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Chapter 8: Problem 26

Simplify each expression to a single complex number. $$ i^{11} $$

Short Answer

Expert verified
The expression simplifies to \( -i \).

Step by step solution

01

Identify the Pattern of Powers of i

The complex number \( i \) has the property that \( i^2 = -1 \). By continuing this pattern, we find that \( i^3 = i^2 \cdot i = -1 \cdot i = -i \) and \( i^4 = i^3 \cdot i = -i \cdot i = -i^2 = 1 \). Since \( i^4 = 1 \), the powers of \( i \) cycle every four numbers: \( i, -1, -i, 1 \).
02

Find the Remainder of the Exponent Divided by 4

To simplify \( i^{11} \), we can use the fact that the powers of \( i \) cycle every four terms. We calculate the remainder when 11 is divided by 4. Performing this division, we get \( 11 \div 4 = 2 \) with a remainder of 3. This means \( i^{11} = i^3 \).
03

Use the Cycle to Simplify the Expression

According to the pattern identified, \( i^3 = -i \). Therefore, \( i^{11} = i^3 = -i \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Powers of i
Complex numbers often involve the imaginary unit denoted by the symbol \( i \). The equation \( i^2 = -1 \) is fundamental to understanding how \( i \) behaves. To explore further, let's consider higher powers of \( i \):
  • \( i^1 = i \)
  • \( i^2 = -1 \)
  • \( i^3 = i^2 \cdot i = -1 \cdot i = -i \)
  • \( i^4 = i^3 \cdot i = -i \cdot i = -(-1) = 1 \)
Notice how after \( i^4 \), the pattern starts again with \( i^5 = i \). This repeating sequence is key when performing calculations involving complex numbers. By understanding powers of \( i \), you can simplify complex expressions easily.
Simplifying Expressions
Simplifying expressions involving complex numbers often requires reducing higher powers of \( i \) to their simplest form. Following the cyclic pattern of powers of \( i \), you can easily determine the simplest equivalent. For instance, if you encounter \( i^{11} \), you don't need to calculate it step-by-step up to \( i^{11} \). Instead, knowing that the powers cycle every four terms, you can determine that:
  • \( i^5 \) is equivalent to \( i \)
  • \( i^6 \) simplifies to \(-1 \)
Continue this pattern to verify the simplification for any power of \( i \). This approach makes handling complex numbers much more straightforward.
Remainders
The use of remainders is crucial when simplifying powers of \( i \). This comes into play when dividing the exponent by 4, as the powers of \( i \) repeat every four steps. To simplify, follow these straightforward steps:
  • Divide the exponent by 4.
  • Find the remainder, which will be between 0 and 3.
  • Match this remainder against the known cycle: 0 for 1, 1 for \( i \), 2 for \(-1 \), and 3 for \(-i \).
For example, when simplifying \( i^{11} \), divide 11 by 4, resulting in a remainder of 3. This tells you that \( i^{11} = i^3 = -i \). Using remainders allows you to quickly reduce complex expressions to their basic form.
Cyclic Patterns
A deep understanding of cyclic patterns allows for simplification of complex numbers without tedious calculations. The powers of \( i \) produce a clear sequence that repeats every four terms: \( i, -1, -i, 1 \). Recognizing and utilizing this cycle helps in managing complex number operations efficiently. Here's how you can apply cyclic patterns:
  • Identify the cycle: Always cycle through \( i, -1, -i, 1 \).
  • Apply the cycle by using remainders of the exponent's division by 4.
  • Simplify redundant calculations by recognizing where in the cycle the expression lies.
Understanding cyclic patterns provides a strategic shortcut in algebra involving complex numbers, ensuring accuracy and saving time.

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