Powers of i Calculator

The imaginary unit i is defined by the equation i² = -1. That single definition extends the real number system into the complex numbers and creates a remarkably simple pattern for integer powers. Starting at exponent zero, the values are 1, i, -1, and -i. Multiplying once more by i returns to 1, so the same four results repeat forever. This calculator finds the correct point in that cycle without carrying out a long chain of multiplication. To evaluate i raised to an integer n, divide n by four and use the remainder. A remainder of zero gives 1, a remainder of one gives i, a remainder of two gives -1, and a remainder of three gives -i. For example, 27 leaves remainder three when divided by four, so i²⁷ equals -i. The quotient does not matter because every complete group of four contributes i⁴, which equals 1 and therefore does not change the product. Negative exponents follow the same cycle. By the reciprocal rule, i to the power -1 equals 1 divided by i, which simplifies to -i. A mathematically normalized modulo operation maps every negative integer to a remainder from zero through three. Thus -6 maps to remainder two and i raised to -6 equals -1. The calculator performs that normalization automatically, avoiding the negative-remainder behavior found in some programming languages. Powers of i appear when simplifying complex-number expressions, solving polynomial equations, studying electrical circuits, and working with waves or rotations. Euler's formula connects complex exponentials to sine and cosine, while multiplication by i corresponds geometrically to a counterclockwise quarter-turn in the complex plane. Four multiplications produce a full rotation, which gives an intuitive explanation for the four-step cycle. Only integer exponents use this unique single-valued cycle. Fractional and general complex powers require roots and logarithms, which can have multiple branches, so they are outside this calculator's scope. Enter zero, a positive integer, or a negative integer. The displayed remainder shows exactly why the result was selected and gives a quick method you can reproduce by hand on an exam. When simplifying a larger expression, reduce each power separately and then combine ordinary coefficients and like terms. For instance, i¹¹ plus i¹⁴ becomes -i plus -1. Keeping the four-value table nearby is often enough for small exponents, but modular reduction is faster and less error-prone when exponents are large or negative.