Questions tagged [rational-numbers]

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.

A rational number is any number that can be expressed as the quotient or fraction $\frac pq$ of two integers, with the denominator $q$ not equal to zero. Since $q$ may be 1, every integer is a rational number. The set of all rational numbers is usually denoted by $\Bbb Q$; it was thus named in 1895 by Peano after quoziente, Italian for "quotient".

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Proving that any union of dedekind cuts is itself a dedekind cut as long as it is not the set of rational numbers

If you could help me to feedback on this proof and vet it it would be a great help to me! A Dedekind Cut is defined as a set $D \subset \mathbb{Q}$ such that a) $D \neq \mathbb{Q}$ and $D \neq \emptyset$ b) Let $x \in D, $ if $y \in \mathbb{Q}$ and…
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Help with a proof involving Dedekind Cuts

I have not much experience with proofs, so any feedback will be welcome! If you could me with the last step it would be great! A Dedekind Cut is defined as a set $D \subset \mathbb{Q}$ such that $D \neq \mathbb{Q}$ and $D \neq \emptyset$ Let $x \in…
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Finding the set of integers within the set of rational numbers

Sorry if this is pretty basic, I just want to know whether my proof works! I would love any feedback on it. What I wanted to do in this proof was try to identity the set $\mathbb{Z}$ with a subset of $\mathbb{Q}$ such that the identification would…
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What are the rational points of $8y^{3}-12n^{2}y=3j^{2}x+x^{3}$?

I've been working on finding the rational points of the equation $A^3+B^3=C^3+D^3$ and managed to reduce it (using a method I've been developing) to $8y^{3}-12n^{2}y=3j^{2}x+x^{3},n,j\in\mathbb Q$. Any rational parametrization of $(x,y)$ and methods…
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Show that all rational numbers Q fit into the Hilbert hotel

Can I prove this with help of simple induction grounded on the basic axioms of number theory and a linearity pattern? $$\frac{-x}{y} \neq \frac{x}{y}$$ $$\frac{-x}{-y} = \frac{x}{y}$$ The complete set of rational numbers according to me. $$P =…
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Algebraic closure of $\mathbb{Q}$ and $\mathbb{R}$

I am trying to prove 2 statements: $\bar{\mathbb{Q}} _\mathbb{R}\subset \mathbb{R}$ $\bar{\mathbb{R}} _\mathbb{C}=\mathbb{C}$ For (1), I have come up with a few examples such as $\sqrt{2},\sqrt{3}\in\mathbb{R}$ which are algebraic in $\mathbb{Q}$,…
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difference between rational and fraction

my question is are we doing the same thing in $\frac{6}{-7}$ and $\frac{-6}{7}$? my argument is they are giving the same answer but at first we are dividing by $-7$ and second by $7$, so we are actually making changes to get the same answer, am I…
ARIGHNA
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Counterexample to implications of mean value theorem over the rationals that can be extended to a differentiable function on the reals

This is a follow-up question to a previous question of mine. The previous one was answered (in fact it was a duplicate), but I still didn't feel like my inquiry was over. The original question was about the mean value theorem on a space which is not…
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Counterexamples to implications of mean value theorem without completeness

By using the mean value theorem (for which we require completeness) we can show for a function $f\colon [a,b]\to\mathbb{R}$ differentiable on $(a,b)$ that $$f \;\text{monotone increasing}\iff \forall x\in (a,b)\colon f'(x)\ge 0.$$ Now,…
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What do we call a number transformation or function that outputs the input number without its decimal?

This question is looking for any established terminology, function, transformation or whatever the generalized term is for transmuting one collection of things into the same or different collection of things, whereby the english definition of said…
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Can somebody construct multiplication of rational numbers?

Imagine you being an ancient person, developing a theory of rational numbers from scratch, and suppose, you've discovered all known properties of integers (you've extended, previously found by you, natural numbers for operating with a notion of…
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Density of rationals and real numbers clamped between close rational numbers

I was watching MIPT lection on math in russian It was stated there that $$\forall \alpha,\beta\in\mathbb{R}, \alpha<\beta, \exists r\in\mathbb{Q}: \alpha
avi9526
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Is there a method for finding two irrational numbers that multiply together to form a rational number

I know multiplying $x*\frac{1}{x}$ gives us the rational number 1 as long as x is nonzero. I also know the square of rational numbers multiplied by itself gives us a rational number. Are these the only way of finding irrational numbers that multiply…
zxcv
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Sum and product of numerator of rational number doubt

Given the sum rule: $$ \frac{n}{m}+\frac{r}{s}=\frac{n \cdot s+m \cdot r}{m \cdot s} $$ the product rule: $$ \frac{n}{m} \cdot \frac{r}{s}=\frac{n \cdot r}{m \cdot s} $$ and the quotient rule: $$ \frac{\frac{n}{m}}{\frac{r}{s}}=\frac{n}{m} \cdot…
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