# Tagged Questions

Questions about numbers expressible as the quotient of two integers. For questions on determining whether a number is rational, use the (rationality-testing) tag instead.

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### Pugh's exercise on Dedekind cuts addition

I am trying to solve the following exercise: Let $x=A|B$ and $x'=A'|B'$ be cuts in $\mathbb{Q}$. Show that although $B+B'$ is disjoint from $A+A'$, it may happen in degenerate cases that $\mathbb{Q}$ ...
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### How to Find a rational number between two irratonal number? [on hold]

Find the rational number between $\sqrt 2$ and $\sqrt3$. I try to solve by using some methods in my book but can not understand steeps.
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### Rational numbers problem [on hold]

I have a problem with rational numbers, how do i find the numbers behind the following equation: 0,a(b) = a/b, using the following rationale: a/b*10=a,(...
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### Are there any natural proofs of irrationality using the decimal characterization?

Mathematicians typically define rational number to mean quotient of two integers. It is not hard to show that a number is rational by that definition if and only if its decimal expansion terminates ...
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### Proof that every repeating decimal is rational

Wikipedia claims that every repeating decimal represents a rational number. According to the following definition, how can we prove that fact? Definition: A number is rational if it can be written ...
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### Why must the decimal representation of a rational number in any base always either terminate or repeat?

Wikipedia makes the following statement about rational numbers. The decimal expansion of a rational number always either terminates after a finite number of digits or begins to repeat the same ...
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### Pairs of irreducible fractions that add up to a given irreducible fraction

Given the irreducible fraction $\frac a b$, with $a, b \in \mathbb N$, what is the expression that enumerates all the irreducible fractions of integers that add up to $\frac a b$? Namely, an ...
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### Algorithm that calculates decimal places of number

My background. I am a school student. Recently, we learned about rational numbers and irrational numbers. For example, we were told that rational numbers can always be written as a repeating decimal, ...
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### Is $\pi e$ irrational? [duplicate]

During our ongoing research, we need to prove that $\pi e<\lceil \pi e\rceil$. Is $\pi e$ irrational? How to prove it? Thanks- mike
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### Is $x^x$ rational for $x=\sqrt{2}^\sqrt{2}$

This might be naive. Is $x^x$ a rational number for $x=\sqrt{2}^\sqrt{2}$ ? I remember reading somewhere a long time ago that such $x^x$ is a rational number, as an example of issues with non-...
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### Rational root coefficient

I saw this question in my exam recently, If a, b, c are distinct rational roots of $x^3+ax^2+bx+c=0$, find the values of a, b, c. Can someone give me a hint or answer? I tried factoring it and ...
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### nearest approximation for a/b with denominator less than n

Given a rational number a/b what is the closest ration c/d such that d I would like a formula for c and d in terms of a,b and n if possible but if no mathematical solution exists, an algorithm for ...
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### is there any convergent sub-sequence of a sequence of all rational numbers?

Let $(a_n)$ be a sequence of rational numbers, where all rational numbers are terms. (i.e. enumeration of rational numbers) Then, is there any convergent sub-sequence of $(a_n)$?
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### General Conic and its Rational Solutions

Suppose you have a rational conic $ax^2+bxy+cy^2+dx+ey+f=0$. There is a theorem that states if a conic has 1 rational solution it has infinitely many rational solutions. How can you prove this ...
Given a rational $r \in \mathbb Q$, how to find the irreducible fraction $\frac a b = r$? Any direct formula based on the digits of $r$, instead of successive approximations by increasing ...