√2Is approximately equal to 1.41 and √3 is approximately equal to 1.73. Use this idea to find two fractions between √2 and√3

To find two fractions between \(\sqrt{2}\) and \(\sqrt{3}\), we can use the approximate values you provided: \(\sqrt{2} \approx 1.41\) and \(\sqrt{3} \approx 1.73\). We can select fractions within this range.

Let's choose fractions with small denominators to make calculations easier:

1. The fraction \(\frac{7}{5} = 1.4\), which is slightly less than \(\sqrt{2}\), so it doesn't work.
2. The fraction \(\frac{8}{5} = 1.6\), which is between \(\sqrt{2}\) and \(\sqrt{3}\).
3. The fraction \(\frac{5}{3} = 1.666\), which is also between \(\sqrt{2}\) and \(\sqrt{3}\).
4. The fraction \(\frac{9}{5} = 1.8\), which is slightly more than \(\sqrt{3}\), so it doesn't work.

Thus, two fractions between \(\sqrt{2}\) and \(\sqrt{3}\) are:

1. \(\frac{8}{5} = 1.6\)
2. \(\frac{5}{3} = 1.666\)

These fractions satisfy the condition \( \sqrt{2} < \frac{8}{5} < \frac{5}{3} < \sqrt{3} \).

Would you like more details or have any questions?

Here are some related questions:

1. What is the decimal representation of \(\sqrt{2}\) to more decimal places?
2. What is the decimal representation of \(\sqrt{3}\) to more decimal places?
3. How do we rationalize the denominator of a fraction?
4. What is the continued fraction representation of \(\sqrt{2}\)?
5. What is the continued fraction representation of \(\sqrt{3}\)?
6. How can we find more fractions between \(\sqrt{2}\) and \(\sqrt{3}\)?
7. What are other methods to approximate square roots?
8. How do we determine if a fraction is closer to \(\sqrt{2}\) or \(\sqrt{3}\)?

**Tip:** When approximating square roots, using fractions with small denominators helps in making quick and simple estimates.

Discover how to find two fractions between √2 and √3 using approximate values. Learn about the approximation of square roots and the concept of fraction selection within a specified range. Suitable for students in Grades 7-9.

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