Complex numbers
Graphing complex numbers worksheets
Plot complex numbers and read a + bi from points on the complex plane. Choose six plotting questions, six identification questions or a balanced mixture.
Preview and downloadTwo components. One number.
Complex numbers worksheets
Connect a + bi with a point on the plane. Find its distance from the origin, calculate with i and use a conjugate to simplify a quotient.
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Connect position and distance
Read the two components, locate the point and calculate its distance from the origin.
Complex numbers
Absolute value of complex numbers worksheets
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
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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Calculate with i
Combine like terms, expand products and use conjugates to simplify quotients.
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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Complex numbers
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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Keep one idea in focus before combining skills. Each link opens the same generator with a useful selection already set.
Plot from a + bi
Use each pair of signed components to place a point on its own plane.
Open this practice →Read the point
Write a complex number from a plotted location.
Open this practice →Combine like terms
Keep the focus on real and imaginary sums and differences.
Open this practice →Use i squared
Simplify products of imaginary factors and distribute one over a binomial.
Open this practice →Expand complex products
Practice binomial products, squares and products with an imaginary factor.
Open this practice →Start with an imaginary denominator
Rewrite a real number divided by a nonzero multiple of i.
Open this practice →Use a conjugate
Make a two-term complex denominator real, then reduce the quotient.
Open this practice →Keep three different quantities distinct
For z = −4 + 3i, the real part is −4, the imaginary part is 3, and the modulus is 5. The coefficient 3 is a real number even though it is called the imaginary part.
The point (−4, 3) gives a position. The distance √(16 + 9) gives a magnitude. Complex numbers with different positions can have the same magnitude.
A quick diagnostic is to ask for all three quantities before handing out a mixed review sheet.
Build a short lesson
- Locate. Plot one complex number and explain what each component controls.
- Measure. Connect that point with a right triangle and find its modulus.
- Calculate. Expand a product and show exactly where i² becomes −1.
- Check. Multiply a simplified quotient by its original denominator.
Use a focused set when one step needs attention. The combined review checks position and distance; the operations and rationalization sheets practice a different skill.
Teaching questions
Why is i² equal to −1?
The imaginary unit i is defined by this property. Complex arithmetic extends real arithmetic while keeping distribution and the usual addition and multiplication rules.
Is the imaginary part 3 or 3i?
For a + 3i, the imaginary part is the real coefficient 3. The imaginary term is 3i. The vertical coordinate is therefore 3.
Should answers be decimals?
These worksheets use exact answers. Leave a modulus as a simplified radical when necessary, and keep rationalized quotients as reduced fractions.
How do mixed graph sheets work?
The graph worksheet has three plotting and three identification questions. The review has four modulus questions followed by two of each graph type.
What grade should use these?
Course placement differs across school systems. Choose by readiness: signed coordinates, radicals, distribution and basic algebra. The Algebra 2 guide connects these resources with prerequisites.
Can I change the paper or language?
Custom sheets support Letter/A4 and English, German, Albanian, Spanish, Italian, French, Turkish and Norwegian Bokmål. Ready-made downloads are English.
Review a prerequisite
Roots and radicals
Simplify square roots before using them to express a modulus exactly.
Practice square roots →Binomial products
Practice distribution with real variables before adding the identity i² = −1.
Practice binomial multiplication →Algebra 2
Choose a learning sequence and return to a foundation when a specific step needs work.
Explore Algebra 2 practice →