Mathematics Stack Exchange
Mathematics Stack Exchange

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 .

1389 questions
188
votes
10 answers

Is $0$ a natural number?

Is there a consensus in the mathematical community, or some accepted authority, to determine whether zero should be classified as a natural number? It seems as though formerly $0$ was considered in the set of natural numbers, but now it seems more…
bryn
  • 10,045
131
votes
9 answers

Produce an explicit bijection between rationals and naturals

I remember my professor in college challenging me with this question, which I failed to answer satisfactorily: I know there exists a bijection between the rational numbers and the natural numbers, but can anyone produce an explicit formula for such…
Alex Basson
  • 4,331
122
votes
12 answers

Is there a domain "larger" than (i.e., a supserset of) the complex number domain?

I've been teaching my 10yo son some (for me, anyway) pretty advanced mathematics recently and he stumped me with a question. The background is this. In the domain of natural numbers, addition and multiplication always generate natural numbers,…
user1324
70
votes
9 answers

Given real numbers: define integers?

I have only a basic understanding of mathematics, and I was wondering and could not find a satisfying answer to the following: Integer numbers are just special cases (a subset) of real numbers. Imagine a world where you know only real numbers. How…
Daniel A.A. Pelsmaeker
  • 1,109
62
votes
0 answers

Does the average primeness of natural numbers tend to zero?

Note 1: This questions requires some new definitions, namely "continuous primeness" which I have made. Everyone is welcome to improve the definition without altering the spirit of the question. Click here for a somewhat related question. A number…
Nilotpal Sinha
  • 24,086
56
votes
3 answers

Bijection $f\colon\mathbb{N}\to\mathbb{N}$ with $f(0)=0$ and $|f(n)-f(n-1)|=n$

Let $\mathbb{N}=\{0,1,2,\ldots\}$. Does there exist a bijection $f\colon\mathbb{N}\to\mathbb{N}$ such that $f(0)=0$ and $|f(n)-f(n-1)|=n$ for all $n\geq1$? The values $f(1)=1$, $f(2)=3$, and $f(3)=6$ are forced. After that, you might choose to…
Thomas Browning
  • 4,541
56
votes
8 answers

Why not to extend the set of natural numbers to make it closed under division by zero?

We add negative numbers and zero to natural sequence to make it closed under subtraction, the same thing happens with division (rational numbers) and root of -1 (complex numbers). Why this trick isn't performed with division by zero?
lithuak
  • 1,255
45
votes
6 answers

The best symbol for non-negative integers?

I would like to specify the set $\{0, 1, 2, \dots\}$, i.e., non-negative integers in an engineering conference paper. Which symbol is more preferable? $\mathbb{N}_0$ $\mathbb{N}\cup\{0\}$ $\mathbb{Z}_{\ge…
Ari
  • 561
28
votes
7 answers

Are there any two numbers such that multiplying them together is the same as putting their digits next to each other?

I have two natural numbers, A and B, such that A * B = AB. Do any such numbers exist? For example, if 20 and 18 were such numbers then 20 * 18 = 2018. From trying out a lot of different combinations, it seems as though putting the digits of the…
Pro Q
  • 943
25
votes
2 answers

Demonstration that 0 = 1

I have been proposed this enigma, but can't solve it. So here it is: $$\begin{align} e^{2 \pi i n} &= 1 \quad \forall n \in \mathbb{N} && (\times e) \tag{0} \\ e^{2 \pi i n + 1} &= e &&(^{1 + 2 \pi i n})\ \text{(raising both sides to the $2\pi in+1$…
astabada
  • 533
25
votes
2 answers

How does Peano Postulates construct Natural numbers only?

I am beginning real analysis and got stuck on the first page (Peano Postulates). It reads as follows, at least in my textbook. Axiom 1.2.1 (Peano Postulates). There exists a set $\Bbb N$ with an element $1 \in \Bbb N$ and a function $s:\Bbb N \to…
Solomon Tessema
  • 361
  • 3
  • 6
24
votes
2 answers

Every natural number is covered by consecutive numbers that sum to a prime power.

Conjecture. For every natural number $n \in \Bbb{N}$, there exists a finite set of consecutive numbers $C\subset \Bbb{N}$ containing $n$ such that $\sum\limits_{c\in C} c$ is a prime power. A list of the first few numbers in $\Bbb{N}$ has several…
Daniel Donnelly
  • 22,288
23
votes
3 answers

How is addition defined?

I've been reading On Numbers and Games and I noticed that Conway defines addition in his number system in terms of addition. Similarly in the analysis and logic books that I've read (I'm sure that this is not true of all such books) how addition…
JSchlather
  • 15,745
22
votes
3 answers

List of powers of Natural Numbers

Greatings,   Some time ago a friend of mine showed me this astonishing algorithm and recently i tried to find some information about it on the internet but couldn't find anything... Please help. Pseudocode: Consider that 1 is the starting index of…
mr-fotev
  • 507
20
votes
2 answers

Set theoretic construction of the natural numbers

I'm trying to tie some loose ends here. My lecturer didn't bother to go into details, so I have to work it out myself. I usually hate to be pedantic, but these questions have been bugging me for a while. First, what's the proper definition of the…
dkdsj93
  • 367
1
2 3
…
92 93