ALGEBRAAlgebraMathematics Calculator
ℂ

Complex Number Arithmetic

Complex numbers a+bi extend the reals with i²=−1. Add and subtract by combining parts; multiply using (a+bi)(c+di)=(ac−bd)+(ad+bc)i. Polar form z=r·e^(iθ) simplifies powers and roots.

Concept Fundamentals
−1
i²
√(a²+b²)
|z|
−1
e^(iπ)
z^(1/n) has n values
n roots

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Euler's formula: e^(iθ)=cos θ+i sin θ connects exponentials to trig. De Moivre: (re^(iθ))ⁿ=rⁿ·e^(inθ) for integer n. nth roots of z are equally spaced on a circle of radius |z|^(1/n).

Key quantities
−1
i²
Key relation
√(a²+b²)
|z|
Key relation
−1
e^(iπ)
Key relation
z^(1/n) has n values
n roots
Key relation

Ready to run the numbers?

Why: Complex numbers unify algebra (polynomial roots), trigonometry (Euler's formula), and physics (quantum states, AC circuits). Every polynomial has roots in ℂ.

How: Rectangular: add parts, multiply with i²=−1. Division: multiply by conjugate. Polar: z=r(cos θ+i sin θ)=re^(iθ). Powers: zⁿ=rⁿ·e^(inθ). Roots: n values at angles θ/n + 2πk/n.

Euler's formula: e^(iθ)=cos θ+i sin θ connects exponentials to trig.De Moivre: (re^(iθ))ⁿ=rⁿ·e^(inθ) for integer n.
Sources:Khan Academy — Complex NumbersWolfram MathWorld — Complex Number

Run the calculator when you are ready.

Compute with Complex NumbersAdd, multiply, divide, powers, roots

📌 Example Scenarios — Click to Load

First Complex Number z₁

Second Complex Number z₂

complex_number
CALCULATED
$ compute --z1="3 + 2i" --z2="1 + 4i" --op=add
Result
4 + 6i
Modulus |z|
7.2111
Argument θ
56.31°
Polar
7.2111(cos 56.31° + i sin 56.31°)
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Component Comparison

Result: Real vs Imaginary

📐 Calculation Steps

ADDITION
Add real parts
3 + 1 = 4
ext{Re}(z_{1}) + ext{Re}(z_{2})
Add imaginary parts
2 + 4 = 6
ext{Im}(z_{1}) + ext{Im}(z_{2})
POLAR
Modulus |z|
7.2111
√( ext{Re}^{2} + ext{Im}^{2})
Argument θ
56.31°
atan2( ext{Im}, ext{Re})
Polar form
7.2111(cos 56.31° + i sin 56.31°)
r(\text{cos} \text{theta} + i \text{sin} \text{theta} )

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

📐

e^(iπ)+1=0 links five fundamental constants.

— Euler

⚡

AC circuits: impedance Z=R+jX, voltage V=I·Z.

— EE

📋 Key Takeaways

  • • Add/subtract: combine real and imaginary parts separately
  • • Multiply: use (a+bi)(c+di) = (ac−bd) + (ad+bc)i, with i² = −1
  • • Divide: multiply numerator and denominator by the conjugate of the denominator
  • • Polar form: z = r(cos θ + i sin θ) = re^(iθ) where r = |z|, θ = arg(z)
  • • Euler's formula: e^(iθ) = cos θ + i sin θ — connects exponentials to trigonometry

💡 Did You Know?

🔄i^n cycles every 4: i¹=i, i²=−1, i³=−i, i⁴=1, then repeats. So i^2026 = i² = −1Source: Powers of i
📐Multiplication by e^(iθ) rotates a complex number by θ radians counterclockwiseSource: Geometric interpretation
⚡AC circuit analysis: impedance Z = R + jX combines resistance and reactance as complex numbersSource: Electrical engineering
🌀The Mandelbrot set uses z → z² + c; each iteration is complex multiplication plus additionSource: Fractals
📜Fundamental Theorem of Algebra: every polynomial has at least one complex root (counting multiplicity)Source: Mathematics
🔢An n-th degree polynomial has exactly n complex roots (counting multiplicity)Source: Algebra

📖 How Complex Arithmetic Works

Complex numbers extend the real numbers with i = √(−1). They form a field: you can add, subtract, multiply, and divide (except by zero).

Rectangular vs Polar

Rectangular: z = a + bi. Good for addition and subtraction.

Polar: z = r(cos θ + i sin θ) = re^(iθ). Good for multiplication and powers: |z₁z₂| = |z₁||z₂|, arg(z₁z₂) = arg(z₁)+arg(z₂).

Euler's Formula

e^(iπ) + 1 = 0 connects five fundamental constants. More generally, e^(iθ) = cos θ + i sin θ.

🎯 Expert Tips

💡 Addition

Add like terms: (3+2i)+(1+4i) = (3+1)+(2+4)i = 4+6i.

💡 Multiplication

FOIL and use i²=−1. Or use polar: multiply moduli, add arguments.

💡 Division

Multiply top and bottom by conjugate of denominator to get real denominator.

💡 Powers

De Moivre: (re^(iθ))^n = r^n e^(inθ). Convert to polar first.

📊 Reference: i^n Cycle

n mod 4i^nExample
01i⁴, i⁸, i¹²
1ii¹, i⁵, i⁹
2−1i², i⁶, i¹⁰
3−ii³, i⁷, i¹¹

❓ FAQ

How do I add two complex numbers?

Add the real parts and the imaginary parts separately: (a+bi)+(c+di) = (a+c)+(b+d)i.

How do I multiply complex numbers?

Use FOIL: (a+bi)(c+di) = ac + adi + bci + bdi² = (ac−bd) + (ad+bc)i, since i² = −1.

What is the polar form of a complex number?

z = r(cos θ + i sin θ) = re^(iθ), where r = |z| = √(a²+b²) and θ = atan2(b,a).

What is Euler's formula?

e^(iθ) = cos θ + i sin θ. The special case e^(iπ) = −1 is famous.

How many nth roots does a complex number have?

Exactly n distinct roots, equally spaced around a circle in the complex plane.

Why are complex numbers used in electrical engineering?

AC circuits use phasors (complex numbers) to represent voltage and current. Impedance Z = R + jX combines resistance and reactance.

📈 Quick Reference

i² = −1
Definition
|z| = √(a²+b²)
Modulus
θ = atan2(b,a)
Argument
e^(iπ)=−1
Euler

⚠️ Disclaimer: This calculator is for educational purposes. For critical engineering or scientific work, verify with specialized software. Angles in degrees.

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