name the set or sets of numbers to which each real number belongs. use:
- n for natural numbers
- w for whole numbers
- z for integers
- q for rational numbers
- p for irrational numbers
ex. 3 → n, w, z, q
1. 0.66666666...
2. $\frac{5}{8}$
3. 2
4. -57

Question

Accepted Answer

### Problem 1: $0.66666666\ldots$
# Explanation:
## Step 1: Recall number set definitions
Natural numbers ($N$): positive integers (1, 2, 3, ...). Whole numbers ($W$): non - negative integers (0, 1, 2, ...). Integers ($Z$): positive/negative whole numbers and 0. Rational numbers ($Q$): numbers that can be expressed as $\frac{a}{b}$ ($b
eq0$, $a,b\in Z$). Irrational numbers ($P$): non - repeating, non - terminating decimals, cannot be expressed as $\frac{a}{b}$.
The number $0.66666666\ldots=\frac{2}{3}$ (since $0.\overline{6}=\frac{2}{3}$), so it can be written as a fraction of two integers. It is not a natural number (not a positive integer starting from 1 in the natural number sense here, as it's a non - integer decimal), not a whole number (not a non - negative integer), not an integer (not a whole number or its negative). But it is a rational number.
## Step 2: Determine the set
Since $0.66666666\ldots=\frac{2}{3}$, it belongs to the set of rational numbers ($Q$). It does not belong to $N$ (not a positive integer), $W$ (not a non - negative integer), $Z$ (not an integer), or $P$ (it is a repeating decimal, hence rational).
# Answer: $Q$

### Problem 2: $\frac{5}{8}$
# Explanation:
## Step 1: Recall number set definitions
As before, rational numbers ($Q$) are numbers that can be written as $\frac{a}{b}$ with $b
eq0$ and $a,b\in Z$. $\frac{5}{8}$ is in the form $\frac{a}{b}$ where $a = 5$, $b = 8$, both are integers and $b
eq0$.
It is not a natural number (not a positive integer), not a whole number (not a non - negative integer), not an integer (not a whole number or its negative), and not an irrational number (since it can be expressed as a fraction of two integers).
## Step 2: Determine the set
Since $\frac{5}{8}$ can be expressed as a fraction of two integers, it belongs to the set of rational numbers ($Q$).
# Answer: $Q$

### Problem 3: $2$
# Explanation:
## Step 1: Recall number set definitions
Natural numbers ($N$): positive integers (1, 2, 3, ...). Whole numbers ($W$): non - negative integers (0, 1, 2, ...). Integers ($Z$): positive/negative whole numbers and 0. Rational numbers ($Q$): numbers that can be expressed as $\frac{a}{b}$ ($b
eq0$, $a,b\in Z$).
The number $2$ is a positive integer, so it is a natural number ($N$), a whole number ($W$) (since it is a non - negative integer), an integer ($Z$) (since it is a whole number), and it can be written as $\frac{2}{1}$, so it is also a rational number ($Q$). It is not an irrational number ($P$) because it is a rational number.
## Step 2: Determine the set
$2$ belongs to $N$ (positive integer), $W$ (non - negative integer), $Z$ (integer), and $Q$ (can be written as $\frac{2}{1}$).
# Answer: $N, W, Z, Q$

### Problem 4: $- 57$
# Explanation:
## Step 1: Recall number set definitions
Integers ($Z$) include negative whole numbers, positive whole numbers, and 0. Rational numbers ($Q$) can be written as $\frac{a}{b}$ ($b
eq0$, $a,b\in Z$). Natural numbers ($N$) are positive integers (so $-57$ is not in $N$). Whole numbers ($W$) are non - negative integers (so $-57$ is not in $W$).
$-57$ is a negative integer, so it is an integer ($Z$). Also, it can be written as $\frac{- 57}{1}$, so it is a rational number ($Q$). It is not an irrational number ($P$) as it is a rational number.
## Step 2: Determine the set
$-57$ belongs to $Z$ (integer) and $Q$ (rational number, since $-57=\frac{-57}{1}$).
# Answer: $Z, Q$

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QUESTION IMAGE

Question

Explanation:

Problem 1: \(0.66666666\ldots\)

Step 1: Recall number set definitions

Step 2: Determine the set

Step 1: Recall number set definitions

Step 2: Determine the set

Step 1: Recall number set definitions

Step 2: Determine the set

Answer:

Problem 2: \(\frac{5}{8}\)