Imaginary Number Calculator
Perform operations on complex numbers with real and imaginary parts
Complex Number Operations
First Complex Number (z₁ = a + bi)
z₁ Properties
0
0.000
0.000°
Second Complex Number (z₂ = c + di)
z₂ Properties
0
0.000
0.000°
Display Settings
Complex Number Operations Results
Basic Operations
Advanced Operations
Operation Formulas
Addition: (a + bi) + (c + di) = (a + c) + (b + d)i
Subtraction: (a + bi) - (c + di) = (a - c) + (b - d)i
Multiplication: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
Division: (a + bi)/(c + di) = [(ac + bd) + (bc - ad)i]/(c² + d²)
Magnitude: |z| = √(a² + b²)
Phase: φ = tan⁻¹(b/a)
Natural Log: ln(z) = ln|z| + iφ
Example Calculation
Example: Complex Number Operations
Problem: Find the natural logarithm of z = 5 + 7i
Given: z = 5 + 7i
Solution
Step 1: Find magnitude |z| = √(5² + 7²) = √(25 + 49) = √74 = 8.602
Step 2: Find phase φ = tan⁻¹(7/5) = tan⁻¹(1.4) = 0.951 rad = 54.46°
Step 3: Apply natural log formula ln(z) = ln|z| + iφ
Step 4: ln(5 + 7i) = ln(8.602) + i(0.951) = 2.152 + 0.951i
Complex Number Forms
Rectangular Form
z = a + bi
Standard form with real and imaginary parts
Polar Form
z = |z|e^(iφ)
Magnitude and phase representation
Trigonometric Form
z = |z|(cos φ + i sin φ)
Using trigonometric functions
Imaginary Unit
i² = -1
Fundamental imaginary unit
Calculation Tips
Complex numbers extend real numbers to solve x² + 1 = 0
Imaginary numbers have zero real part (pure imaginary)
Use polar form for multiplication and division
Phase angle is measured from positive real axis
Understanding Complex and Imaginary Numbers
What are Complex Numbers?
Complex numbers are numbers that consist of a real part and an imaginary part, written in the form z = a + bi, where a and b are real numbers and i is the imaginary unit with the property that i² = -1.
Real-World Applications
- •Electrical engineering (AC circuits and impedance)
- •Quantum mechanics and wave functions
- •Signal processing and Fourier transforms
- •Control systems and stability analysis
- •Computer graphics and 2D rotations
Key Properties
Imaginary Unit: i² = -1, i³ = -i, i⁴ = 1
Complex Conjugate: z* = a - bi
Magnitude: |z| = √(a² + b²)
Argument: arg(z) = tan⁻¹(b/a)
Operation Rules
- • Addition: Combine like terms
- • Multiplication: Use FOIL and i² = -1
- • Division: Multiply by conjugate of denominator
- • Powers: Best computed in polar form
- • Logarithms: ln(z) = ln|z| + i arg(z)
Note: Every polynomial equation has solutions in the complex numbers (Fundamental Theorem of Algebra)