Simplify each power of i. i^1001

Key Concepts

Imaginary Unit (i)

The imaginary unit 'i' is defined as the square root of -1. It is a fundamental concept in complex numbers, allowing for the extension of the real number system to include solutions to equations that do not have real solutions, such as x^2 + 1 = 0. Powers of 'i' cycle through a predictable pattern: i^1 = i, i^2 = -1, i^3 = -i, and i^4 = 1.

Powers of i

The powers of 'i' exhibit a cyclical behavior every four terms. Specifically, i^1 = i, i^2 = -1, i^3 = -i, and i^4 = 1. This cycle repeats, meaning that to simplify higher powers of 'i', one can reduce the exponent modulo 4. For example, to simplify i^1001, we calculate 1001 mod 4, which helps determine the equivalent lower power.

Modulo Operation

The modulo operation finds the remainder of division of one number by another. In the context of simplifying powers of 'i', we use modulo 4 to determine which power of 'i' corresponds to a larger exponent. For instance, calculating 1001 mod 4 gives a remainder of 1, indicating that i^1001 is equivalent to i^1, which simplifies to 'i'.