b - d)i

        = (1 - 3)

          + (2 - -4)i

        = -2 + 6i

|A - B| = 6.32456, arg(A-B) = 108.435°

 

Complex Product:

  Let A = a + bi, and B = c + di, then

  A × B = (ac - bd) + (ad + bc)i

        = (1 × 3 - 2 × -4)

          + (1 × -4 + 2 × 3)i

        = 11 + 2i

|A × B| = 11.1803, arg(A × B) = 10.3048°

 

Complex Division:

  Let A = a + bi, and B = c + di,

so |B|² = c² + d²

        = 3² + -4²

        = 25

  A ÷ B = [(ac + bd)/|B|²] + [(bc - ad)/|B|²]i

        = [ (1 × 3

               + 2 × -4)/25 ]

           + [ (2 × 3

               - 1 × -4)/25 ]i

        = -0.2 + 0.4i

|A ÷ B| = 0.447214, arg(A ÷ B) = 116.565°

How to make changes to your complex settings

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'.