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Questions tagged [natural-numbers]

For question about natural numbers $\Bbb N$, their properties and applications

In mathematics, the natural numbers are those used for counting ("there are six coins on the table") and ordering ("this is the third largest city in the country"). These purposes are related to the linguistic notions of and , respectively (see English numerals). A later notion is that of a nominal number, which is used only for naming.

Properties of the natural numbers related to , such as the distribution of , are studied in . Problems concerning counting and ordering, such as partition enumeration, are studied in .

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Find the numbers of the form $\overline{abcd}$ for which the relations are checked simultaneously.

my question Find the numbers of the form $\overline{abcd}$ for which the relations are checked simultaneously: i) $\overline{ab}$ and $\overline{cd}$ are consecutive natural numbers ii) $(2*\overline{ab}+3)(2*\overline{ab}+7)=\overline{abcd}$. my…
IONELA BUCIU
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Why does Shilov exclude $0$ from natural numbers?

In the book "Linear Algebra" by Georgi E. Shilov, Chapter I the exclusion of $0$ from the numbers considered is justified by a note stating: Given two elements $N$ and $E$, say, we can construct a field by the rules $N+N=E, N+E=E, E+E=N, N\cdot…
AlexanderCar
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Let $m,n,o \in \mathbb{N} $, when $o$ is $m+n$?!

let $ \mathbb{N}:=\{ \emptyset, (\emptyset)^+, ((\emptyset)^+)^+,...,(...((\emptyset)^+)^+...)^+,...\} $, $n \in \mathbb{N} $ with $ I_n:=\{t \in \mathbb{N}|t \leq n\} $ $ I_n^*:=I_n\backslash \{0\} $ in this case $ 0:=\emptyset $, $ t \leq n $ if $…
mle
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Strict Ordering with Multiplication on Natural Numbers

I'm trying to prove that for all $a,b,c,d\in\mathbb{N}$ we have that \begin{align*}\tag{*} (a
PAT
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How to properly represent natural numbers less than n?

What is the better representation for natural numbers $x$ less equal a number $n>0$ in a scientific paper? $x \in \textbf{N}_{\le n}$ or $x \in \{1, \ldots, n\}$ or $0 \lt x \le n$?
CLRW97
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Is $\frac{n!}{\left(n/2\right)!\left(n/2\right)!}$ a natural number for $n$ even?

Probably this is a easy question, but I was unable to solve it. Let $n$ be a even natural number. Is true that the following number is natural for all $n$? $$\frac{n!}{\left(n/2\right)!\left(n/2\right)!}$$ I can see, for example, that $(n/2)!$…
Tomás
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Does an open Interval of natural numbers have a minimum and maximum? (unlike one of rational numbers)

Like for example (1, 9) The maximum would be 8, because we are talking about natural numbers, so the problem of an undefined maximum (the number right before 9) doesn't exist? Am I right?
tafelwasser123
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bijection between naturals

in https://www.coursehero.com/file/69596247/hw0pdf/ problem 3 asks to find a bijection between $\mathbb{N}$ and $\mathbb{N} \times \mathbb{N}=\mathbb{N}^2$. Recalling Cantor diagonal proof it is easy to show that such bijection exists. I was…
user730712
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Proving inverse direction of cancellation laws on natural numbers

I already proved the following using proof by induction: For all $a,b,c\in\mathbb{N}$ we have: \begin{align*} a+c=b+c \Rightarrow a=b \end{align*} and \begin{align*} a\cdot c = b\cdot c \Rightarrow a=b . \end{align*} Now I want to show that for all…
PAT
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Natural Numbers as $0$ and its Successors

I've been thinking about the propositions one can prove using the Peano axioms lately and there's this one question that crossed my mind. I understand that the axiom of induction was introduced to remove 'unwanted' elements from $\mathbb{N}$. These…
ghost
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Greatest Common Divisor question

Let $m$ and $n$ be natural numbers. How many pairs $x$ and $y$ of natural numbers with $m\cdot x+n\cdot y=m\cdot n$ are there ?
andrew
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How can I find the smallest possible of full miles to get full kilometers?

$1 \textrm{mile} = 1.609344 \textrm{km}$ I know that using $1000000$ miles I can move the decimal point and get a full number of $1609344 $km. But how can I find the smallest amount of full miles that is equal to full kilometers and do the same for…
Sven
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Let $ \mathbb{N}$, $a \in \mathbb{N} \to a+1 \in \mathbb{N}$

I need to prove the following: "Let $ \mathbb{N}$, if $a \in \mathbb{N} \to a+1 \in \mathbb{N}$" thanks in advance!!
mle
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Axiomatic construction of natural numbers and its properties from Zorich's book

I am reading the book of V.A.Zorich Mathematical analysis and trying to follow the formal construction of natural numbers and its properties: Definition 1: The set $X\subset \mathbb{R}$ is called inductive, if for any $x\in R$ the element $x+1 \in…
RFZ
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3 Elements of Sℕ that are finite sets? (Natural Numbers)

So i have Sℕ = P(ℕ) 1. I need 3 elements of Sℕ that are finite sets. My idea: Sℕ = {x | x < 4) does that work? 2. I need to write down an element of Sℕ that is an infinite set. Well, ℕ itself is already infinite ℕ=(1, 2,3,4,5......n) So my…
Prometheus
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