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MathidiaMathematics Academy
Complex numbers
Lesson 2 of 75
Easy practice

Cartesian form

Objective. Add, multiply and divide complex numbers.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Cartesian form writes a complex number as z = a + bi, with a its real part and b its imaginary part. Addition and subtraction combine real and imaginary parts separately, multiplication expands brackets using i² = −1, and division multiplies numerator and denominator by the conjugate of the denominator to make it real. For instance, (2+3i)(1−i) = 2 − 2i + 3i − 3i² = 2 + i + 3 = 5 + i. Conjugates a − bi are central: z z̄ = a² + b² is always real and non-negative, which is exactly what makes division work cleanly.

Worked example

Express (3 + 4i) ÷ (1 − 2i) in Cartesian form a + bi.

-1124578+4
Number line from -1 to 8 with a jump of 4 forwards from 3
  1. 1Multiply numerator and denominator by the conjugate 1 + 2i
  2. 2Denominator: (1−2i)(1+2i) = 1 + 4 = 5
  3. 3Numerator: (3+4i)(1+2i) = 3 + 6i + 4i + 8i² = 3 + 10i − 8 = −5 + 10i
  4. 4Divide by 5: (−5+10i)/5 = −1 + 2i

Answer: −1 + 2i

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Common mistake. A common error is dividing complex numbers term by term, e.g. writing (4+2i)/(1+i) as 4+2i with the i cancelled. Instead multiply top and bottom by the conjugate 1−i before simplifying.

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