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Proof that there is an irrational number between any two rational numbers Video Lecture - Engineering Mathematics

FAQs on Proof that there is an irrational number between any two rational numbers Video Lecture - Engineering Mathematics

1. How can we prove that there is an irrational number between any two rational numbers?
Ans. One way to prove this is by contradiction. Assume that there are two rational numbers, a and b, such that there is no irrational number between them. We can represent a and b as fractions (a = p/q and b = r/s) where p, q, r, and s are integers. Now, we can find the average of a and b (c = (a+b)/2) which is also a rational number. However, c is not equal to a or b, so it must be different from both. This means that c is either greater than a or less than b. Without loss of generality, let's assume c > a. Since there is no irrational number between a and b, this means that c must be a rational number that is greater than a. But this contradicts the fact that c is the average of a and b and should be between them. Therefore, there must be an irrational number between any two rational numbers.
2. Can you give an example of an irrational number between two rational numbers?
Ans. Sure, let's consider the rational numbers 1/3 and 1/2. An irrational number between them is √2. We know that 1/3 < √2 < 1/2. Therefore, √2 is an irrational number between 1/3 and 1/2.
3. Are there infinitely many irrational numbers between any two rational numbers?
Ans. Yes, there are infinitely many irrational numbers between any two rational numbers. This is because the set of rational numbers is countable, while the set of irrational numbers is uncountable. Therefore, there are infinitely more irrational numbers than rational numbers, and hence, infinitely many irrational numbers between any two rational numbers.
4. Can we find the exact value of the irrational number between two given rational numbers?
Ans. It depends on the specific rational numbers given. In some cases, it is possible to find the exact value of the irrational number between two given rational numbers. For example, if the two rational numbers are consecutive integers, then the irrational number between them can be found using the square root function. However, in general, finding the exact value of the irrational number between two given rational numbers may not be possible without additional information.
5. Can we find all the irrational numbers between two given rational numbers?
Ans. No, it is not possible to find all the irrational numbers between two given rational numbers. This is because the set of irrational numbers is uncountable, while the set of rational numbers is countable. Therefore, there are infinitely many irrational numbers between any two rational numbers, and it is not possible to list or describe all of them.
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