Simplify each power of i. i¹⁰⁰¹

Verified step by step guidance

1. Recall that the imaginary unit \( i \) has the property \( i^2 = -1 \). Powers of \( i \) cycle every 4 steps: \( i^1 = i \), \( i^2 = -1 \), \( i^3 = -i \), and \( i^4 = 1 \).
2. To simplify \( i^{1001} \), find the remainder when 1001 is divided by 4, since the powers of \( i \) repeat every 4.
3. Calculate \( 1001 \mod 4 \) to determine the equivalent smaller exponent.
4. Use the remainder from the previous step to rewrite \( i^{1001} \) as one of \( i^0, i^1, i^2, \) or \( i^3 \), which correspond to \( 1, i, -1, \) or \( -i \) respectively.
5. Express the simplified form of \( i^{1001} \) based on the equivalent power found.

Key Concepts

Imaginary Unit i

The imaginary unit i is defined as the square root of -1, satisfying i² = -1. It is the fundamental unit used to extend the real number system to complex numbers, allowing for the representation of numbers involving the square roots of negative values.

Powers of i and Their Cyclic Pattern

Powers of i repeat in a cycle of four: i¹ = i, i² = -1, i³ = -i, and i⁴ = 1. This pattern repeats for higher powers, so simplifying i raised to any integer power involves finding the remainder when the exponent is divided by 4.

Modular Arithmetic for Exponent Simplification

Modular arithmetic helps simplify large exponents by reducing them modulo a certain number—in this case, 4. By calculating the exponent modulo 4, we determine the equivalent power of i within the four-step cycle, making simplification straightforward.