Question

section a: number classification (3 pts each)
classify each number — list all sets that apply (n, w, z, q, irrational, r):
a) -12
b) \\(\sqrt{16}\\)
c) \\(\frac{2}{7}\\)
d) \\(-\sqrt{3}\\)
e) 0.777...
f) 0
g) \\(-\sqrt{25}\\)
h) 3.14159...

1. which is not rational? (a) 0.333... (b) -4/5 (c) \\(\sqrt{11}\\) (d) \\(\sqrt{25}\\)
2. true or false: every integer is also a rational number. explain.
3. name one irrational number that becomes rational when squared.
4. order from least to greatest: \\(|-7|\\), \\(|3|\\), \\(|-1|\\), \\(|0|\\), \\(|-5|\\)
5. name the property: \\(7 \times (x + 3) = 7x + 21\\)
6. name the property: \\((m + n) + p = m + (n + p)\\)
7. evaluate: \\(-3 \times |-4| + |-2|\\)
8. is \\(\sqrt{49}\\) rational or irrational? is \\(\sqrt{48}\\) rational or irrational? explain.

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 9,
"skills_matched": [
"step_cot",
"explain_then_answer",
"tutor"
],
"needs_drawing": false,
"concepts_used": [],
"new_concepts": [
"Number Classification",
"Real Number Properties",
"Absolute Value"
],
"current_concepts": [
"Number Classification",
"Real Number Properties",
"Absolute Value"
]
}
</pre_analysis>

<reasoning>

Classify the numbers in Section A

We classify each given number into the standard sets: Natural numbers \(\mathbb{N}\), Whole numbers \(\mathbb{W}\), Integers \(\mathbb{Z}\), Rational numbers \(\mathbb{Q}\), Irrational numbers, and Real numbers \(\mathbb{R}\).

• a) \(-12\): An integer, rational, and real. Sets: \(\mathbb{Z}, \mathbb{Q}, \mathbb{R}\).
• b) \(\sqrt{16} = 4\): A natural, whole, integer, rational, and real. Sets: \(\mathbb{N}, \mathbb{W}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}\).
• c) \(\frac{2}{7}\): A rational and real. Sets: \(\mathbb{Q}, \mathbb{R}\).
• d) \(-\sqrt{3}\): An irrational and real. Sets: Irrational, \(\mathbb{R}\).
• e) \(0.777\dots = \frac{7}{9}\): A repeating decimal, which is rational and real. Sets: \(\mathbb{Q}, \mathbb{R}\).
• f) \(0\): A whole number, integer, rational, and real. Sets: \(\mathbb{W}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}\).
• g) \(-\sqrt{25} = -5\): An integer, rational, and real. Sets: \(\mathbb{Z}, \mathbb{Q}, \mathbb{R}\).
• h) \(3.14159\dots = \pi\): A non-repeating, non-terminating decimal, which is irrational and real. Sets: Irrational, \(\mathbb{R}\).

Solve Questions 1 to 3

• Question 1: Identify which number is NOT rational.
• (A) \(0.333\dots = \frac{1}{3}\) (Rational)
• (B) \(-\frac{4}{5}\) (Rational)
• (C) \(\sqrt{11}\) (Irrational, since 11 is not a perfect square)
• (D) \(\sqrt{25} = 5\) (Rational)

Therefore, the answer is (C).

• Question 2: "Every integer is also a rational number."
• This is True. Any integer \(a\) can be written as a fraction \(\frac{a}{1}\), which fits the definition of a rational number \(\frac{p}{q}\) where \(p, q \in \mathbb{Z}\) and \(q

eq 0\).

• Question 3: Name one irrational number that becomes rational when squared.
• Consider \(\sqrt{2}\). It is irrational, but \((\sqrt{2})^2 = 2\), which is rational.

Solve Questions 4 to 6

• Question 4: Order from least to greatest: \(|-7|, |3|, |-1|, |0|, |-5|\).
• First, evaluate the absolute values:

\[

-7

= 7,\quad

3

= 3,\quad

-1

= 1,\quad

0

= 0,\quad

-5

= 5

\]

• Order the values: \(0 < 1 < 3 < 5 < 7\).
• Write in original form: \(|0| < |-1| < |3| < |-5| < |-7|\).
• Question 5: Name the property: \(7 \times (x + 3) = 7x + 21\).
• This is the Distributive Property of Multiplication over Addition.
• Question 6: Name the property: \((m + n) + p = m + (n + p)\).
• This is the Associative Property of Addition.

Solve Questions 7 and 8

• Question 7: Evaluate \(-3 \times |-4| + |-2|\).
• Simplify absolute values: \(|-4| = 4\) and \(|-2| = 2\).
• Substitute and compute:

\[
-3 \times 4 + 2 = -12 + 2 = -10
\]

• Question 8: Is \(\sqrt{49}\) rational or irrational? Is \(\sqrt{48}\) rational or irrational? Explain.
• \(\sqrt{49} = 7\), which is a terminating integer, so it is rational.
• \(\sqrt{48} = 4\sqrt{3}\). Since 48 is not a perfect square, its square root is a non-terminating, non-repeating decimal, so it is irrational.

</reasoning>

<answer>

Section A: Number Classification

• a) \(\mathbb{Z}, \mathbb{Q}, \mathbb{R}\)
• b) \(\mathbb{N}, \mathbb{W}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}\)
• c) \(\mathbb{Q}, \mathbb{R}\)
• d) Irrational, \(\mathbb{R}\)
• e) \(\mathbb{Q}, \mathbb{R}\)
• f) \(\mathbb{W}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}\)
• g) \(\mathbb{Z}, \mathbb{Q}, \mathbb{R}\)
• h) Irrational, \(\mathbb{R}\)

Question 1

<mcq-option>(A) 0.333...</mcq-option>
<mcq-option>(B) -4/5</mcq-option>
<mcq-correct>(C) sqrt(11)</mcq-correct>
<mcq-option>(D) sqrt(25)</mcq-option>

Question 2

True. Every integer \(a\) can be expressed as the fraction \(\frac{a}{1}\), which satisfies the definition of a rational number.

Question 3

\(\sqrt{2}\) (since \((\sqrt{2})^2 = 2\), which is rational).

Question 4

\(|0|, |-1|, |3|, |-5|, |-7|\)

Question 5

Distributive Property

Question 6

Associative Property of Addition

Question 7

\(-10\)

Question 8

\(\sqrt{49}\) is rational because \(\sqrt{49} = 7\), which is an integer. \(\sqrt{48}\) is irrational because 48 is not a perfect square, so its square root cannot be written as a simple fraction.
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multi-part",
"knowledge_point": [
"Mathematics",
"Algebra",
"Number Classification"
]
}
</post_analysis>