Powers of i Calculator

Name: Powers of i Calculator
Author: Jitendra Kumar

Calculate i raised to any integer power with modulo-4 reduction, cycle steps, nearby powers, and complex a + bi output.

Last Updated: May 2026

Power Result

-1

Remainder Mod 4

Reduction

i^2 = i^2

Complex Form

-1 + 0i

Powers of i Input

Enter any integer exponent. The calculator reduces it modulo 4 and returns the exact value in the repeating imaginary-unit cycle.

Exponent (n)

Use an integer such as 2, 17, -5, or 2026.

Four-Value Cycle

Remainder	Reduced power	Value	Pattern
n mod 4 = 0	i^0	1	Examples: i^0, i^4, i^8
n mod 4 = 1	i^1	i	Examples: i^1, i^5, i^9
n mod 4 = 2	i^2	-1	Examples: i^2, i^6, i^10
n mod 4 = 3	i^3	-i	Examples: i^3, i^7, i^11

Reduction Steps

Step	Calculation	Result
Input exponent	2	Use the integer power n.
Cycle length	4	The values repeat every four powers.
Modulo reduction	2 mod 4	2
Reduced power	i^2	-1

Nearby Powers

Power	Modulo check	Value	Note
i^-2	-2 mod 4 = 2	-1	Nearby cycle value
i^-1	-1 mod 4 = 3	-i	Nearby cycle value
i^0	0 mod 4 = 0	1	Nearby cycle value
i^1	1 mod 4 = 1	i	Nearby cycle value
i^2	2 mod 4 = 2	-1	Selected exponent
i^3	3 mod 4 = 3	-i	Nearby cycle value
i^4	4 mod 4 = 0	1	Nearby cycle value
i^5	5 mod 4 = 1	i	Nearby cycle value
i^6	6 mod 4 = 2	-1	Nearby cycle value

Complex Number Notice

This calculator handles integer powers of the imaginary unit i. It does not evaluate general complex expressions, fractional powers, or symbolic simplification beyond the i power cycle.

Reviewed For Methodology, Labels, And Sources

Every CalculatorWallah calculator is published with visible update labeling, linked source references, and review of formula clarity on trust-sensitive topics. Use results as planning support, then verify institution-, policy-, or jurisdiction-specific rules where they apply.

Reviewed By

Jitendra Kumar, Founder & Editorial Standards Lead, reviews methodology, labels, assumptions, and trust-sensitive publishing decisions for this topic area.

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Topic Ownership

Sales tax and tax-sensitive estimate tools, Education and GPA planning calculators, Health, protein, and screening-formula pages, Platform-wide publishing standards and methodology

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Methodology & Updates

Page updated May 2026. Trust-critical pages are reviewed when official rates or rules change. Evergreen calculator guides are checked on a recurring quarterly or annual cycle depending on topic volatility.

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How to Use the Powers of i Calculator

Enter any integer exponent n. The calculator evaluates i^n exactly, including negative exponents and very large integer exponents.

Read the modulo-4 reduction first, then compare the result against the four-value cycle and nearby powers table.

Step 1: Enter an integer exponent
Use values such as 2, 17, -5, 0, or 2026.
Step 2: Reduce modulo 4
The calculator finds the exponent remainder in the repeating four-value cycle.
Step 3: Read the simplified value
The result is always one of 1, i, -1, or -i.
Step 4: Review nearby powers
Use the table to see how the selected power fits into the cycle.

How This Powers of i Calculator Works

The imaginary unit is defined by i^2 = -1. From that rule, i^3 = -i and i^4 = 1. Once i^4 returns to 1, the same four values repeat indefinitely.

The calculator reduces the exponent n modulo 4. A remainder of 0 gives 1, a remainder of 1 gives i, a remainder of 2 gives -1, and a remainder of 3 gives -i.

Negative exponents fit the same cycle because the calculator uses a nonnegative modulo remainder. For example, -1 mod 4 is 3, so i^-1 = -i.

Powers of i Guide

Four-Power Cycle

Power	Value	Remainder	Meaning
i^0	1	0 mod 4	Returns to the real number 1.
i^1	i	1 mod 4	The imaginary unit itself.
i^2	-1	2 mod 4	The defining rule for i.
i^3	-i	3 mod 4	Multiply i^2 by i.
i^4	1	0 mod 4	The cycle restarts.

Examples

Power	Reduction	Result
i^17	17 mod 4 = 1	i
i^2026	2026 mod 4 = 2	-1
i^-1	-1 mod 4 = 3	-i
i^-17	-17 mod 4 = 3	-i
i^0	0 mod 4 = 0	1

Complex Number Context

Powers of i are a compact introduction to complex numbers. Each result can be written in a + bi form: 1 is 1 + 0i, i is 0 + 1i, -1 is -1 + 0i, and -i is 0 - 1i.

The modulo pattern is also a useful bridge to modular arithmetic. Instead of expanding a large power, only the exponent remainder modulo 4 is needed.

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