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Powers of i Calculator

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

2

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.

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

Four-Value Cycle

RemainderReduced powerValuePattern
n mod 4 = 0i^01Examples: i^0, i^4, i^8
n mod 4 = 1i^1iExamples: i^1, i^5, i^9
n mod 4 = 2i^2-1Examples: i^2, i^6, i^10
n mod 4 = 3i^3-iExamples: i^3, i^7, i^11

Reduction Steps

StepCalculationResult
Input exponent2Use the integer power n.
Cycle length4The values repeat every four powers.
Modulo reduction2 mod 42
Reduced poweri^2-1

Nearby Powers

PowerModulo checkValueNote
i^-2-2 mod 4 = 2-1Nearby cycle value
i^-1-1 mod 4 = 3-iNearby cycle value
i^00 mod 4 = 01Nearby cycle value
i^11 mod 4 = 1iNearby cycle value
i^22 mod 4 = 2-1Selected exponent
i^33 mod 4 = 3-iNearby cycle value
i^44 mod 4 = 01Nearby cycle value
i^55 mod 4 = 1iNearby cycle value
i^66 mod 4 = 2-1Nearby 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.

Checked by Jitendra Kumar

Powers of i Calculator is checked for formula labels, source links, and result limits.

Jitendra Kumar, Founder & Editorial Standards Lead. Updated May 2026. Scope: math calculators.

Sources & methodology · Review standards

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.

  1. Step 1: Enter an integer exponent

    Use values such as 2, 17, -5, 0, or 2026.

  2. Step 2: Reduce modulo 4

    The calculator finds the exponent remainder in the repeating four-value cycle.

  3. Step 3: Read the simplified value

    The result is always one of 1, i, -1, or -i.

  4. 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

PowerValueRemainderMeaning
i^010 mod 4Returns to the real number 1.
i^1i1 mod 4The imaginary unit itself.
i^2-12 mod 4The defining rule for i.
i^3-i3 mod 4Multiply i^2 by i.
i^410 mod 4The cycle restarts.

Examples

PowerReductionResult
i^1717 mod 4 = 1i
i^20262026 mod 4 = 2-1
i^-1-1 mod 4 = 3-i
i^-17-17 mod 4 = 3-i
i^00 mod 4 = 01

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.

Keep the research moving with Scientific Calculator, Power Mod Calculator, Modulo Calculator, and Polish Notation Converter.

Frequently Asked Questions

The imaginary unit i is defined by i^2 = -1. It is used to build complex numbers in the form a + bi.

Multiplying by i cycles through 1, i, -1, and -i. After four powers, the value returns to 1, so the pattern repeats every 4 exponents.

Reduce the exponent modulo 4. The remainder determines whether the result is 1, i, -1, or -i.

Yes. The same modulo-4 cycle works for negative integer exponents when using a nonnegative remainder from 0 through 3.

i^0 equals 1, following the usual nonzero-base zero-exponent rule.

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Sources & References

  1. 1.Wolfram MathWorld - Imaginary Number(Accessed May 2026)
  2. 2.Wolfram MathWorld - Complex Number(Accessed May 2026)
  3. 3.Khan Academy - Powers of the Imaginary Unit(Accessed May 2026)