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 between two given irrational numbers, we need to find a rational number whose square falls between the squares of the given irrational numbers.
First, we square both the given irrational numbers:
(√7)^2 = 7
(√8)^2 = 8
We can see that 7 < 7.5^2 < 8, so we need to find a rational number whose square is 7.5^2.
To simplify the square, we note that 7.5 is half of 15, so 7.5^2 = (15/2)^2 = 15^2 / 2^2 = 225/4.
Therefore, a rational number with a denominator of 7 that is between √7 and √8 is:
225/4 ÷ 7 = 225/28.
So, the answer is 225/28.
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)
To find a rational number with a denominator of 7 between √7 and √8, we need to consider the decimal values of these numbers.
Calculating √7, we find that it is approximately 2.646.
Calculating √8, we find that it is approximately 2.828.
Now, let's consider the decimal values between these two numbers that can be written with a denominator of 7.
2.646 is equal to 2 5/7 as an improper fraction.
2.828 is equal to 2 4/7 as an improper fraction.
Hence, a rational number with a denominator of 7 between √7 and √8 would be 2 5/7.
To find a rational number between √7 and √8 with a denominator of 7, we need to consider the decimal equivalents of both numbers and their corresponding fractions.
First, let's find the decimal equivalents of √7 and √8:
√7 ≈ 2.645751311
√8 ≈ 2.828427125
Now, let's write these numbers as fractions:
√7 ≈ 2.645751311 ≈ 2 + (0.645751311)
≈ 2 + (0.645751311 * 1/1)
√8 ≈ 2.828427125 ≈ 2 + (0.828427125)
≈ 2 + (0.828427125 * 1/1)
Next, let's find the fractions that correspond to the decimals above:
√7 ≈ 2 + (0.645751311 * 1/1)
√7 ≈ 2 + (645751311/1000000000)
√7 ≈ (2 * 1000000000 + 645751311) / 1000000000
√7 ≈ 5645751311/1000000000
√8 ≈ 2 + (0.828427125 * 1/1)
√8 ≈ 2 + (828427125/1000000000)
√8 ≈ (2 * 1000000000 + 828427125) / 1000000000
√8 ≈ 2828427125/1000000000
Now, we have the fractions 5645751311/1000000000 and 2828427125/1000000000. We need to find a rational number between these two fractions with a denominator of 7.
To do this, we find the average of the numerators:
(5645751311 + 2828427125) / 2 = 8474178436 / 2 = 4237089218
Our rational number is 4237089218/1000000000. However, we need to simplify this fraction to be in its lowest terms.
To simplify, we can divide both the numerator and denominator by their greatest common divisor (GCD):
GCD(4237089218, 1000000000) = 2
Dividing the numerator and the denominator by 2:
4237089218 ÷ 2 = 2118544609
1000000000 ÷ 2 = 500000000
Therefore, the rational number between √7 and √8 with a denominator of 7 is:
2118544609/500000000
Note: This fraction is already in its lowest terms (no common factors between the numerator and denominator besides 1). It is an improper fraction since the numerator is greater than the denominator.