Simplify each power of i. i^29

Key Concepts

Imaginary Unit (i)

The imaginary unit, denoted as '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

The powers of 'i' follow a cyclical pattern: i^1 = i, i^2 = -1, i^3 = -i, and i^4 = 1. This cycle repeats every four powers, which means any power of 'i' can be simplified by finding the remainder when the exponent is divided by 4.

Modular Arithmetic

Modular arithmetic is a system of arithmetic for integers, where numbers wrap around after reaching a certain value, known as the modulus. In the context of simplifying powers of 'i', we use modulus 4 to determine the equivalent lower power of 'i' by calculating the exponent modulo 4.