How Powers of i Are Calculated
The imaginary unit i is defined as the square root of -1, so i² = -1. Because multiplying by i four times brings you back to 1, every integer power of i falls into a repeating 4-value cycle: i¹ = i, i² = -1, i³ = -i, i⁴ = 1. This calculator finds where a given integer exponent lands in that cycle and reports the simplified result.
Formula and method
To simplify iⁿ for any integer n, divide n by 4 and keep the remainder: n = 4q + r, where r is 0, 1, 2, or 3. Because i⁴ = 1, the quotient q contributes nothing — i^(4q) = (i⁴)^q = 1^q = 1 — so iⁿ = i^r. Look up r in the four-value cycle: r = 0 gives 1, r = 1 gives i, r = 2 gives -1, r = 3 gives -i. The same rule holds for negative exponents; just make sure the remainder you use is between 0 and 3 (for example, -1 mod 4 = 3, so i⁻¹ = i³ = -i).
Common mistakes
- Negative-exponent remainders: in many languages, -1 % 4 evaluates to -1, not 3 — always adjust so the remainder stays between 0 and 3 before looking up the result.
- Treating i like an ordinary real base: i² is not "i squared as a positive value" — by definition it equals -1.
- Assuming the pattern breaks for large n: the 4-value cycle holds for every integer exponent, no matter how large in magnitude.
Real-world applications
- Electrical engineering uses the imaginary unit (commonly written j) to represent phase-shifted quantities in AC circuit and impedance calculations.
- Signal processing and Fourier analysis use powers of i to rotate complex exponentials by 90-degree increments.
- Control theory and quantum mechanics rely on the same cyclical property when analyzing eigenvalues and phase behavior.
Frequently Asked Questions
What is the imaginary unit i?
i is defined as the square root of -1, meaning i² = -1. It is not a real number, but it lets mathematicians work with square roots of negative numbers and forms the basis of complex numbers written as a + bi.
Why do powers of i repeat every 4 steps?
Because i⁴ = i² × i² = (-1) × (-1) = 1, multiplying by i four times returns to 1. Every later power just repeats the same 4 values (i, -1, -i, 1) since i^(n+4) = i^n × i⁴ = i^n × 1.
How do you calculate i raised to a negative power?
Use the same mod-4 rule, but keep the remainder positive: compute n mod 4 and add 4 if the result is negative. For example, i⁻³ has remainder -3 + 4 = 1, so i⁻³ = i¹ = i.
What is i to the power of 0?
Any nonzero number raised to the power 0 equals 1, and i follows the same rule: i⁰ = 1.
Common Tip Percentages
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| Restaurant (sit-down) | 15–20% | 20%+ for exceptional service |
| Food delivery | 10–20% | Consider extra for long distances |
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General Household Budget Ratios (50/30/20 Rule)
| Category | % of Net Income | Examples |
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| Needs | 50% | Rent, groceries, utilities, insurance |
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Common Measurement Conversions for Everyday Use
| Quantity | US / Imperial | Metric Equivalent |
|---|
| 1 cup | 8 fl oz | 237 mL |
| 1 pint | 16 fl oz | 473 mL |
| 1 quart | 32 fl oz | 946 mL |
| 1 gallon | 128 fl oz | 3.785 L |
| 1 lb | 16 oz | 453.6 g |
| 1 ft | 12 in | 30.48 cm |