Powers of i Calculator
Calculate any power of the imaginary unit i using the cyclic pattern of complex numbers
Calculate Powers of i
Enter any integer (positive, negative, or zero)
Result
Cyclic Pattern
Properties
Complex Number Properties
Real powers: When n ≡ 0 or 2 (mod 4)
Imaginary powers: When n ≡ 1 or 3 (mod 4)
i² = -1 (fundamental definition)
Cycle length: 4 (pattern repeats every 4)
Example Calculations
Positive Powers
Large Powers
100 ÷ 4 = 25 remainder 0
So i¹⁰⁰ = i⁰ = 1
Negative Powers
Zero Power
Any non-zero number to the power of 0 equals 1
Quick Reference
The Four Values
Quick Method
1. Find n mod 4
2. Use remainder to get result
3. Pattern repeats every 4
Mathematical Properties
Understanding Powers of i
What is the Imaginary Unit i?
The imaginary unit i is defined as a number that satisfies the equation i² = -1. Sometimes written informally as i = √(-1), it's the fundamental building block of complex numbers.
The Cyclic Pattern
Powers of i follow a repeating cycle of length 4. This means that i^n depends only on the remainder when n is divided by 4, making calculations much simpler.
Pattern Table
How to Calculate Powers of i
Step 1: Find the Remainder
Divide the exponent by 4 and find the remainder
Step 2: Apply the Pattern
Use the remainder to determine the result from the cycle
Step 3: Handle Negatives
For negative exponents, calculate as 1/i^|n|
Applications
- • Electrical Engineering: AC circuit analysis
- • Physics: Quantum mechanics, wave functions
- • Mathematics: Complex analysis, Fourier transforms
- • Computer Science: Signal processing, graphics