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Chapter 1: Problem 64

List three rational numbers greater than zero.

Short Answer

Expert verified
Examples: \( \frac{1}{2} \), \( \frac{3}{4} \), \( \frac{5}{6} \).

Step by step solution

01

Understanding Rational Numbers

Rational numbers are numbers that can be expressed as the quotient or fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b eq 0 \). Positive rational numbers are greater than zero.
02

Selecting Rational Numbers

To find three rational numbers greater than zero, we must choose numerators and denominators that will create positive fractions. Examples of such fractions are \( \frac{1}{2} \), \( \frac{3}{4} \), and \( \frac{5}{6} \).
03

Verifying Positivity

Verify that each of the chosen numbers is indeed a positive rational number: \( \frac{1}{2} = 0.5 \), \( \frac{3}{4} = 0.75 \), and \( \frac{5}{6} \approx 0.833 \). All these values are greater than zero, confirming they are valid answers.

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Key Concepts

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

Positive Rational Numbers
Positive rational numbers are simply rational numbers that are greater than zero. Remember that rational numbers themselves are defined as numbers that can be expressed as a fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers, and \( b eq 0 \).
For a rational number to be positive, both the numerator and denominator must ensure the fraction results in a positive value.

Positive fractions can be achieved in a couple of ways:
  • Both the numerator and the denominator are positive integers, e.g., \( \frac{3}{4} \).
  • Both are negative integers, e.g., \( \frac{-3}{-4} \), which also gives a positive result because a double negative makes a positive.
In exercises like choosing rational numbers greater than zero, selecting a positive numerator and denominator is the simplest path to success.
Fractions
Fractions are representations of parts of a whole. They are written in the form \( \frac{a}{b} \) where \( a \) is called the numerator and \( b \) is called the denominator.
The concept of fractions is fundamental in understanding rational numbers. All rational numbers can be written as fractions.

Here's what you should keep in mind about fractions:
  • The numerator (\( a \)) denotes how many parts of the whole are being considered.
  • The denominator (\( b \)) tells into how many parts the whole is divided.
Fractions can represent values less than, equal to, or greater than one, depending on the relationship between the numerator and the denominator.
Numerators and Denominators
To fully grasp rational numbers and fractions, it's important to understand the roles of numerators and denominators.
The numerator is the top part of a fraction, and it indicates how many parts of a certain size we have.
In \( \frac{3}{4} \), 3 is the numerator.

The denominator is the bottom part, showing the total number of equal parts that the whole is divided into.
In \( \frac{3}{4} \), 4 is the denominator.

Some vital points regarding numerators and denominators include:
  • When the numerator is smaller than the denominator, the fraction represents a number less than one.
  • If the numerator is equal to the denominator, the fraction's value is one.
  • When the numerator is larger than the denominator, the fraction stands for a number greater than one.
Understanding these components helps in verifying and forming positive rational numbers effectively.

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