The imaginary unit \( i \) has the property that \( i^2 = -1 \). Therefore, powers of \( i \) follow a repeating cycle every 4 terms:

\[
\begin{aligned}
i^1 &= i, \\
i^2 &= -1, \\
i^3 &= -i, \\
i^4 &= 1.
\end{aligned}
\]

To compute \( i^{100} \), we can use the fact that powers of \( i \) repeat every 4 steps by finding the remainder when 100 is divided by 4.

\[
100 \div 4 = 25 \text{ remainder } 0.
\]

Since the remainder is 0, we conclude that \( i^{100} = i^0 \). And by definition, \( i^0 = 1 \).

Thus, \( i^{100} = 1 \).

Would you like further details, or do you have any questions?

Here are some related questions:
1. What is \( i^{101} \)?
2. How can we extend this cycle to negative powers of \( i \)?
3. Can powers of complex numbers other than \( i \) also repeat in cycles?
4. What are the applications of imaginary numbers in real-world scenarios?
5. How can Euler's formula relate to powers of \( i \)?

**Tip:** When working with imaginary numbers, always remember their cyclic nature, especially for higher powers.

Learn how to calculate i^100 using the cyclic nature of the powers of the imaginary unit. This problem explores key concepts in complex numbers and modular arithmetic, perfect for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation