Show an example of a rational number that is between √10 and √11
One rational number that is between √10 and √11 is 3.2.
To show that 3.2 is rational and lies between √10 and √11, we can square both numbers:
(√10)^2 = 10
(√11)^2 = 11
So, we know that √10 < 3.2 < √11.
To demonstrate that 3.2 is rational, we can express it as a fraction by finding the quotient of two integers:
3.2 = 32/10 = 16/5
Therefore, 3.2 is a rational number between √10 and √11.
To find a rational number between √10 and √11, we can take their average. The average of two numbers lies between the two numbers.
√10 ≈ 3.1622
√11 ≈ 3.3166
Now, we can find the average:
(3.1622 + 3.3166) / 2 ≈ 3.2394
So, 3.2394 is a rational number that lies between √10 and √11.