Problem 297 - Intermediate Algebra

Question

In the following exercises, list the (a) whole numbers, (b) integers, \(\odot\) rational numbers, \(@\) irrational numbers, \(\Theta\) real numbers for each set of numbers. $$ -8,0,1.95286 \ldots, \frac{12}{5}, \sqrt{36}, 9 $$

Step-by-Step Solution

Answer

(a) 0, 6, 9 (b) -8, 0, 6, 9 (c) -8, 0, 2.4, 6, 9 (d) 1.95286... (e) All numbers are real

Step 1 - Identify whole numbers

Whole numbers are non-negative numbers without fractions or decimals. From the given set, list the numbers that fit this description.

Step 2 - Identify integers

Integers include all whole numbers and their negative counterparts. Identify the numbers from the set that are integers.

Step 3 - Identify rational numbers

Rational numbers are numbers that can be expressed as the ratio of two integers. In this case, include both fractions and decimals that terminate or repeat.

Step 4 - Identify irrational numbers

Irrational numbers cannot be expressed as the ratio of two integers; their decimals are non-terminating and non-repeating. Identify any such numbers.

Step 5 - Identify real numbers

Real numbers include all rational and irrational numbers. Summarize all numbers from previous steps to identify all real numbers in the set.

Key Concepts

whole numbers

Whole numbers are the simplest kind of numbers to understand. They include all positive numbers starting from zero and go on infinitely.
These numbers do not include fractions, decimals, or negative numbers.
For example, in the given set \(-8, 0, 1.95286 ... , \frac{12}{5}, \sqrt{36}, 9\)\, the whole numbers are \(0\) and \(9\).\

integers

Integers expand on whole numbers to include their negative counterparts.
This means all positive and negative numbers without decimal or fractional parts.
In the given set, the integers are -8, 0, and \(\sqrt{36}\ = 6\).\

rational numbers

Rational numbers are numbers that can be expressed as a fraction \(\frac{a}{b} \) where 'a' and 'b' are integers and 'b' is not zero.
Rational numbers can be whole numbers, integers, terminating decimals, and repeating decimals.
From the set, \(-8, 0, \frac{12}{5}, \sqrt{36}, 9\) are rational numbers.
Here, \(1.95286 ...\) is not included because it is a non-terminating, non-repeating decimal.\

irrational numbers

Irrational numbers are the opposite of rational numbers.
They cannot be expressed as a simple fraction, and their decimal expansion is non-terminating and non-repeating.
In this problem, \(1.95286 ...\)\ is classified as an irrational number.
This is because its decimal expansion neither terminates nor repeats in a predictable way.\

real numbers

Real numbers encompass both rational and irrational numbers.
Essentially, any number that you can place on a number line is a real number.
This includes all whole numbers, integers, fractions, terminating decimals, repeating decimals, and non-repeating non-terminating decimals.
So from the given set, \(-8, 0, 1.95286 ..., \frac{12}{5}, \sqrt{36}, 9\) are all real numbers.
In summary, real numbers wrap up everything: whole numbers, integers, rational, and irrational numbers.
They cover just about any number you can think of.