Powers of i Calculator
Calculation Result
About Powers of i
The imaginary unit i is defined as the square root of -1 (i = √-1). When raised to different powers, i follows a repeating pattern of four values: i, -1, -i, and 1.
Pattern of Powers:
| Power | i⁰ | i¹ | i² | i³ | i⁴ | i⁵ | i⁶ | i⁷ |
|---|---|---|---|---|---|---|---|---|
| Value | 1 | i | -1 | -i | 1 | i | -1 | -i |
Key Properties:
- i² = -1
- i³ = -i
- i⁴ = 1
- i^n = i^(n mod 4)
- i^(-n) = 1/i^n
Example Calculations:
| Power | Calculation | Result | Explanation |
|---|---|---|---|
| i⁵ | i⁴ × i | i | i⁴ = 1, so i⁵ = 1 × i = i |
| i⁶ | i⁴ × i² | -1 | i⁴ = 1, i² = -1, so i⁶ = 1 × -1 = -1 |
Applications:
- Complex number calculations
- Electrical engineering
- Signal processing
- Quantum mechanics
- Control systems
Tips:
- Use the pattern: i, -1, -i, 1
- For large powers, use modulo 4
- Remember i² = -1
- Negative powers can be converted to positive
- Practice with different powers
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