# Proofs & Rational Number Btwn $\sqrt2$ & $\sqrt2 + \frac{1}{1000}$

• KelvinMa
In summary, the conversation discusses finding a rational number between any two real numbers, determining the intervals on which a given function is positive and negative, and finding the max and min points of a function. The first and third questions are currently unresolved, while the second question can be solved by finding the max and min points.
KelvinMa
1. Show that between any two real numbers there is a rational number.
You may assume that for each a>0 and b>0 in R there is a positive integer satisfying b<ma.

2.Find the intervals on which f(x) = (3x -2)(x 1)(x + 1) is positive, and
the intervals on which it is negative.

3.Find a rational number p/q between root of 2 and root of 2 + (1/1000)

Last edited:
Welcome to PF!

Hi KelvinMa! Welcome to PF!

Show us what you've tried, and where you're stuck, and then we'll know how to help!

thx tiny-tim : D
i really don't have any ideas about the first question and the third question
i ve looked up on my notes and i couldn't find any clues
the second question i ll find the max and min points but is there any other way to do it?

## 1. What is a proof?

A proof is a logical argument that demonstrates the truth or validity of a mathematical statement or theorem. It is a step-by-step explanation using established rules and definitions to show that a statement is true.

## 2. How do you prove that a number is rational?

A number is rational if it can be expressed as a ratio of two integers. To prove that a number is rational, we must show that it can be written in the form of a/b, where a and b are integers. This can be done by finding a common factor or by using the Euclidean algorithm.

## 3. What is the difference between a rational and an irrational number?

A rational number is any number that can be expressed as a ratio of two integers, while an irrational number cannot be expressed as a simple fraction. Irrational numbers have decimal expansions that never repeat or terminate, such as pi or the square root of 2.

## 4. How do you prove that a number is between two other numbers?

To prove that a number is between two other numbers, we must show that it is greater than the lower number and less than the higher number. This can be done by comparing decimal or fractional representations of the numbers, or by using inequalities.

## 5. How can you prove that a number is between $\sqrt2$ and $\sqrt2 + \frac{1}{1000}$?

One way to prove that a number is between $\sqrt2$ and $\sqrt2 + \frac{1}{1000}$ is to find its decimal representation and show that it falls between the decimal representations of $\sqrt2$ and $\sqrt2 + \frac{1}{1000}$. Another way is to use the fact that $\sqrt2$ and $\sqrt2 + \frac{1}{1000}$ are consecutive terms in a sequence of rational approximations of $\sqrt2$ and show that the number in question is closer to $\sqrt2$ than $\sqrt2 + \frac{1}{1000}$.

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