In Exercises 35–38, 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. {-11, -5/6, 0, 0.75, √5, π, √64}
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Step 1: Identify natural numbers from the set. Natural numbers are positive integers starting from 1. Check each element in the set to see if it is a positive integer.
Step 2: Identify whole numbers from the set. Whole numbers include all natural numbers and zero. Check each element in the set to see if it is a non-negative integer.
Step 3: Identify integers from the set. Integers include all whole numbers and their negative counterparts. Check each element in the set to see if it is a whole number or a negative integer.
Step 4: Identify rational numbers from the set. Rational numbers can be expressed as a fraction of two integers. Check each element in the set to see if it can be written as a fraction.
Step 5: Identify irrational numbers from the set. Irrational numbers cannot be expressed as a simple fraction. Check each element in the set to see if it is a non-repeating, non-terminating decimal or a non-perfect square root.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Natural Numbers
Natural numbers are the set of positive integers starting from 1 and going upwards (1, 2, 3, ...). They do not include zero or any negative numbers. In the context of the given set, identifying natural numbers involves recognizing that only positive whole numbers qualify.
Rational numbers are numbers that can be expressed as a fraction of two integers, where the denominator is not zero. This includes integers, fractions, and finite or repeating decimals. In contrast, irrational numbers cannot be expressed as simple fractions; they have non-repeating, non-terminating decimal expansions, such as √5 and π.
Real numbers encompass all the numbers on the number line, including both rational and irrational numbers. This means that real numbers include natural numbers, whole numbers, integers, fractions, and decimals. Understanding this concept is crucial for categorizing the numbers in the provided set accurately.