Powers of i Calculator
Enter any integer exponent n and this calculator evaluates i^n instantly. It shows the exact symbolic result (1, i, -1, or -i), the real and imaginary components, the remainder that drives the calculation, and a step-by-step derivation. Negative exponents and large exponents are handled automatically via the 4-cycle rule.
Formula
Worked example
i^7: divide 7 by 4 to get quotient 1 remainder 3. So i^7 = (i^4)^1 x i^3 = 1 x (-i) = -i. Check with Euler: cos(7 x 90 degrees) + i sin(7 x 90 degrees) = cos(630 degrees) + i sin(630 degrees) = 0 + i(-1) = -i. Confirmed.
What are powers of i?
The imaginary unit i is defined as the square root of -1. When you raise i to successive integer powers you get a repeating sequence of exactly four distinct values: i^0 = 1, i^1 = i, i^2 = -1, i^3 = -i, and then i^4 = 1 again, starting the cycle over. This 4-periodicity is one of the most elegant patterns in mathematics. Every integer power of i, no matter how large or how negative, reduces to one of these four numbers.
How to calculate i^n for any integer n
The rule is simple: divide n by 4 and find the remainder r (always 0, 1, 2, or 3). Then i^n equals i^r. For example, i^123: 123 / 4 = 30 remainder 3, so i^123 = i^3 = -i. For negative exponents the same idea applies with a positive-modulo adjustment: i^(-1) has remainder (-1 + 4) = 3, giving -i. You can also verify any result with Euler's formula: i^n = cos(n x 90 degrees) + i sin(n x 90 degrees), since i = e^(i times pi/2) and each full rotation of 360 degrees brings you back to the start.
Negative and fractional exponents
For negative integer exponents, compute n mod 4 using the positive-remainder rule (add 4 if the ordinary remainder is negative). i^(-3) has raw remainder -3, adjusted to 1, so i^(-3) = i. You can cross-check this algebraically: i^(-3) = 1/i^3 = 1/(-i) = i (multiply numerator and denominator by i). Fractional exponents of i produce values on the unit circle in the complex plane but are not purely real or purely imaginary; they require the full complex exponential and are outside the integer scope of this calculator.
Applications of powers of i
The cyclic behavior of i is not just a curiosity. In electrical engineering, AC circuit analysis uses j (the engineering symbol for i) to represent phase rotations of 90 degrees. Each multiplication by j corresponds to a 90-degree phase shift, so j^4 = 1 means four such shifts complete a full rotation back to the original phase. In quantum mechanics, complex amplitudes involve powers of i in wave-function decompositions. In signal processing, the discrete Fourier transform uses complex exponentials that cycle through the same four-quadrant pattern. Understanding the 4-cycle of i is a foundational skill for all these fields.
The 4-cycle of powers of i
| Exponent (n) | i^n | Real part | Imaginary part | Equivalent angles |
|---|---|---|---|---|
| 0 | 1 | 1 | 0 | 0 degrees |
| 1 | i | 0 | 1 | 90 degrees |
| 2 | -1 | -1 | 0 | 180 degrees |
| 3 | -i | 0 | -1 | 270 degrees |
| 4 | 1 (cycle repeats) | 1 | 0 | 360 degrees = 0 degrees |
| -1 | -i (same as i^3) | 0 | -1 | -90 degrees = 270 degrees |
| -2 | -1 (same as i^2) | -1 | 0 | -180 degrees = 180 degrees |
| -3 | i (same as i^1) | 0 | 1 | -270 degrees = 90 degrees |
Every integer power of i reduces to one of four values. The pattern repeats with period 4 for both positive and negative exponents.
Frequently asked questions
What are the only possible values of i^n for integer n?
Exactly four: 1, i, -1, and -i. Because i^4 = 1, the sequence cycles with period 4. No matter what integer you choose for n, i^n will always equal one of these four values.
How do I compute i^n by hand?
Divide n by 4 and keep the remainder r (use positive remainder even for negative n). Then use the lookup: remainder 0 gives 1, remainder 1 gives i, remainder 2 gives -1, remainder 3 gives -i. For example, i^50: 50 / 4 = 12 remainder 2, so i^50 = -1.
What is i^0?
i^0 = 1, following the universal rule that any non-zero number raised to the power 0 equals 1. The remainder of 0 divided by 4 is 0, which maps to 1 in the cycle.
What is i^(-1)?
i^(-1) = -i. You can verify this algebraically: 1/i times i/i = i/i^2 = i/(-1) = -i. Using the positive-remainder rule, -1 mod 4 = 3, and i^3 = -i, which confirms the result.
Can i^n ever be a real number?
Yes, when n is even. If n is divisible by 2, then i^n is either 1 (when n is divisible by 4) or -1 (when n is even but not divisible by 4). When n is odd, i^n is purely imaginary: either i or -i.
What is i^2?
i^2 = -1. This is actually the definition of the imaginary unit: i is the number whose square equals -1. It is the foundational identity from which all other powers of i follow.
How does Euler's formula relate to powers of i?
Euler's formula states that e^(i*theta) = cos(theta) + i*sin(theta). Since i = e^(i*pi/2), we have i^n = e^(i*n*pi/2) = cos(n*pi/2) + i*sin(n*pi/2). At n=0 this gives cos(0)+i*sin(0)=1; at n=1 it gives cos(90 degrees)+i*sin(90 degrees) = i; at n=2 it gives cos(180 degrees)+i*sin(180 degrees) = -1; and at n=3 it gives cos(270 degrees)+i*sin(270 degrees) = -i. The 4-cycle is simply the geometry of quarter-turns on the unit circle.