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Chapter 5: Problem 47

For Exercises \(47-58,\) identify the number as rational or irrational. $$-\frac{2}{5}$$

Short Answer

Expert verified
-2/5 is a rational number because it can be expressed as a fraction of two integers where the denominator is not zero.

Step by step solution

01

Understanding Rational and Irrational Numbers

A rational number can be expressed as the quotient or fraction \(\frac{a}{b}\) of two integers, where \(a\) and \(b\) are integers and \(b eq 0\). An irrational number cannot be expressed as a simple fraction.
02

Identify the Number Type

Consider the given number \(-\frac{2}{5}\). This number is in the form of \(\frac{a}{b}\) where \(a = -2\) and \(b = 5\). Both \(a\) and \(b\) are integers, and \(b eq 0\).
03

Determine the Classification

Since \(-\frac{2}{5}\) can be expressed as a fraction of two integers, it meets the criteria for a rational number.

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Key Concepts

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

Rational Numbers
A rational number is any number that can be written as a fraction. This means we can express it as \(\frac{a}{b}\), where both \(a\) and \(b\) are integers and \(b eq 0 \). Common examples of rational numbers include:
  • Whole numbers like 3 (which can be written as \(\frac{3}{1}\))
  • Fractions like \(\frac{3}{4}\)
  • Negative fractions like \(-\frac{2}{5}\)
One key thing about rational numbers is they can be written with a finite number of decimal places, or as repeating decimals. For example, 0.75 (which is \(\frac{3}{4}\)) and 0.666... (which is \(\frac{2}{3}\)) are both rational numbers. If a number fits these criteria, it's rational.
Irrational Numbers
An irrational number is a number that cannot be written as a simple fraction. This means it can't be expressed as \(\frac{a}{b}\) where \(a\) and \(b\) are integers. Instead, irrational numbers have infinite, non-repeating decimal parts. Some well-known examples include:
  • Pi (蟺), approximately 3.14159...
  • The square root of 2 (鈭�2), approximately 1.41421...
  • Euler's number (e), approximately 2.71828...
These numbers go on forever without repeating, making them impossible to express as exact fractions. Unlike rational numbers, you can't write irrational numbers using a finite number of digits.
Fractions
Fractions are a way to represent parts of a whole. They consist of a numerator and a denominator, written as \(\frac{a}{b}\). The numerator (\(a\)) indicates how many parts we have, and the denominator (\(b\)) shows the number of equal parts the whole is divided into. Some details about fractions:
  • A fraction where the numerator and denominator are both integers and the denominator is not zero, for example, \(\frac{3}{4}\).
  • If the numerator is larger than the denominator, it's called an improper fraction, like \(\frac{5}{3}\).
  • If the numerator is zero, the fraction is equal to zero (\frac{0}{b})
Fractions can represent rational numbers if their numerators and denominators are integers. Therefore, any fraction where the numerator and denominator are whole numbers (integers) represents a rational number.

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Most popular questions from this chapter

Which numbers when divided by 5.3 will produce a quotient greater than 15.8? Circle all that apply. a. 84.8 b. 84.27 c. 83.21 d. 79.5

For Exercises \(71-82,\) simplify by using the order of operations. $$15.06-1.92-0.4^{2}$$

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