Decide whether each statement is true or false. If false, correct the right side of the equation. i^12 = 1

Key Concepts

Complex Numbers and Imaginary Unit

The imaginary unit 'i' is defined as the square root of -1. In complex numbers, 'i' is used to express numbers that cannot be represented on the real number line. Understanding how 'i' operates is crucial for evaluating powers of 'i' and recognizing its cyclical nature.

Powers of i

The powers of 'i' follow a specific 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 reducing the exponent modulo 4. This concept is essential for determining the value of i raised to any integer exponent.

Modular Arithmetic

Modular arithmetic involves calculations with remainders after division. In the context of powers of 'i', it helps simplify exponents by finding their equivalent within a smaller range, specifically modulo 4 for 'i'. This technique is vital for quickly determining the value of higher powers of 'i' without extensive calculations.