Rational numbers can have decimals and even an infinite decimals, BUT any rational number's decimals will have a repeating pattern at some point whether it be like $$ \frac23 = 0.666... $$ or $$\frac{92}{111000} = 0.000\hspace{2px}828\hspace{2px}828\hspace{2px}828... $$ or $$\frac32 = 1.500 \hspace{2px} 000 \hspace{2px} 000...$$ The reason why ...

2014年11月18日 · Between any two rational numbers there exist another rational number. For example 1/2 and 1/4 are two rational numbers, but there exist another rational number 1/3 between the two above.In the case of other subsets of numbers in real numbers for instance,integers,there cannot exist another integers between any two.

2017年8月13日 · The rational number system is inadequate for many purposes, both as a field and as an ordered set. Addition and multiplication of rational numbers are commutative and associative, and multiplication is distributive over addition. Both 'zero' and 'one' exist. Plus, as I recall, rational numbers are the smallest subfield of $\mathbb{C}$.

2020年5月3日 · If the rationals were an open set, then each rational would be in some open interval containing only rationals. Therefore $\mathbb{Q}$ is not open. If $\mathbb{Q}$ were closed, then its complement would be open. Then each irrational number would be in some interval containing only irrational numbers. That doesn't happen either.

2018年9月29日 · Fix an irrational number $\alpha\in (-\epsilon,\epsilon)$ and let $\{x_n\}_{n\in\mathbb N}$ be a sequence of rational numbers in $[-\epsilon,\epsilon]$ converging (in $\mathbb R)$ to $\alpha$. Then $\{x_n\}$ is an infinite sequence with no $\mathbb Q$-convergent subsequence, and therefore $[-\epsilon,\epsilon]\cap\mathbb Q$ is non-compact ...

I have been looking at examples showing that the set of all rationals have Lebesgue measure zero. In examples, they always cover the rationals using an infinite number of open intervals, then compu...

2017年5月1日 · Rational numbers are ordered but unlike real numbers are not complete. The incompleteness is due to the existence of gaps in the rational numbers. When we say gaps for this ordered objects, doesn't it intuitively mean that there should exist a pair of rational numbers with a gap in between them. There exist infinitely many rational numbers in ...

It depends on the topology we adopt. In the standard topology or $\mathbb{R}$ it is $\operatorname{int}\mathbb{Q}=\varnothing$ because there is no basic open set (open interval of the form $(a,b)$) inside $\mathbb{Q}$ and $\mathrm{cl}\mathbb{Q}=\mathbb{R}$ because every real number can be written as the limit of a sequence of rational numbers.

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The gamma function, shown with a Greek capital gamma $\Gamma$, is a function that extends the factorial function to all real numbers, except to the negative integers and zero, for which it is not defined. $\Gamma(x)$ is related to the factorial …