Question

Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. Name the integers and the rational numbers in the set. $$ \left\\{-3,0, \sqrt{2}, \frac{6}{5}, \pi\right\\} $$

Short answer

Integers: \[-3, 0\]. Rational numbers: \[-3, 0, \(\frac{6}{5}\)\].

Step-by-step solution

Identify the Given Set

Examine the given set: \[-3, 0, \(\frac{6}{5}\), \(\frac{6}{5}\), π \].

Define Integers

Recall that integers are whole numbers that can be positive, negative, or zero. Look for elements in the set that match this definition.

List Integers

From the set, identify the integers: \[-3, 0\].

Define Rational Numbers

Recall that rational numbers are numbers that can be expressed as a fraction of two integers, where the denominator is not zero. This includes integers and proper fractions.

List Rational Numbers

Identify the rational numbers in the set: \[-3, 0, \(\frac{6}{5}\)\].

Key concepts

Integers

Integers are key components of the number system. They include all whole numbers and their negative counterparts, encompassing \(-3, -2, -1, 0, 1, 2, 3\), and so on.

In the set \{-3, 0, \sqrt{2}, \frac{6}{5}, \pi\}, we identify integers by looking for whole numbers, which in this case are \(-3\) and \(0\). These two numbers can stand alone without involving any fractions or decimals.

Understanding integers is straightforward because they do not involve complex parts like fractions or decimals, making them easy to spot in a set of numbers.

Rational Numbers

Rational numbers broaden our perspective by including any number that can be written as a fraction. This means both the numerator and denominator are integers, and the denominator is not zero.

Examples of rational numbers are \(\frac{6}{5}\) and \(-3\) because they can be expressed as \(\frac{6}{5}\) and \(\frac{-3}{1}\), respectively. Even \(0\) is a rational number, written as \(\frac{0}{1}\).

Rational numbers encompass a wide range of types, including:

• Whole numbers like \(0\) and \(-3\)
• Fractions like \(\frac{6}{5}\)

This category helps in bridging the gap between whole numbers and more detailed fractions, allowing for a richer number system.

Number Sets Analysis

When analyzing number sets, it's crucial to understand how different number types are categorized. This involves identifying and classifying elements into types like integers and rational numbers.

Take the set \{-3, 0, \sqrt{2}, \frac{6}{5}, \pi\}. Start by clearly defining properties for each category. Integers are straightforward as whole numbers, while rational numbers include any number expressible as a fraction with integers in the numerator and denominator.

A detailed analysis of this set reveals:

• Integers: \(0\) and \(-3\)
• Rational numbers: \(0\), \(-3\), and \(\frac{6}{5}\)

Identifying where \sqrt{2}\ and \pi\ fall helps clarify boundaries within number sets since they do not fit into either of the first two categories, which means they belong to other types like irrational numbers. This systematic approach ensures a comprehensive understanding of the number system.