Question

fill in the blank in the sentence below so that the resulting statement is true.

the set \\(\\{..., -4, -3, -2, -1, 0, 1, 2, 3, 4, ...\\}\\) is called the set of ______.

the set \\(\\{..., -4, -3, -2, -1, 0, 1, 2, 3, 4, ...\\}\\) is called the set of

• rational numbers.
• irrational numbers.
• whole numbers.
• integers.

Explanation:

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<reasoning>

Analyze the given set

The set is written as:
\[
\{\dots, -4, -3, -2, -1, 0, 1, 2, 3, 4, \dots\}
\]
This set contains all positive whole numbers, negative whole numbers, and zero.

Evaluate the definitions of the options

We compare the given set with standard mathematical number systems:

• Rational numbers: Numbers that can be written as a fraction \(\frac{a}{b}\) where \(a\) and \(b\) are integers and \(b

eq 0\).

• Irrational numbers: Real numbers that cannot be expressed as a simple fraction (e.g., \(\pi\), \(\sqrt{2}\)).
• Whole numbers: The set of non-negative integers \(\{0, 1, 2, 3, \dots\}\).
• Integers: The set of all positive and negative whole numbers, including zero.

Match the set to the correct term

The set \(\{\dots, -4, -3, -2, -1, 0, 1, 2, 3, 4, \dots\}\) matches the definition of integers.
</reasoning>

<answer>
<mcq-option>rational numbers.</mcq-option>
<mcq-option>irrational numbers.</mcq-option>
<mcq-option>whole numbers.</mcq-option>
<mcq-correct>integers.</mcq-correct>
</answer>

<post_analysis>
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"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
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Answer:

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"direct_cot",
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"new_concepts": [
"Number Systems",
"Integers"
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"Number Systems",
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</pre_analysis>

<reasoning>

Analyze the given set

The set is written as:
\[
\{\dots, -4, -3, -2, -1, 0, 1, 2, 3, 4, \dots\}
\]
This set contains all positive whole numbers, negative whole numbers, and zero.

Evaluate the definitions of the options

We compare the given set with standard mathematical number systems:

• Rational numbers: Numbers that can be written as a fraction \(\frac{a}{b}\) where \(a\) and \(b\) are integers and \(b

eq 0\).

• Irrational numbers: Real numbers that cannot be expressed as a simple fraction (e.g., \(\pi\), \(\sqrt{2}\)).
• Whole numbers: The set of non-negative integers \(\{0, 1, 2, 3, \dots\}\).
• Integers: The set of all positive and negative whole numbers, including zero.

Match the set to the correct term

The set \(\{\dots, -4, -3, -2, -1, 0, 1, 2, 3, 4, \dots\}\) matches the definition of integers.
</reasoning>

<answer>
<mcq-option>rational numbers.</mcq-option>
<mcq-option>irrational numbers.</mcq-option>
<mcq-option>whole numbers.</mcq-option>
<mcq-correct>integers.</mcq-correct>
</answer>

<post_analysis>
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"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Number Systems"
]
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