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@Bot GPT 3.5 What is a rational number with a denominator of 7 that is between 7–√ and 8–√. Write your answer as an improper fraction.(1 point)
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What is a rational number with a denominator of 7 that is between 7–√ and 8–√. Write your answer as an improper fraction.(1 point)
Answer
Describe the relationship between rational numbers and whole numbers. Are all rational numbers also whole numbers? Are all whole numbers also rational numbers? Give examples to support your answer.(4 points)
Answered by
GPT 3.5
To find a rational number between 7–√ and 8–√ with a denominator of 7, we can use the fact that (√2)^2 = 2.
Let's subtract (√2)^2 = 2 from both sides of the inequality:
7 - √2 - 2 < x < 8 - √2 - 2
This simplifies to:
5 - √2 < x < 6 - √2
Now, let's rationalize the denominator by multiplying both sides of the inequality by 7:
7(5 - √2) < 7x < 7(6 - √2)
This gives:
35 - 7√2 < 7x < 42 - 7√2
Divide all three sides by 7:
35/7 - 7√2/7 < x < 42/7 - 7√2/7
Simplifying further:
5 - √2 < x < 6 - √2
Therefore, a rational number between 7–√ and 8–√ with a denominator of 7 is in the range 5 - √2 < x < 6 - √2.
However, it is not possible to write this as an improper fraction with a denominator of 7, since the range is in terms of √2 and cannot be expressed as a simple fraction.
Let's subtract (√2)^2 = 2 from both sides of the inequality:
7 - √2 - 2 < x < 8 - √2 - 2
This simplifies to:
5 - √2 < x < 6 - √2
Now, let's rationalize the denominator by multiplying both sides of the inequality by 7:
7(5 - √2) < 7x < 7(6 - √2)
This gives:
35 - 7√2 < 7x < 42 - 7√2
Divide all three sides by 7:
35/7 - 7√2/7 < x < 42/7 - 7√2/7
Simplifying further:
5 - √2 < x < 6 - √2
Therefore, a rational number between 7–√ and 8–√ with a denominator of 7 is in the range 5 - √2 < x < 6 - √2.
However, it is not possible to write this as an improper fraction with a denominator of 7, since the range is in terms of √2 and cannot be expressed as a simple fraction.
Answered by
GPT 3.5
To find a rational number with a denominator of 7 between 7–√ and 8–√, we can first find the difference between these two numbers.
The difference between 7–√ and 8–√ can be obtained by subtracting 7 from 8 and √ from √, which gives:
(8 - √) - (7 - √) = 8 - √ - 7 + √ = 1
Therefore, the difference between the two numbers is 1.
Since we're looking for a rational number with a denominator of 7, we can express the difference of 1 as a fraction with a denominator of 7 as follows:
1/7
Hence, a rational number with a denominator of 7 between 7–√ and 8–√ is 1/7, written in improper fraction form.
The difference between 7–√ and 8–√ can be obtained by subtracting 7 from 8 and √ from √, which gives:
(8 - √) - (7 - √) = 8 - √ - 7 + √ = 1
Therefore, the difference between the two numbers is 1.
Since we're looking for a rational number with a denominator of 7, we can express the difference of 1 as a fraction with a denominator of 7 as follows:
1/7
Hence, a rational number with a denominator of 7 between 7–√ and 8–√ is 1/7, written in improper fraction form.
Answered by
GPT 3.5
The relationship between rational numbers and whole numbers is as follows:
1. All whole numbers are rational numbers: Whole numbers are a subset of rational numbers. Rational numbers include all numbers that can be expressed as a fraction p/q, where p and q are integers and q is not equal to zero. Whole numbers can be expressed in this form by considering them as the quotient of the number divided by 1. For example, 5 can be expressed as 5/1, and 0 can be expressed as 0/1. Therefore, all whole numbers are rational.
2. Not all rational numbers are whole numbers: Rational numbers include both fractions and whole numbers. Fractions that are not whole numbers are also considered rational numbers. For example, 3/4 or 7/8 are rational numbers that are not whole numbers.
In summary, while all whole numbers are rational numbers, not all rational numbers are whole numbers. Whole numbers are a specific subset of rational numbers that can be expressed as fractions where the denominator is 1.
1. All whole numbers are rational numbers: Whole numbers are a subset of rational numbers. Rational numbers include all numbers that can be expressed as a fraction p/q, where p and q are integers and q is not equal to zero. Whole numbers can be expressed in this form by considering them as the quotient of the number divided by 1. For example, 5 can be expressed as 5/1, and 0 can be expressed as 0/1. Therefore, all whole numbers are rational.
2. Not all rational numbers are whole numbers: Rational numbers include both fractions and whole numbers. Fractions that are not whole numbers are also considered rational numbers. For example, 3/4 or 7/8 are rational numbers that are not whole numbers.
In summary, while all whole numbers are rational numbers, not all rational numbers are whole numbers. Whole numbers are a specific subset of rational numbers that can be expressed as fractions where the denominator is 1.
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