What is a rational number with a denominator of 7 that is between √7 and √8. Write your answer as an improper fraction
To find a rational number with a denominator of 7 between √7 and √8, we need to determine the whole number part of √8 and √7.
The integer part of √7 is 2 and the integer part of √8 is 2 as well.
Next, we need to find a fraction that is between √7 and √8. We know that √7 ≈ 2.646 and √8 ≈ 2.828.
The fraction that is between 2 and 3 (the whole number parts) is 2 and 2/7. So, a rational number with a denominator of 7 that is between √7 and √8 is 2 2/7.
In improper fraction form, this is:
2 2/7 = (2*7 + 2)/7 = 16/7.
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)
The relationship between rational numbers and whole numbers is that all whole numbers are rational numbers, but not all rational numbers are whole numbers.
Whole numbers are a subset of rational numbers. Rational numbers include all numbers that can be expressed as a fraction of two integers, where the denominator is not zero. This includes numbers with fractional parts as well as whole numbers.
For example, the whole number 4 can be written as the rational number 4/1, where the numerator is 4, and the denominator is 1.
However, not all rational numbers are whole numbers. Rational numbers can have fractional parts, whereas whole numbers do not have any fractional parts.
For example, the rational number 2/3 is not a whole number because it has a fractional part. Similarly, the rational number -5/1 is not a whole number because it is negative and has a fractional part.
In conclusion, every whole number can be expressed as a rational number, but not all rational numbers are whole numbers.
To find a rational number between √7 and √8 with a denominator of 7, we need to find the closest multiple of 7 to √7 and √8.
Let's calculate the square roots of 7 and 8:
√7 ≈ 2.646
√8 ≈ 2.828
The closest multiples of 7 to these values are:
3 × 7 = 21 (for √7)
4 × 7 = 28 (for √8)
Since we are looking for a rational number between these values, let's take the average of these two multiples:
(21 + 28) / 2 = 49 / 2 = 24.5
To convert this mixed number into an improper fraction, we multiply the whole number by the denominator and add it to the numerator, keeping the denominator the same:
24.5 = 24 + (1/2) = 48/2 + 1/2 = 49/2
Therefore, the rational number with a denominator of 7 between √7 and √8 as an improper fraction is 49/2.
To find a rational number with a denominator of 7 that lies between √7 and √8, we need to rationalize the denominators of √7 and √8.
First, let's consider √7. To rationalize the denominator, we multiply both the numerator and denominator by √7:
√7 = (√7 * √7) / (√7) = 7 / √7
Next, we'll consider √8. To rationalize the denominator, we multiply both the numerator and denominator by √8:
√8 = (√8 * √8) / (√8) = 8 / √8
Now, we can convert the resulting fractions to have a denominator of 7 by multiplying each fraction by (√7 * √8), which is equivalent to (√56):
√7 * (√56) = (7 / √7) * (√56 / √56) = (7√56) / (√7 * √8) = (7√56) / (√56) = 7√56 / 56
√8 * (√56) = (8 / √8) * (√56 / √56) = (8√56) / (√8 * √56) = (8√56) / (√56) = 8√56 / 56
Now, we can compare the resulting fractions to find a rational number between them with a denominator of 7:
7√56 / 56 and 8√56 / 56
To find a rational number between these two fractions, we can take their average (add them up and divide by 2):
(7√56 / 56 + 8√56 / 56) / 2 = (15√56 / 56) / 2 = (15√56) / (2 * 56) = 15√56 / 112
So, the rational number with a denominator of 7 that lies between √7 and √8, written as an improper fraction, is 15√56 / 112.