Complex Number Calculator

What you can do

• Use i to represent the imaginary unit, with i² = −1.
• Add, multiply and divide supported complex expressions.
• Work with magnitude, conjugate, argument and selected transcendental functions.
• Compare rectangular and polar representations when studying phasors.

Formula and interpretation

For z = a + bi, the modulus is √(a²+b²). Complex multiplication follows i² = −1, so (a+bi)(c+di)=(ac−bd)+(ad+bc)i. Arguments and complex logarithms require attention to branches; the principal result may not represent every mathematically valid branch.

Worked multiplication example

Multiply (2 + 3i)(4 − i). Expanding gives 8 − 2i + 12i − 3i² = 11 + 10i. Its modulus is √(11²+10²)=√221≈14.866. Confirm the arithmetic in Complex mode, then check that the expression uses i rather than a variable name.

How to use the workspace

1. Open the scientific calculator and select its supported Complex mode.
2. Enter (2+3i)*(4-i) using the calculator's complex-number input rules.
3. Check that the real part is 11 and the imaginary part is 10.
4. For roots and logarithms, clarify whether your assignment needs the principal value or all branches.

Assumptions and common errors

• Mode selection: real-number mode may not accept complex roots.
• Angle conventions: principal arguments depend on the selected angular units.
• All roots: a principal complex square root is one value, not the full solution set of z² = a.

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