Complex Number Operations
Work with imaginary and complex numbers using standard mathematical notation. The imaginary unit i represents the square root of -1.
Input Formats
Enter complex numbers in multiple ways:
3+2i → 3 + 2i
5-3i → 5 - 3i
i → i (pure imaginary)
-2i → -2i
Rectangular Notation
Use parentheses with comma for (real, imaginary):
(3,4) → 3 + 4i
(-2,5) → -2 + 5i
rect(3,4) → 3 + 4i
Polar Notation
Use semicolon for (magnitude;angle):
(5;45) → polar, 45 degrees
polar(2,90) → 2i
Angle uses current deg/rad mode
Addition & Subtraction
Add and subtract complex numbers:
2+3i + 3+2i → 5 + 5i
(3,4) + (1,2) → 4 + 6i
5+2i - 3+i → 2 + i
Multiplication
Multiply complex numbers:
i * i → -1
(3+2i) * (1-i) → 5 - i
2i * 3i → -6
Division
Divide complex numbers:
(3+4i) / (1+i) → 3.5 + 0.5i
1 / i → -i
(6+3i) / 3 → 2 + i
Powers & Roots
Calculate powers and square roots:
(1+i)^2 → 2i
sqrt(-1) → i
sqrt(-4) → 2i
Helper Functions
Extract components of complex numbers:
re(3+4i) → 3 (real part)
im(3+4i) → 4 (imaginary part)
conj(3+4i) → 3 - 4i (conjugate)
Magnitude & Argument
Get polar form components:
mag(3+4i) → 5
arg(1+i) → 45 (in degrees)
abs(3+4i) → 5
Trigonometric Functions
Complex-valued trig functions:
sin(i) → complex result
cos(1+i) → complex result
Works with all trig functions
Logarithms & Exponentials
Complex logarithms and exponentials:
ln(-1) → complex result
exp(i*3.14159) ≈ -1
(Euler's formula: e^(iπ) = -1)
Other Features
This is a full scientific calculator! Press AC to clear and access all calculator functions including graphing, fractions, and more.