Which expression is equivalent to i 233?

Step 1: Define the Problem

Identify the key components of the question that need to be addressed:

• Determine the equivalent expression for $i^{233}$.

Step 2: Break Down the Problem

Divide the problem into smaller, manageable parts:

• Understand the properties of the imaginary unit $i$.
• Use the cyclical nature of powers of $i$.

Step 3: Apply Relevant Concepts

Use appropriate academic theories or formulas to solve each part of the problem:

• The powers of $i$ cycle every four:
  • $i^1 = i$
  • $i^2 = -1$
  • $i^3 = -i$
  • $i^4 = 1$
• Therefore, $i^n$ can be determined by finding $n \mod 4$.

Step 4: Analysis, Verify and Summarize

Analysis each part in details if necessary carry out the calculations by steps. Verify the calculations result and summarize the findings:

• Calculate $233 \mod 4$: $233 \div 4 = 58 \text{ remainder } 1$ So, $233 \equiv 1 \mod 4$.

Final Answer

Present the final result in a concise manner:

• Therefore, $i^{233} = i^1 = i$.