For any positive integers $a$ and $b$, what relation must hold between them to ensure that $\dfrac{a}{b}$ is an irrational number?

Question

When thinking about $\pi$, I came across this question myself, and don't know how to start approaching this problem.

I first thought that $a$ and $b$ must be prime numbers, but a simple calculation ($\dfrac{7}{3}$) tells me that this is not the answer.

Answer 1

According to the definition of an irrational number, a number is irrational if and only if it can't be expressed as $\dfrac ab$, where $a$ and $b$ are integers. Therefore it's impossible to find integers $a$ and $b$ such that $\dfrac ab$ is irrational

Answer 2

It is simply not possible since the ratio of two positive integers is by definition, rational.

Answer 3

Any in form of p/q would be rational with q is not equal to zero. So there cannot exist a irratioal number in form ofp/q.and 22/7 is approx value of pi this is rational but actualvalue of pi is irrational

Answer 4

This question has sense, depending on what you think it's an irrational number.

Consider the following definition:

An irrational number is a real number whose decimal expansion is infinite and has no any repetitive patterns.

This is the first definition one uses to see in school, since we understand better an assertion ("whose expansion is infinite...") than a negation ("that can not be written as $a/b$..."). In this case, it makes sense to wonder:

Can I get an irrational number from a quotient of integers?

The answer to this question is no, simply because any quotient of integers will yield a real number whose decimal expansion is either finite or has a repetitive pattern (that we call a period). However, this fact must be proved, and for instance, it is not so trivial in the infinite periodic case (this uses infinite series!).