Complex Number Calculator (Exact, With Steps)
Add, subtract, multiply or divide complex numbers with exact fractions, and get the conjugate, modulus, argument, polar form and powers — with the working shown.
Tool inputs stay in your browser · optional AI search sends your query
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Good to know: Enter the real and imaginary parts as whole numbers, fractions (1/3) or decimals. The four operations stay exact — 1/3 never becomes 0.333333 — because the parts are kept as rational numbers. Each result shows the rectangular form, the decimal, the polar form r∠θ, the modulus and the argument, plus the algebra that produced it. Division also multiplies back as a check.
How to use it (step by step)
- 1Type the real part and the imaginary part of z₁ (and z₂ if the operation needs two numbers). Leave the “i” off — a 4 in the imaginary box means 4i.
- 2Pick an operation: add, subtract, multiply, divide, conjugate, modulus, argument, or a whole-number power.
- 3Read the answer in rectangular form (a + bi) and in polar form (r∠θ), with the modulus and argument underneath.
- 4The “How it was calculated” box shows the algebra with your own numbers. Division also multiplies back, so you can see z₁ ÷ z₂ × z₂ = z₁.
Words to know
- Fraction / ratio
- A fraction is a part of a whole (3/4 = three of four parts). A ratio compares two amounts (2:1 = twice as much).
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