To find a rational number between \(7-\sqrt{2}\) and \(8-\sqrt{2}\) with a denominator of 7, we can first express \(7-\sqrt{2}\) and \(8-\sqrt{2}\) with a common denominator of 7.
\[
7-\sqrt{2} = \frac{7}{1} - \frac{\sqrt{2}}{1} = \frac{7}{1} - \frac{\sqrt{2}}{1} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{7}{1} - \frac{\sqrt{2}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{1} = \frac{7}{1} - \frac{\sqrt{4}}{1} = \frac{7}{1} - \frac{2}{1} = \frac{5}{1}
\]
\[
8-\sqrt{2} = \frac{8}{1} - \frac{\sqrt{2}}{1} = \frac{8}{1} - \frac{\sqrt{2}}{1} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{8}{1} - \frac{\sqrt{2}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{1} = \frac{8}{1} - \frac{\sqrt{4}}{1} = \frac{8}{1} - \frac{2}{1} = \frac{6}{1}
\]
So, we need to find a rational number between \(\frac{5}{1}\) and \(\frac{6}{1}\) with a denominator of 7. To find such a number, we can take the average of \(\frac{5}{1}\) and \(\frac{6}{1}\) which gives us \(\frac{5+6}{2} = \frac{11}{2}\).
To write this as an improper fraction, we divide the numerator 11 by the denominator 2, which gives us:
\[
\frac{11}{2} = \boxed{ \frac{11}{2}}
\]
What is a rational number with a denominator of 7 that is between 7–√ and 8–√. Write your answer as an improper fraction.
4 answers
AAAaannndd the bot gets it wrong yet again!
√7 ≈ 2.645 and 4/7 < 2.645 < 5/7
√8 ≈ 2.828 and 5/7 < 2.828 < 6/7
so √7 < 19/7 < √8
√7 ≈ 2.645 and 4/7 < 2.645 < 5/7
√8 ≈ 2.828 and 5/7 < 2.828 < 6/7
so √7 < 19/7 < √8
Apologies for the previous incorrect response. You are absolutely right.
To find a rational number between \(7 - \sqrt{2}\) and \(8 - \sqrt{2}\) with a denominator of 7, we can compare these numbers to the rational numbers with a denominator of 7:
\(\sqrt{7} \approx 2.645\) and \(\sqrt{8} \approx 2.828\).
We know that \(\frac{4}{7} < 2.645 < \frac{5}{7}\) and \(\frac{5}{7} < 2.828 < \frac{6}{7}\).
Therefore, a rational number with a denominator of 7 that lies between \(7 - \sqrt{2}\) and \(8 - \sqrt{2}\) is \(\frac{19}{7}\).
So, the answer is \(\frac{19}{7}\).
To find a rational number between \(7 - \sqrt{2}\) and \(8 - \sqrt{2}\) with a denominator of 7, we can compare these numbers to the rational numbers with a denominator of 7:
\(\sqrt{7} \approx 2.645\) and \(\sqrt{8} \approx 2.828\).
We know that \(\frac{4}{7} < 2.645 < \frac{5}{7}\) and \(\frac{5}{7} < 2.828 < \frac{6}{7}\).
Therefore, a rational number with a denominator of 7 that lies between \(7 - \sqrt{2}\) and \(8 - \sqrt{2}\) is \(\frac{19}{7}\).
So, the answer is \(\frac{19}{7}\).
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)
1 answer