Powers of i Calculator
Reduce any positive, zero, or negative integer exponent to one of the four exact values in the repeating imaginary-unit cycle.
Input
Enter a signed whole number, such as 42, 0, or – 7. Up to 1,000 digits are supported exactly.
The calculator uses Euclidean division: n = 4q + r, where the normalized remainder r is always 0, 1, 2, or 3.
Exact result
Simplified value
i¹²³ = – i
Normalized remainder
3
Complete four-step cycles
30
Number type
Pure imaginary
Real coefficient
0
Imaginary coefficient
– 1
123 = 4 × 30 + 3, so i¹²³ = i³ = – i.
Four-value cycle lookup
The highlighted row is selected by the current normalized remainder.
| Remainder r | Simplified value | Number type | Real coefficient | Imaginary coefficient |
|---|---|---|---|---|
| 0 | 1 | Real | 1 | 0 |
| 1 | i | Pure imaginary | 0 | 1 |
| 2 | – 1 | Real | – 1 | 0 |
| 3 | – i | Pure imaginary | 0 | – 1 |
For negative exponents, the same table applies because the remainder is normalized into the range 0 through 3. For example, – 1 mod 4 is treated as 3, giving i⁻¹ = – i.
How to use the powers of i calculator
What this calculator does
This calculator simplifies i raised to any integer exponent. The imaginary unit i is defined by i² = – 1, and its integer powers repeat every four steps: 1, i, – 1, – i, then back to 1. The tool identifies the exponent's position in that cycle, returns the exact simplified value, and shows the real and imaginary coefficients of the result. It is an exact algebraic identity for integer exponents; it does not evaluate fractional or complex exponents, which require branch choices and a broader complex-analysis model. For a foundation on the notation and the complex number system, see the OpenStax introduction to complex numbers.
When to use it
Use the calculator when simplifying algebraic expressions that contain large powers of i, checking homework involving imaginary numbers, reducing negative integer powers, or verifying an intermediate step before solving a quadratic or polynomial equation. It is especially useful when the exponent is too large to expand manually but its remainder after division by four is easy to interpret.
How to calculate
- The calculator opens with the demonstration exponent 123, so the result and the Excel workbook are ready immediately.
- Replace 123 in Exponent n with a signed whole number. Digits may be preceded by + or – . Do not use commas, decimal points, spaces inside the number, or scientific notation.
- Read Simplified value first, then use Normalized remainder and Complete four-step cycles to see why that value was selected.
- Review the coefficient cards and the highlighted row in Four-value cycle lookup when you need the result in the standard form a + bi.
- Select Download Excel to export the current exact input, result, decomposition, and cycle table. Select Reset to clear the demonstration data. Reset may disable the download until a complete valid integer is entered again.
Input guide
Exponent n is the only required input. Enter a base-10 integer with an optional leading plus or minus sign; examples include 42, 0, – 7, and 123. The calculator accepts up to 1,000 digits and evaluates the cycle exactly, without converting the exponent to a floating-point number. Increasing or decreasing n does not make the magnitude grow – the magnitude of every integer power of i remains 1 – but it can move the result to another place in the four-step cycle. A common mistake is entering a decimal such as 2.5, a grouped value such as 1,000, or scientific notation such as 1e3; those formats are rejected because this calculator is intentionally limited to integer powers.
Output guide
Simplified value is the exact identity iⁿ ∈ {1, i, – 1, – i}. Normalized remainder is n mod 4 expressed as 0, 1, 2, or 3, including for negative n. Complete four-step cycles is the integer q in n = 4q + r; it may be negative when n is negative. Number type says whether the result is real or pure imaginary. Real coefficient and Imaginary coefficient give a and b in a + bi, so – i is shown as a = 0 and b = – 1. Exact decomposition displays the Euclidean-division step used to select the answer. In the lookup table, the columns Remainder r, Simplified value, Number type, Real coefficient, and Imaginary coefficient show all four possible states; the highlighted row is the current state. Zero is a valid exponent and gives i⁰ = 1.
Worked example
With the startup value n = 123, divide 123 by 4. This gives 30 complete cycles and remainder 3:
123 = 4 × 30 + 3Only the remainder determines the power, so i¹²³ = i³. The third position in the cycle is – i. Therefore the displayed exact result is i¹²³ = – i, the normalized remainder is 3, the number type is pure imaginary, the real coefficient is 0, and the imaginary coefficient is – 1. The same cycle logic follows from the standard definition of the imaginary unit.
Why powers of i repeat
Because i² = – 1, squaring once more gives i⁴ = (i²)² = ( – 1)² = 1. Multiplying by i⁴ therefore leaves any power unchanged: iⁿ⁺⁴ = iⁿ × i⁴ = iⁿ. That identity is why the cycle length is four and why very large exponents can be reduced to a small remainder calculation.
The same pattern can also be viewed through Euler's formula, eⁱˣ = cos x + i sin x. Since i = eⁱπ⁄², an integer power rotates by another quarter-turn on the complex plane. Four quarter-turns complete one full revolution. The MathWorld explanation of Euler's formula gives the broader connection between trigonometry, exponential notation, and complex numbers.
Negative exponents and interpretation
Negative powers do not create new values because i has multiplicative inverse – i: i × ( – i) = 1. Thus i⁻¹ = – i, i⁻² = – 1, i⁻³ = i, and i⁻⁴ = 1. Normalized modulo arithmetic captures this backward movement cleanly. For example, – 7 = 4 × ( – 2) + 1, so the normalized remainder is 1 and i⁻⁷ = i.
Remember that the output is exact, not a decimal approximation. A real result has imaginary coefficient 0, while a pure imaginary result has real coefficient 0. The magnitude is always 1, so a larger exponent changes direction in the complex plane rather than size.