Questions: List all numbers from the given set that are a. natural numbers, b. whole numbers, c. integers, d. rational numbers, e. irrational numbers, and f. real numbers. [ left-13,-frac37, 0,0.2, sqrt7, pi, sqrt64right ] Drag each of the numbers given above into the appropriate category below. Numbers may be members of more than one subset of real numbers. natural numbers whole numbers integers rational numbers irrational numbers real numbers

$List all numbers from the given set that are a. natural numbers, b. whole numbers, c. integers, d. rational numbers, e. irrational numbers, and f. real numbers. [ left-13,-frac37, 0,0.2, sqrt7, pi, sqrt64right ] Drag each of the numbers given above into the appropriate category below. Numbers may be members of more than one subset of real numbers. natural numbers whole numbers integers rational numbers irrational numbers real numbers$

Transcript text: List all numbers from the given set that are a. natural numbers, b. whole numbers, c. integers, d. rational numbers, e. irrational numbers, and f. real numbers. \[ \left\{-13,-\frac{3}{7}, 0,0.2, \sqrt{7}, \pi, \sqrt{64}\right\} \] Drag each of the numbers given above into the appropriate category below. Numbers may be members of more than one subset of real numbers. natural numbers whole numbers integers rational numbers irrational numbers real numbers

Solution

Solution Steps

To categorize the numbers, we need to understand the definitions of each type of number:

Natural numbers are positive integers starting from 1.
Whole numbers are natural numbers including 0.
Integers include all whole numbers and their negative counterparts.
Rational numbers are numbers that can be expressed as a fraction of two integers.
Irrational numbers cannot be expressed as a simple fraction.
Real numbers include all rational and irrational numbers.

We will iterate through the given set and check each number against these definitions to categorize them.

To solve the problem, we need to categorize each number from the given set into the appropriate subsets of real numbers. Let's go through the steps to determine which numbers belong to each category.

Step 1: Identify the Numbers in the Set

The given set of numbers is: \[ \left\{-13, -\frac{3}{7}, 0, 0.2, \sqrt{7}, \pi, \sqrt{64}\right\} \]

Step 2: Define Each Category

Natural Numbers: Positive integers starting from 1 (e.g., 1, 2, 3, ...).
Whole Numbers: Non-negative integers (e.g., 0, 1, 2, 3, ...).
Integers: Whole numbers and their negatives (e.g., ..., -2, -1, 0, 1, 2, ...).
Rational Numbers: Numbers that can be expressed as a fraction of two integers (e.g., $-\frac{3}{7}, 0.2$).
Irrational Numbers: Numbers that cannot be expressed as a simple fraction (e.g., $\pi, \sqrt{7}$).
Real Numbers: All numbers on the number line, including rational and irrational numbers.

Step 3: Categorize Each Number

$-13$: Integer, Rational, Real
$-\frac{3}{7}$: Rational, Real
$0$: Whole, Integer, Rational, Real
$0.2$: Rational, Real (since $0.2 = \frac{1}{5}$)
$\sqrt{7}$: Irrational, Real
$\pi$: Irrational, Real
$\sqrt{64}$: Natural, Whole, Integer, Rational, Real (since $\sqrt{64} = 8$)

Final Answer

Natural Numbers: $\sqrt{64}$
Whole Numbers: $0, \sqrt{64}$
Integers: $-13, 0, \sqrt{64}$
Rational Numbers: $-13, -\frac{3}{7}, 0, 0.2, \sqrt{64}$
Irrational Numbers: $\sqrt{7}, \pi$
Real Numbers: $-13, -\frac{3}{7}, 0, 0.2, \sqrt{7}, \pi, \sqrt{64}$

\[ \boxed{ \begin{array}{ll} \text{Natural Numbers:} & \sqrt{64} \\ \text{Whole Numbers:} & 0, \sqrt{64} \\ \text{Integers:} & -13, 0, \sqrt{64} \\ \text{Rational Numbers:} & -13, -\frac{3}{7}, 0, 0.2, \sqrt{64} \\ \text{Irrational Numbers:} & \sqrt{7}, \pi \\ \text{Real Numbers:} & -13, -\frac{3}{7}, 0, 0.2, \sqrt{7}, \pi, \sqrt{64} \\ \end{array} } \]

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