Simplify each power of i. i^23

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^23, we calculate 23 mod 4, which equals 3, leading to i^23 = i^3 = -i.

Modulo Operation

The modulo operation finds the remainder of division of one number by another. In the context of simplifying powers of 'i', it helps determine the equivalent lower power by reducing the exponent. For instance, when calculating i^23, we use 23 mod 4 to find the remainder, which is 3, indicating that i^23 is equivalent to i^3. This technique is essential for efficiently handling large exponents.