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Chapter 7: Problem 26

Write each expression in the form \(a+b i,\) where a and b are real numbers. \(i^{1003}\)

Short Answer

Expert verified
The simplified expression is \(0 - i\), where \(a = 0\) and \(b = -1\).

Step by step solution

01

Identify a pattern in powers of i

We will first find a pattern in powers of \(i\), which will help us simplify \(i^{1003}\): \(i^1 = i\) \(i^2 = -1\) \(i^3 = i^3 = i^2 \cdot i^1 = (-1)i = -i\) \(i^4 = i^4 = i^2 \cdot i^2 = (-1)(-1) = 1\) \(i^5 = i^5 = i^4 \cdot i^1 = (1)i = i\) We can observe that the pattern repeats itself every 4 powers of \(i\).
02

Simplify the given expression using the pattern

Knowing that the pattern repeats every 4 powers of \(i\), we can simplify \(i^{1003}\) as follows: \(i^{1003} = i^{1000} \cdot i^3\) Since \(i^4 = 1\), we can write \(i^{1000}\) as: \(i^{1000} = (i^4)^{250} = 1^{250} = 1\) Now we substitute this back into the original expression: \(i^{1003} = 1 \cdot i^3 = i^3 = -i\)
03

Write the expression in the form a + bi

Finally, we rewrite the simplified expression in the form of \(a + bi\): \(i^{1003} = -i = 0 - 1i = 0 - i\) So, in the form of \(a + bi\), it is equal to \(0 - i\), where \(a = 0\) and \(b = -1\).

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Key Concepts

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

Powers of Imaginary Unit
In the world of complex numbers, the imaginary unit, denoted as \(i\), is a fundamental concept. The imaginary unit \(i\) is defined as the square root of \(-1\), i.e., \(i^2 = -1\). Understanding the powers of \(i\) is essential in simplifying complex expressions.
To comprehend the behavior of \(i\), it's helpful to observe how its powers progress:
  • \(i^1 = i\)
  • \(i^2 = -1\)
  • \(i^3 = -i\)
  • \(i^4 = 1\)
The pattern here is crucial鈥攖hese values repeat every four powers. This cyclical pattern is: \(i, -1, -i, 1\). Recognizing this cycle allows us to simplify higher powers of \(i\) easily by determining the remainder when divided by four. For example, since \(1003 \mod 4 = 3\), it follows that \(i^{1003} = i^3 = -i\).
Simplification of Expressions
Simplifying expressions involving powers of \(i\) relies heavily on identifying and applying the cycle of powers of \(i\). Once this cycle is understood, you can reduce large powers into their simplest form.
For the given example of \(i^{1003}\), we approach the problem by expressing the power as a combination of full cycles.
  • First, find the closest multiple of 4 to the power, which is \(i^{1000}\).
  • Since \(i^4 = 1\), you can simplify \(i^{1000}\) as \((i^4)^{250} = 1^{250} = 1\).
  • Then multiply by the remaining parts, given by the remainder: \(i^{1003} = i^{1000} \cdot i^3 = 1 \cdot i^3 = -i\).
This method breaks down complex expressions into manageable pieces, making simplification straightforward and easily achieved.
Patterns in Mathematics
Mathematical patterns simplify calculations and problem-solving by providing predictable sequences or structures. In complex numbers, recognizing patterns like the power cycle of \(i\) allows us to quickly solve what might seem to be daunting calculations at first glance.
Patterns minimize computation by revealing shortcuts, like repeating cycles or symmetry, which help in efficiently simplifying expressions.
  • For instance, the cycle \(i, -1, -i, 1\) repeats, allowing rapid simplification of the powers of \(i\).
  • Understanding how to leverage these patterns in numerical computations can significantly reduce the time and effort needed.
Exploring and identifying such patterns is a key skill in mathematics, and it helps build a deeper understanding of the underlying principles, enabling students to tackle more complex problems with ease and confidence.

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