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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Commutativity of multiplication in $\mathbb{N}$

I'm trying to prove that $a\cdot b=b\cdot a$ when $a$ and $b$ are two natural numbers. In the rest of this question I'm using $a'$ for the successor of $a$. Addition is defined as: $a+0=a$ $a+b'=(a+b)'$ Multiplication is defined as: $a\cdot…
user112679
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Existence of a sequence that has every element of $\mathbb N$ infinite number of times

I was wondering if a sequence that has every element of $\mathbb N$ infinite number of times exists ($\mathbb N$ includes $0$). It feels like it should, but I just have a few doubts. Like, assume that $(a_n)$ is such a sequence. Find the first $a_i…
Valtteri
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Writing numbers with fewer symbols using expressions with powers

For example, it takes 7 symbols to write the natural number $n=9999999$ but we can also write it with 5 symbols as $n=10^7-1$. (Of course, with even larger exponents we can save even more symbols.) Another example: $13841304697 = 7^{12}+8*3^7$. Here…
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Can $(\Bbb N,\leq)$ have an $\aleph_0$-categorical theory (in a larger language)?

One of the nicer consequences of compactness is that there is no statement in first-order logic whose content "$\leq$ is a well-order". So we can show that there are countable structure $(M,\leq)$ which are elementarily equivalent to $\Bbb N$, but…
Asaf Karagila
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Should $\mathbb{N}$ contain $0$?

This is a classical question, that has led to many a heated argument: Should the symbol $\mathbb{N}$ stand for $0,1,2,3,\dots$ or $1,2,3,\dots$? It is immediately obvious that the question is not quite well posed. This convention, as many others,…
Jakub Konieczny
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Prove that product of four consecutive natural numbers can not be a perfect cube

The question is Prove that product of four consecutive natural numbers can not be a perfect cube . I really dont know what actually do to proceed with the question. However after seeing this related result for the product of four consecutive…
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Is there a specialized formula for Lagrangian interpolation on equispaced points?

If we know $f(0),f(1),f(2),\cdots f(n)$, is there a specialized version of the Lagrangian interpolation formula and a shortcut to compute the coefficients ? (Stability is not a concern.)
user65203
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If the order in a set doesn’t matter, can we change order of, say, $\Bbb{N}$?

I’m given to understand that the order of the elements of a set doesn’t matter. So can I change the order of the set of natural numbers or any set of numbers ( $\mathbb{W,Z,Q,R}$ for that matter) as follows? $$ \mathbb{N} = \{ 1,2,5,4,3,\cdots \}…
William
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Is the product of uniformly distributed numbers, uniformly distributed too?

My question is simple, I think. If we took two random natural numbers $a$ and $b$ uniformly distributed in a specific range $[c,d]$, is $ab$ a uniformly distributed too? What if $a$ and $b$ are not natural numbers, but real numbers? What if $a$ is a…
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How can I define $\mathbb{N}$ if I postulate existence of a Dedekind-infinite set rather than existence of an inductive set?

Suppose in the axioms of $\sf ZF$ we replaced the Axiom of infinity There exists an inductive set. with the Axiom of Dedekind-infinite set There exists a set equipollent with its proper subset. How can I define the set of natural numbers…
Vladimir Reshetnikov
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Landau's "Foundations of Analysis" - Addition of natural numbers

At the beginning of his Foundations of Analysis book (translated from German), Landau writes in his Preface for the Teacher : Peano defines $x+y$ for fixed $x$ and all $y$ as follows : $$x+1 = x' \\ x+y' = (x+y)',$$ and he and his successors…
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How do we know that the number $1$ is not equal to the number $-1$?

How do we know that the number $1$ is not equal to the number $-1$? (I am not talking about the multiplicative inverse of an arbitrary field, but the integer/rational, real or complex number $1$.) Is that an axiom? Since someone in the comments…
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Non-linear system of equations over the positive integers — more unknowns than equations

This exercise appeared on a german online tutoring board and caught my attention but stumbled me for hours. The task is to find 6 distinct positive three digit integers satisfying:…
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Prove 24 divides $u^3-u$ for all odd natural numbers $u$

At our college, a professor told us to prove by a semi-formal demonstration (without complete induction): For every odd natural: $24\mid(u^3-u)$ He said that that example was taken from a high school book - maybe I din't get something, but I…
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Bijection between Prime numbers and Natural numbers

We know that if set $S$ is countable then this set and set of all natural numbers are equivalent, which means that there must be some bijection between this two sets $F:S\rightarrow N$. We know that set of all Prime numbers is countable as well as…
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