Add, subtract, multiply or divide two complex numbers step by step, and see the answer’s magnitude (size) and argument (angle).
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Check your inputs—Complete valid inputs to calculate a current result.
The plain number part, such as 3 in 3 + 2i.
The number in front of i, such as 2 in 3 + 2i.
The plain number part, such as 1 in 1 − 4i.
The number in front of i, such as −4 in 1 − 4i.
Complex result
11 − 10i
Magnitude14.8661
Argument-42.2737 degrees
Rectangular complex arithmetic with real-number floating-point calculations. Argument is the principal angle in degrees; zero has no defined argument.
How this was calculated
Multiply every part of the first number by every part of the second, like expanding brackets. Remember that i × i = −1, so the i² part turns into a minus.(a + bi)(c + di) = (ac − bd) + (ad + bc)i.Put in your numbers: (3 + 2i)(1 − 4i)
The real part is a × c take away b × d.real part = ac − bdPut in your numbers: 3 × 1 − 2 × (−4) = 3 − (−8)Answer: 11
The imaginary part is a × d plus b × c.imaginary part = ad + bcPut in your numbers: 3 × (−4) + 2 × 1 = −12 + 2Answer: −10i
Put the real part and the imaginary part together to get the answer.answer = real part + imaginary part × iAnswer: 11 − 10i
The magnitude is the size of the answer (its distance from 0), found with Pythagoras.|z| = √(real² + imaginary²)Put in your numbers: √(11² + (−10)²)Answer: 14.8660687473
Complex numbers in simple words
A complex number has two parts: a plain number (the real part) and a number times i (the imaginary part). For example, 3 + 2i has real part 3 and imaginary part 2.
The letter i stands for a special number where i × i = −1. No ordinary number does that, which is why it gets its own letter.
To add, add the real parts together and the i parts together: (3 + 2i) + (1 − 4i) = 4 − 2i.
To multiply, multiply every part by every part, then swap each i × i for −1: (3 + 2i)(1 − 4i) = 3 − 12i + 2i − 8i² = 3 − 10i + 8 = 11 − 10i.
Formula and method
For multiplication, (a + bi)(c + di) = (ac − bd) + (ad + bc)i. Division multiplies by the divisor’s complex conjugate.
Worked example
(3 + 2i)(1 − 4i) = 11 − 10i. The magnitude is √221, approximately 14.8661.
Frequently asked questions
What is the argument of zero?
It is undefined because zero has no direction in the complex plane. The calculator labels it explicitly instead of giving a misleading angle.
Last updated . Results are estimates for informational purposes only.