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Chapter 2: Problem 3

\(i^{4}\)

Short Answer

Expert verified
\(i^{4} = 1\)

Step by step solution

01

Identify the value of i

Understand that the imaginary unit, denoted as 'i', is defined by the property: \( i^{2} = -1 \).
02

Simplify the exponent

Recognize that raising 'i' to higher powers is cyclical. Specifically, \( i^{4} = (i^{2})^{2} \).
03

Compute i squared

Since \( i^{2} = -1 \), compute \( i^{4} = (-1)^{2} \).
04

Conclude the calculation

Since \( (-1)^{2} = 1 \), we conclude that \( i^{4} = 1 \).

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Key Concepts

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

Imaginary Unit
The imaginary unit, represented as 'i', is a fundamental concept in complex numbers. It is defined by the property that its square is equal to -1. In mathematical terms, we write this as: \[ i^{2} = -1 \] This definition allows us to extend the real number system to include solutions to equations that don't have real solutions. For instance, the equation \(x^2 + 1 = 0\) has no real solution because the square of a real number cannot be negative. However, using the imaginary unit, we can say the solutions are \(x = i\) and \(x = -i\).
Powers of i
When dealing with the powers of the imaginary unit 'i', it is useful to recognize a pattern. This pattern can be summarized as follows:
  • \( i^{1} = i \)
  • \( i^{2} = -1 \)
  • \( i^{3} = -i \)
  • \( i^{4} = 1 \)
When you calculate higher powers of 'i', the results repeat every four exponents. For example: \( i^{5} = i^{4+1} = i \)Understanding this cyclical nature makes simplifying expressions that involve powers of 'i' much more manageable.
Cyclical Nature of i
The cyclical nature of 'i' is a valuable property because it simplifies computations. After every four operations, the powers of 'i' repeat. This cyclic pattern can be used to break down higher powers of 'i' into simpler, more recognizable forms. For example, to simplify \( i^{58} \), we can use the fact that the powers of 'i' repeat every four terms: \( i^{58} = i^{4 \times 14 + 2} = (i^{4})^{14} \times i^{2} = 1^{14} \times -1 = -1 \) By recognizing and utilizing this cyclical pattern, complex calculations become easier and more intuitive.

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Most popular questions from this chapter

In optics, the following expression needs to be evaluated in calculating the intensity of light transmitted through a film after multiple reflections at the surfaces of the film: $$ \left(\sum_{n=0}^{\infty} r^{2 n} \cos n \theta\right)^{2}+\left(\sum_{n=0}^{\infty} r^{2 n} \sin n \theta\right)^{2} $$ Show that this is equal to \(\left|\sum_{n=0}^{x} r^{2 n} e^{i n \theta}\right|^{2}\) and so evaluate it assuming \(|r|<1\) ( \(r\) is the fraction of light reflected each time).

In the following integrals express the sines and cosines in exponential form and then integrate to show that: $$ \int_{-\pi}^{\pi} \sin 3 x \cos 4 x d x=0 $$

Show that the absolute value of a product of two complex numbers is equal to the product of the absolute values. Also show that the absolute value of the quotient of two complex numbers is the quotient of the absolute values. Hint : : Write the numbers in the \(r e^{i \theta}\) form.

Find each of the following in the \(x+i y\) form. $$ \cosh \left(\frac{i \pi}{2}-\ln 3\right) $$

Find and plot the complex conjugate of each number. \(2 i\)
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