Problem 35 - Algebra and Trigonometry

Question

List all numbers from the given set that are a. natural numbers, b. whole numbers, c. integers, d. rational numbers, e. irrational numbers, f. real numbers. $$\left\\{-9,-\frac{4}{5}, 0,0.25, \sqrt{3}, 9.2, \sqrt{100}\right\\}$$

Step-by-Step Solution

Answer

From the given set: (a) the natural number is \( \sqrt{100} \); (b) the whole numbers are \( 0 \) and \( \sqrt{100} \); (c) the integers are \( -9 \), \( 0 \), and \( \sqrt{100} \); (d) the rational numbers are \( -9 \), \( -4/5 \), \( 0 \), \( 0.25 \), \( 9.2 \), and \( \sqrt{100} \); (e) the irrational number is \( \sqrt{3} \); (f) all numbers in the set are real numbers.

Step 1: Classify as Natural Numbers

Natural numbers are countable numbers starting from 1. From the set, the only natural number is \( \sqrt{100} \) since \( \sqrt{100} = 10 \).

Step 2: Classify as Whole Numbers

Whole numbers include all natural numbers and 0. So, from the given set, \( 0 \) and \( \sqrt{100} \) are whole numbers.

Step 3: Classify as Integers

Integers include all negative and positive whole numbers including 0. Thus, \( -9 \), \( 0 \), and \( \sqrt{100} \) are integers.

Step 4: Classify as Rational Numbers

Rational numbers are numbers that can be represented as a fraction \( a/b \) where both \( a \) and \( b \) are integers and \( b \neq 0 \). The numbers \( -9 \), \( -4/5 \), \( 0 \), \( 0.25 \) (which is \( 1/4 \)), \( 9.2 \) (which is \( 92/10 \)) and \( \sqrt{100} \) are rational numbers.

Step 5: Classify as Irrational Numbers

Irrational numbers can't be expressed as a simple fraction. They're numbers that when expressed in decimal form will not terminate or repeat in a pattern. From the set, \( \sqrt{3} \) is an irrational number.

Step 6: Classify as Real Numbers

All numbers from the given set are real numbers, since every number that is either a rational or an irrational number is a real number.