Question

fill in the blank.

the set of numbers in the form \\(\frac{a}{b}\\), where a and b are integers and \\(b \
eq 0\\), is called the set of ______ numbers.

real
irrational
whole
rational

Explanation:

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<reasoning>

Identify the given definition

The problem asks us to identify a specific set of numbers. We are given the mathematical definition:
The set of numbers in the form \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b
eq 0\).

Analyze the options

Using the Number Systems and Integers knowledge points, let's evaluate the given choices:

• real: Real numbers include all rational and irrational numbers. This definition is too specific to represent all real numbers.
• irrational: Irrational numbers cannot be written in the form \(\frac{a}{b}\) where \(a\) and \(b\) are integers.
• whole: Whole numbers are non-negative integers \(\{0, 1, 2, 3, \dots\}\), which do not generally require a fractional representation with a non-zero denominator.
• rational: By definition, any number that can be expressed as the quotient or fraction \(\frac{a}{b}\) of two integers, a numerator \(a\) and a non-zero denominator \(b\), is a rational number.

Select the correct term

The definition perfectly matches the mathematical definition of rational numbers. Therefore, the blank should be filled with "rational".
</reasoning>

<answer>
Fill in the blank.

The set of numbers in the form \(\frac{a}{b}\), where a and b are integers and \(b
eq 0\), is called the set of <blank>rational</blank> numbers.
</answer>

<post_analysis>
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"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Algebra",
"Number Systems"
]
}
</post_analysis>

Answer:

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"blank_filling_formatter",
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"concepts_used": [
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"Integers"
],
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"Rational Numbers"
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"Integers",
"Rational Numbers"
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</pre_analysis>

<reasoning>

Identify the given definition

The problem asks us to identify a specific set of numbers. We are given the mathematical definition:
The set of numbers in the form \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b
eq 0\).

Analyze the options

Using the Number Systems and Integers knowledge points, let's evaluate the given choices:

• real: Real numbers include all rational and irrational numbers. This definition is too specific to represent all real numbers.
• irrational: Irrational numbers cannot be written in the form \(\frac{a}{b}\) where \(a\) and \(b\) are integers.
• whole: Whole numbers are non-negative integers \(\{0, 1, 2, 3, \dots\}\), which do not generally require a fractional representation with a non-zero denominator.
• rational: By definition, any number that can be expressed as the quotient or fraction \(\frac{a}{b}\) of two integers, a numerator \(a\) and a non-zero denominator \(b\), is a rational number.

Select the correct term

The definition perfectly matches the mathematical definition of rational numbers. Therefore, the blank should be filled with "rational".
</reasoning>

<answer>
Fill in the blank.

The set of numbers in the form \(\frac{a}{b}\), where a and b are integers and \(b
eq 0\), is called the set of <blank>rational</blank> numbers.
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Algebra",
"Number Systems"
]
}
</post_analysis>