How to add, subtract, multiply and divide complex numbers.
Goal:
Theory:
Part 1
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 + bdi²
But i² = -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.
Download the free support file… We have created a Maths Helper Plus document containing the completed example from this topic. You can use this to practice the steps described below, and as a starting point for solving your own problems.
File name: ‘Complex number operations.mhp’ File size: 4kb
Click here to download the file.If you choose` ‘Open this file from its current location’, then Maths Helper Plus should open the document immediately. If not, try the other option: ‘Save this file to disk’, then run Maths Helper Plus and choose the ‘Open’ command from the ‘File’ menu. Locate the saved file and open it. If you do not yet have Maths Helper Plus installed on your computer, click here for instructions.
Method:
Part 2
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
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- Press the F3 key to activate the ‘input box’ for typing (see below):
- Type: com into the input box:
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- Press ENTER to complete the entry
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The ‘options’ box for the complex calculator will display immediately, like this:
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’.
This picture (right) shows you where to double click… 
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.
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