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onthetopo
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How do I find the sup of rationals (p/q, where q is even) that is less than sqrt(10)?
gcd(p,q)=1
gcd(p,q)=1
onthetopo said:we know irrationals are infinite decimals but rationals are finite decimals.
but how to show it rigorously?
AlephZero said:Most rationals are not finite decimals. For example, 1/3.
Werg22 said:Maybe that's what you're looking for:
If we have a supremum x, then sqrt 10 - x > 0. Since Q has the archimedean property, there exist a rational y such as 10 - x > 1/y. This said, we have 10 - (x + 1/y) > 0. This said, 1/y is rational, and hence so is x + 1/y - contradiction.
onthetopo said:thanks a lot for all your reply
I under stand the truncate argument , we know irrationals are infinite decimals but rationals are finite decimals.
onthetopo said:Brilliant, but I thought archimedean property is defined such that 10-x>1/n where n must be an integer. Also the statement sqrt 10 - x > 0 is quite intuitive.
The largest rational number less than the square root of 10 is 3, which can be written as 3/1 or 3.00.
To find the largest rational number less than sqrt(10), we can use the property that the square root of any non-perfect square number is an irrational number. Since 10 is not a perfect square, its square root is irrational. Therefore, the largest rational number less than sqrt(10) must be the greatest whole number less than sqrt(10), which is 3.
Yes, we can prove that 3 is the largest rational number less than sqrt(10) by contradiction. Suppose there exists a rational number x such that 3 < x < sqrt(10). We can write x as a fraction p/q, where p and q are positive integers. Then, 3q < pq < sqrt(10)q. Squaring both sides, we get 9q^2 < p^2q^2 < 10q^2. This means that p^2q^2 is between two consecutive perfect squares, which is impossible since q is a positive integer. Therefore, our assumption is false and 3 must be the largest rational number less than sqrt(10).
Knowing the largest rational number less than sqrt(10) can be useful in various fields of mathematics and science, such as number theory and physics. In number theory, it can be used to prove the irrationality of sqrt(10) and other square roots. In physics, it can be used in calculations involving square roots, such as determining the maximum possible velocity of a projectile.
Yes, there are infinitely many rational numbers that are close to sqrt(10) but not as large as 3. These numbers can be found by taking the square root of any perfect square that is slightly less than 10, such as 9.99, 9.9, 9.999, and so on. For example, sqrt(9.99) = 3.16227766..., sqrt(9.9) = 3.14771548..., and sqrt(9.999) = 3.16227766..., all of which are rational numbers that are close to sqrt(10) but less than 3.