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Math • Complex Numbers Calculator

Complex Numbers Calculator

Add, subtract, multiply complex numbers (a + bi), and compute the modulus and conjugate of a complex number.

Complex Number z₁ (a + bi)
Complex Number z₂ (c + di)
Addition (z₁ + z₂)
4 + 2i
Subtraction (z₁ - z₂)
2 + 6i
Multiplication (z₁ × z₂)
11 + -2i
Modulus |z₁|
5.000
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Calculation Guide & Reference

Complex Number Arithmetic & Modulus Mathematics

Complex numbers calculator adds, subtracts, and multiplies complex numbers (a + bi), and computes the modulus and conjugate.

Standardized Mathematical Formula
(a+bi)(c+di) = (ac−bd) + (ad+bc)i | |a+bi| = √(a²+b²)

Complex number multiplication distributes normally, using i² = −1 to simplify. The modulus (absolute value) treats the real and imaginary parts as coordinates and applies the Pythagorean theorem to find the distance from the origin.

Variables:
a + bi:A complex number with real part a and imaginary part b
i:The imaginary unit, where i² = −1
|z|:Modulus (magnitude) of complex number z
How It Works (Step-by-Step)
  • 1Enter the real and imaginary parts of two complex numbers, z₁ = a + bi and z₂ = c + di.
  • 2View their sum, difference, product, and the modulus and conjugate of z₁.
Real-World Numerical Example
z₁ = 3 + 4i and z₂ = 1 − 2i

Add, multiply, and find the modulus of z₁ = 3 + 4i and z₂ = 1 − 2i.

•Sum: (3+1) + (4+(−2))i = 4 + 2i.
•Product real part: (3×1) − (4×−2) = 3 + 8 = 11.
•Product imaginary part: (3×−2) + (4×1) = −6 + 4 = −2, so product = 11 − 2i.
•Modulus of z₁: √(3² + 4²) = √25 = 5.
Result: z₁ + z₂ = 4 + 2i; z₁ × z₂ = 11 − 2i; |z₁| = 5.
Calculation Best Practices & Tips
Examine the discriminant (b² - 4ac) before solving quadratics to instantly know if roots are real, distinct, repeated, or complex.
In statistical analysis, always check both the mean and median to identify if data distribution is skewed by outliers.
When calculating percentage changes, ensure you divide by the original starting baseline value: ((New - Old) / Old) * 100.

Frequently Asked Questions (FAQ)

The modulus |a+bi| = √(a²+b²) is the distance from the complex number to the origin on the complex plane — geometrically identical to the Pythagorean distance formula.

The conjugate of a+bi is a−bi — flipping the sign of the imaginary part. Multiplying a complex number by its conjugate always yields a real number: (a+bi)(a−bi) = a²+b².

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