i Calculator

Simplify k·i^n, where i is the imaginary unit (i² = −1). Enter any integer exponent and a real coefficient to reduce the expression using the repeating 4-cycle of powers of i.

Quick Facts

Method
Powers of i repeat in a 4-step cycle
i¹=i, i²=−1, i³=−i, i⁴=1, then it repeats — so i^n = i^r where r = n mod 4.

Your Results

Calculated
Simplified result
-
k · i^n reduced to a+bi form
Reduced exponent
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n mod 4 (position in the cycle)
i^n alone
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Base power before scaling by k
Number type
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Real or pure imaginary

Ready

Enter an integer exponent n and a coefficient k, then press Calculate.

How the i Calculator works

The imaginary unit i is defined by i² = −1 (equivalently, i = √−1). Because every integer power of i lands on just one of four values, any power — no matter how large the exponent — can be simplified instantly once you know where it falls in that cycle. This calculator reduces an expression of the form k·i^n, where n is an integer exponent and k is a real coefficient, to its simplest form.

Formula and method

The powers of i repeat forever in a cycle of four: i¹ = i, i² = −1, i³ = −i, and i⁴ = 1. For any integer n (positive, negative, or zero), find the remainder r when n is divided by 4, adjusted so it falls between 0 and 3: r = ((n mod 4) + 4) mod 4. Then i^n = i^r, so:

  • r = 0: i^n = 1 (a real number)
  • r = 1: i^n = i (a pure imaginary number)
  • r = 2: i^n = −1 (a real number)
  • r = 3: i^n = −i (a pure imaginary number)

Multiplying by the coefficient k simply scales that value, so k·i^n works out to k, k·i, −k, or −k·i depending on r.

Common sources of error

  • Confusing i² with i: i² simplifies to the real number −1, not to another imaginary term — this is the single most common slip.
  • Mishandling negative exponents: i⁻¹ = 1/i = −i, not i; negative exponents still land on the same 4-cycle once the remainder is adjusted to be non-negative.
  • Assuming a large exponent can't be simplified: i^103 is not "too big" — only the remainder after dividing by 4 matters (103 mod 4 = 3, so i^103 = −i).

Checking your result

A quick sanity check: the result should always be purely real (no i term) when the exponent's remainder is 0 or 2, and purely imaginary (all i, no separate real part) when the remainder is 1 or 3. If your simplified answer has both a nonzero real part and a nonzero imaginary part, something went wrong — a pure power of i never produces a mixed a+bi result.

Applications

Powers of i show up constantly in algebra and precalculus courses, complex number arithmetic, electrical engineering (where j is often used instead of i to avoid clashing with current), and signal-processing formulas built on Euler's identity. Reducing k·i^n to its simplest form is usually the first step before combining it with other complex terms.

Frequently Asked Questions

What is the imaginary unit i?
i is defined as the square root of −1, so i² = −1. It is not a real number, since no real number squares to a negative value, which is why mathematicians call it "imaginary." Numbers built from i, like 3 + 2i, are called complex numbers.
How do you simplify a large power of i, such as i^103?
Divide the exponent by 4 and keep only the remainder: 103 ÷ 4 = 25 remainder 3, so i^103 = i³ = −i. Because the four powers of i repeat forever (i, −1, −i, 1), only the remainder after dividing by 4 ever matters, no matter how large the exponent is.
Why do powers of i repeat every 4 steps?
Each multiplication by i rotates a complex number 90° counter-clockwise. Four 90° rotations bring you back to the start (360°), so i⁴ = 1 and the pattern i, −1, −i, 1 repeats indefinitely for every exponent that follows.
What does a negative exponent like i⁻⁵ mean?
A negative exponent means dividing: i⁻⁵ = 1 / i⁵. Using the cycle, i⁵ = i¹ = i, so i⁻⁵ = 1/i = −i. Equivalently, adjust −5 mod 4 to the non-negative remainder 3, which gives i⁻⁵ = i³ = −i — the same answer either way.

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