Operations with Complex Numbers (DP IB Analysis & Approaches (AA)): Revision Note

Complex Addition, Subtraction & Multiplication

How do I add and subtract complex numbers in Cartesian Form?

• Adding and subtracting complex numbers should be done when they are in Cartesian form

• When adding and subtracting complex numbers, simplify the real and imaginary parts separately

  • Just like you would when collecting like terms in algebra and surds, or dealing with different components in vectors

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How do I multiply complex numbers in Cartesian Form?

• Complex numbers can be multiplied by a constant in the same way as algebraic expressions:

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• Multiplying two complex numbers in Cartesian form is done in the same way as multiplying two linear expressions:

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  • This is a complex number with real part and imaginary part 

  • The most important thing when multiplying complex numbers is that

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• Your GDC will be able to multiply complex numbers in Cartesian form

  • Practise doing this and use it to check your answers

• It is easy to see that multiplying more than two complex numbers together in Cartesian form becomes a lengthy process prone to errors

  • It is easier to multiply complex numbers when they are in different forms and usually it makes sense to convert them from Cartesian form to either Polar form or Euler’s form first

• Sometimes when a question describes multiple complex numbers, the notation  is used to represent each complex number

How do I deal with higher powers of i?

• Because  this can lead to some interesting results for higher powers of i

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• We can use this same approach of using i² to deal with much higher powers

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  • Just remember that -1 raised to an even power is 1 and raised to an odd power is -1

Complex Conjugation & Division

When dividing complex numbers, the complex conjugate is used to change the denominator to a real number.

What is a complex conjugate?

• For a given complex number , the complex conjugate of is denoted as , where 

• If then 

• You will find that:

  • is always real because 

    • For example:

  • is always imaginary because 

    • For example:

  • is always real because  (as )

    • For example:

How do I divide complex numbers?

• To divide two complex numbers:

  • STEP 1: Express the calculation in the form of a fraction

  • STEP 2: Multiply the top and bottom by the conjugate of the denominator:

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    • This ensures we are multiplying by 1; so not affecting the overall value

  • STEP 3: Multiply out and simplify your answer

    • This should have a real number as the denominator

  • STEP 4: Write your answer in Cartesian form as two terms, simplifying each term if needed

    • OR convert into the required form if needed

• Your GDC will be able to divide two complex numbers in Cartesian form

  • Practise doing this and use it to check your answers if you can