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Chapter 1: Problem 51

Powers Evaluate the power, and write the result in the form \(a+b i\) $$i^{1000}$$

Short Answer

Expert verified
The result is \(1 + 0i\).

Step by step solution

01

Understand the Cycle of i

The imaginary unit, \(i\), satisfies \(i^2 = -1\). Therefore, powers of \(i\) repeat in a cycle: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), \(i^4 = 1\). This cycle repeats every 4 terms. We can use this repeating cycle to simplify higher powers of \(i\).
02

Determine the Remainder

To find \(i^{1000}\), determine where this power falls within the cycle by dividing 1000 by 4. Find the remainder: \[ 1000 \div 4 = 250, \text{ remainder } 0. \]Since the remainder is 0, \(i^{1000}\) corresponds to \(i^4\) in the cycle.
03

Evaluate i(1000)

From the cycle, \(i^4 = 1\). Since the remainder is 0, \(i^{1000} = i^4 = 1\).
04

Write the Result in a + bi Form

Since \(i^{1000} = 1\), it can be expressed in the form \(a + bi\) as \(1 + 0i\).

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Key Concepts

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

Complex numbers
Complex numbers are numbers that consist of a real part and an imaginary part. They are expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit, satisfying the property that \(i^2 = -1\).
  • The real component \(a\) represents a standard real number.
  • The imaginary component \(bi\) involves the imaginary unit.
These numbers broaden the concept of how we handle equations because they allow solutions to equations that have no real number solutions. For instance, the equation \(x^2 + 1 = 0\) has no real solutions, but its solutions are complex numbers \(i\) and \(-i\). Complex numbers have wide applications in engineering, physics, and mathematics as they provide a way to describe quantities that are not easily explained using only real numbers.
Powers of i
The imaginary unit \(i\) is a fundamental component of complex numbers with a special behavior when raised to various powers. Understanding the powers of \(i\) requires recognizing its cyclical nature:
  • \(i^1 = i\)
  • \(i^2 = -1\)
  • \(i^3 = -i\)
  • \(i^4 = 1\)
This pattern repeats every four terms, meaning that any power of \(i\) can be reduced to one of these four results. For example, to solve \(i^{1000}\), you can use this cycle to find that it corresponds to \(i^4\), which equals 1.
This cyclical pattern helps simplify computations and expressions involving high powers of imaginary numbers, providing a clear pathway to solutions in exercises and real-world applications.
Cyclic patterns
Cyclic patterns in mathematics refer to sequences that repeat over a set cycle, which can help simplify complex calculations. The powers of the imaginary unit \(i\) demonstrate such a pattern effectively. As seen, the cycle for powers of \(i\) is four terms long (\(i, -1, -i, 1\)).
To simplify a power like \(i^{1000}\):
  • Divide the exponent (1000) by the cycle length (4).
  • Observe the remainder.
  • This remainder will determine which power in the cycle \(i^{1000}\) matches.
Because 1000 divided by 4 leaves a remainder of 0, you fall back to the first element of the cycle, which is \(i^4 = 1\).
Identifying these cycles is a powerful tool for simplification and can make even complex algebraic expressions more manageable.
Complex plane representation
The complex plane, also known as the Cartesian plane for complex numbers, is a method to visually represent complex numbers. In this system:
  • The horizontal axis (real axis) represents the real part \(a\) of a complex number \(a + bi\).
  • The vertical axis (imaginary axis) represents the imaginary part \(b\).
Each complex number corresponds to a unique point on this plane, giving a visual depiction that is helpful in understanding their properties.Complex plane representation aids in comprehending addition, subtraction, and the concept of magnitude for complex numbers. By plotting these numbers, one can gain better insight into operations involving complex numbers, showcasing the blending of real and imaginary components into this rich numerical system.

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Most popular questions from this chapter

Plot the points \(P(0,3), Q(2,2),\) and \(R(5,3)\) on a coordinate plane. Where should the point \(S\) be located so that the figure \(P Q R S\) is a parallelogram? Write a brief description of the steps you took and your reasons for taking them.

Simplify the expression. (a) \((4 b)^{1 / 2}\left(8 b^{1 / 4}\right)\) (b) \(\left(3 a^{3 / 4}\right)^{2}\left(5 a^{1 / 2}\right)\)

Simplify the expression. (a) \(\left(\frac{x^{8} y^{-4}}{16 y^{4 / 3}}\right)^{-1 / 4}\) (b) \(\left(\frac{4 s^{3} t^{4}}{s^{2} t^{9 / 2}}\right)^{-1 / 2}\)

Write the number indicated in each statement in scientific notation. (a) A light-year, the distance that light travels in one year, is about \(5,900,000,000,000 \mathrm{mi}\) (b) The diameter of an electron is about \(0.0000000000004 \mathrm{cm}\) (c) A drop of water contains more than 33 billion billion molecules.

Evaluate each expression. (a) \(16^{1 / 4}\) (b) \(-8^{1 / 3}\) (c) \(9^{-1 / 2}\)
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