Complex numbers
Complex number operations worksheets
Add, subtract and multiply complex numbers in ten-question sets. Select sums and differences, imaginary monomial products or binomial products and squares.
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Rationalizing complex denominators worksheets
Rewrite ten complex quotients with real denominators. Select real-over-imaginary, binomial-over-imaginary or binomial-over-binomial questions and print worked answers.
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Multiply the numerator and denominator by the same nonzero factor. Simplify to a positive real denominator and reduce common factors.
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Why the conjugate works
Multiplying a + bi by a − bi gives a² + b². The imaginary cross terms cancel, and the remaining denominator is positive whenever the original denominator is nonzero.
Choose the three numerator/denominator groups independently. Worked answers retain exact fractions, with a reduced positive real denominator.
Learning objective
What this activity builds
Use an equivalent fraction and a conjugate to remove the imaginary component from a denominator.
Worked example
Simplify (2 + 3i) / (1 − i).
(−1 + 5i) / 2.
Multiply both parts by 1 + i. The numerator becomes −1 + 5i and the denominator becomes 1 − i² = 2.
Teaching notes
Make the practice count
- Check that the original denominator is nonzero.
- Write the conjugate directly beneath the denominator before multiplying.
- Distribute in the numerator as carefully as in the denominator.
- Check the result by multiplying it by the original denominator.
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Find the modulus of ten complex numbers. Connect real and imaginary components with distance from the origin, then check exact, simplified radical answers.
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Complex numbers review worksheets
Combine four modulus questions with four complex-plane questions. Choose plotting, identifying points or both, with a separate key for the same generated set.
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