Math Problem Statement

√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

Solution

To find two fractions between 2\sqrt{2} and 3\sqrt{3}, we can use the approximate values you provided: 21.41\sqrt{2} \approx 1.41 and 31.73\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 75=1.4\frac{7}{5} = 1.4, which is slightly less than 2\sqrt{2}, so it doesn't work.
  2. The fraction 85=1.6\frac{8}{5} = 1.6, which is between 2\sqrt{2} and 3\sqrt{3}.
  3. The fraction 53=1.666\frac{5}{3} = 1.666, which is also between 2\sqrt{2} and 3\sqrt{3}.
  4. The fraction 95=1.8\frac{9}{5} = 1.8, which is slightly more than 3\sqrt{3}, so it doesn't work.

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

  1. 85=1.6\frac{8}{5} = 1.6
  2. 53=1.666\frac{5}{3} = 1.666

These fractions satisfy the condition 2<85<53<3\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 2\sqrt{2} to more decimal places?
  2. What is the decimal representation of 3\sqrt{3} to more decimal places?
  3. How do we rationalize the denominator of a fraction?
  4. What is the continued fraction representation of 2\sqrt{2}?
  5. What is the continued fraction representation of 3\sqrt{3}?
  6. How can we find more fractions between 2\sqrt{2} and 3\sqrt{3}?
  7. What are other methods to approximate square roots?
  8. How do we determine if a fraction is closer to 2\sqrt{2} or 3\sqrt{3}?

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

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Square Roots
Approximation
Fraction Representation

Formulas

-

Theorems

-

Suitable Grade Level

Grades 7-9