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}

Verified step by step guidance

1. Step 1: Understand the definitions of each type of number: - Natural numbers: positive integers starting from 1 (1, 2, 3, ...). - Whole numbers: natural numbers including zero (0, 1, 2, 3, ...). - Integers: all whole numbers and their negatives (..., -3, -2, -1, 0, 1, 2, 3, ...). - Rational numbers: numbers that can be expressed as a fraction \( \frac{a}{b} \) where \(a\) and \(b\) are integers and \(b \neq 0\). - Irrational numbers: numbers that cannot be expressed as a simple fraction, their decimal expansions are non-repeating and non-terminating. - Real numbers: all rational and irrational numbers combined.
2. Step 2: Analyze each element in the set \( \{-11, -\frac{5}{6}, 0, 0.75, \sqrt{5}, \pi, \sqrt{64} \} \) to determine which categories they belong to: - \(-11\) is a negative integer. - \(-\frac{5}{6}\) is a negative fraction. - \(0\) is zero. - \(0.75\) is a decimal that can be expressed as a fraction. - \(\sqrt{5}\) is an irrational number. - \(\pi\) is an irrational number. - \(\sqrt{64}\) simplifies to an integer.
3. Step 3: Identify natural numbers from the set: - Recall natural numbers are positive integers starting from 1. - Check which numbers are positive integers.
4. Step 4: Identify whole numbers from the set: - Whole numbers include all natural numbers plus zero. - Check which numbers are zero or positive integers.
5. Step 5: Identify integers, rational numbers, irrational numbers, and real numbers: - Integers include all whole numbers and their negatives. - Rational numbers include all numbers that can be written as a fraction (including integers and decimals that terminate or repeat). - Irrational numbers are those that cannot be expressed as fractions. - Real numbers include all rational and irrational numbers.

Key Concepts

Number Sets and Their Definitions

Understanding different types of numbers is essential. Natural numbers are positive counting numbers starting from 1. Whole numbers include all natural numbers plus zero. Integers extend whole numbers to include negative numbers as well.

Rational and Irrational Numbers

Rational numbers can be expressed as a fraction of two integers, including decimals that terminate or repeat. Irrational numbers cannot be written as simple fractions and have non-repeating, non-terminating decimal expansions, such as π and √5.

Real Numbers

Real numbers include all rational and irrational numbers, encompassing every point on the number line. This set covers natural numbers, whole numbers, integers, rational numbers, and irrational numbers, making it the broadest category in this context.