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Chapter 1: Problem 5

Designate each of the given numbers as being an integer, rational, irrational, real, or imaginary. (More than one designation may be correct.) $$3, \quad \sqrt{-4}, \quad-\frac{\pi}{6}, \quad \frac{1}{8}$$

Short Answer

Expert verified
3: integer, rational, real. \( \sqrt{-4} \): imaginary. \( -\frac{\pi}{6} \): irrational, real. \( \frac{1}{8} \): rational, real.

Step by step solution

01

Identify 3

The number 3 is a whole number without any fractional part. It is straightforwardly classified as an integer. Since all integers are also real numbers, 3 is a real number. Furthermore, integers are rational since they can be expressed as a fraction where the denominator is 1, so 3 is a rational number. Thus, 3 is an integer, rational, and real number.
02

Identify \( \sqrt{-4} \)

The expression \( \sqrt{-4} \) involves taking the square root of a negative number. In mathematics, the square root of a negative number involves imaginary units. \( \sqrt{-4} \) can be written as \( 2i \), where \( i \) is the imaginary unit. Therefore, \( \sqrt{-4} \) is an imaginary number.
03

Identify \( -\frac{\pi}{6} \)

The number \( -\frac{\pi}{6} \) is a negative version of a fraction that involves \( \pi \), which is an irrational number. Consequently, \( -\frac{\pi}{6} \) itself is irrational, as any non-zero multiple of an irrational number is also irrational. Furthermore, since it lies on the number line, it is a real number. Thus, \( -\frac{\pi}{6} \) is an irrational and real number.
04

Identify \( \frac{1}{8} \)

The number \( \frac{1}{8} \) is in fraction form where both the numerator and the denominator are integers, and the denominator is not zero. Thus, it can be written as a terminating decimal, making it a rational number. In addition, it is a real number, as all rational numbers are real. Therefore, \( \frac{1}{8} \) is a rational and real number.

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Key Concepts

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

Integers
An integer is a number without any fractional or decimal part. This means integers are whole numbers, and they can be either positive, negative, or zero. Here are some examples of integers:
  • -3 (negative integer)
  • 0 (neutral, also considered an integer)
  • 7 (positive integer)
All integers are also rational numbers and real numbers because they can be expressed as a fraction where the denominator is 1. For instance, the integer 5 can be represented as \( \frac{5}{1} \). Thus, while not having a fractional component themselves, integers naturally fit into the broader categories of rational and real numbers.
Rational Numbers
Rational numbers are numbers that can be written as a ratio, or fraction, of two integers. The key condition is that the denominator is not zero. This means that each rational number can be expressed in the form \( \frac{a}{b} \) where \( a \) and \( b \) are integers and \( b eq 0 \). Some examples:
  • \( \frac{3}{4} \) is rational because both 3 and 4 are integers.
  • 5 can be considered rational as \( \frac{5}{1} \).
  • 0.75 is rational because it can be expressed as \( \frac{3}{4} \).
Rational numbers also include integers since any whole number can be rewritten with a denominator of 1. They are part of the real number system and can be found on the real number line.
Irrational Numbers
Irrational numbers are numbers that cannot be written as a simple fraction of two integers. Their decimal expansions are non-repeating, non-terminating. These numbers often involve roots or constants such as \( \pi \) or \( e \). Examples include:
  • \( \sqrt{2} \) cannot be precisely expressed as a fraction.
  • \( \pi \) is the ratio of the circumference of any circle to its diameter, and it cannot be exactly written as a fraction.
Despite not being expressible as simple fractions, irrational numbers are real numbers. They fill the number line between rational numbers, making the real number system continuous.
Real Numbers
Real numbers encompass all the numbers on the number line, including both rational and irrational numbers. Essentially, any number you can think of that isn't a product of an imaginary unit \( i \) is a real number. This includes:
  • Integers like -5, 0, and 16.
  • Fractions and decimals like \( \frac{1}{4} \) or 0.25.
  • Irrational numbers like \( \sqrt{3} \) or \( \pi \).
Real numbers are fundamental in mathematics as they form the basis for understanding quantities, measurements, and even geometry. They provide a complete representation of quantities we encounter in everyday life.
Imaginary Numbers
Imaginary numbers arise when we take the square root of negative numbers. The imaginary unit \( i \) is defined as \( \sqrt{-1} \). By extension:
  • \( \sqrt{-16} \) equals \( 4i \) since \( \sqrt{-16} = \sqrt{16} \times \sqrt{-1} = 4 \times i \).
  • Any multiple of \( i \) is considered imaginary.
While not found on the traditional number line like real numbers, imaginary numbers are crucial in advanced mathematics and engineering. They are used in complex numbers to analyze phenomena involving two dimensions, such as electrical currents and waves.

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