# What is Natural numbers: Definition and 143 Discussions

In mathematics, the natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country"). In common mathematical terminology, words colloquially used for counting are "cardinal numbers", and words used for ordering are "ordinal numbers". The natural numbers can, at times, appear as a convenient set of codes (labels or "names"), that is, as what linguists call nominal numbers, forgoing many or all of the properties of being a number in a mathematical sense. The set of natural numbers is often denoted by the symbol

N

{\displaystyle \mathbb {N} }
.Some definitions, including the standard ISO 80000-2, begin the natural numbers with 0, corresponding to the non-negative integers 0, 1, 2, 3, ... (sometimes collectively denoted by the symbol

N

0

{\displaystyle \mathbb {N} _{0}}
, to emphasize that zero is included), whereas others start with 1, corresponding to the positive integers 1, 2, 3, ... (sometimes collectively denoted by the symbol

N

1

{\displaystyle \mathbb {N} _{1}}
,

N

+

{\displaystyle \mathbb {N} ^{+}}
, or

N

{\displaystyle \mathbb {N} ^{*}}
for emphasizing that zero is excluded).Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, while in other writings, that term is used instead for the integers (including negative integers).The natural numbers are a basis from which many other number sets may be built by extension: the integers, by including (if not yet in) the neutral element 0 and an additive inverse (−n) for each nonzero natural number n; the rational numbers, by including a multiplicative inverse (1/n ) for each nonzero integer n (and also the product of these inverses by integers); the real numbers by including with the rationals the limits of (converging) Cauchy sequences of rationals; the complex numbers, by including with the real numbers the unresolved square root of minus one (and also the sums and products thereof); and so on. These chains of extensions make the natural numbers canonically embedded (identified) in the other number systems.
Properties of the natural numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics.
In common language, particularly in primary school education, natural numbers may be called counting numbers to intuitively exclude the negative integers and zero, and also to contrast the discreteness of counting to the continuity of measurement — a hallmark characteristic of real numbers.

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1. ### B Cool fact about number of digits in n!

This may have already been found by many people but I discovered the pattern on my own out of curiosity with some coding. There are only 4 natural numbers whose factorial contains the same number of digits as the number itself. That is to say n = digits_in(n!). The trivial case is obviously...
2. ### I Relationship between factorials and squares of natural numbers

Was fooling around and wrote down these two equations today that appear to work. I'm not all that bright and I'm positive these either have some proof or restate some conjecture--probably something in a textbook. Could somebody help me out? \forall n \in \mathbb{N}_0\smallsetminus\{0\} n^2 =...
3. ### Does there exist a surjection from the natural numbers to the reals?

1) Two sets have the same cardinality if there exists a bijection (one to one correspondence) from ##X## to ##Y##. Bijections are both injective and surjective. Such sets are said to be equipotent, or equinumerous. (credit to wiki) 2) ##|A|\leq |B|## means that there is an injective function...
4. ### B Natural Numbers contain all the Primes

w = {0,0 | 1,1 | 2,2...} Let x = number of primes up to w+1 Let y = number of primes up to w-1 Now there's an empty prime box in the 0,0 slot not connected to anything. So I let x = p-1 and y = p+1 p = [p0, p1, p2...] Now p0 becomes 1,0/1 It can be either on or off. For the sake of...
5. ### MHB What story and two-digit Natural Numbers best fit Bayes' Theorem chart?

I screenshot https://math.stackexchange.com/q/4185842.
6. ### MHB Number of natural numbers that have primitive roots

Hey! :giggle: How can we calculate the number of natural numbers between $2$ and $n$ that have primitive roots? Let $m$ be a positive integer. Then $g$ is a primitive root modulo $m$, with $(g,m)=1$, if the modulo of $g\in (Z/m)^{\star}$ is a generator of the group. We have that $g$ is a...
7. ### MHB Finding Even Natural Numbers w/ No Repetition: 0-5 & 6

The number of even natural numbers less than 100000 that can be formed from the digits of the set (0,1,2,3,4,5,6) so that the digits in the number are not repeated is? Here I understand that the even number in the last place is an even number, that is, it has 4 possibilities, but won't the...
8. ### MHB Proof of Triangle Inequality for $n$ Natural Numbers

Prove for all $n\in N$ $\dfrac{|a_1+...a_n|}{1+|a_1+...+a_n|}\leq\dfrac{|a_1|}{1+|a_1|}+...\dfrac{|a_n|}{1+|a_n|}$
9. ### Number of choosing r non-consecutive numbers out of N natural numbers

I have seen a solution for this question which was as follows, first out of 15 elements, take away 5, thus there are 11 gaps created for the remaining 10 numbers (say N) as, _N_N_N_N_N_N_N_N_N_N_ now, now we can insert back the 5 to comply with the non-consecutive stipulation for which, number...
10. ### A Ramanujan Summation, Variations of

Hi, I've seen several videos and documents that state that "the sum of all natural numbers is equal to -1/12". The "proof" in general is using ramanjuan summation and analytic continuation of the riemann function. In this proof, the election of the riemann function in order to perform the...
11. ### Prove A contains all natural numbers ≥ n_0

Homework Statement Prove that if a set A of natural numbers contains n_0 and contains k+1 whenever it contains k, then A contains all natural numbers ≥ n_0Homework EquationsThe Attempt at a Solution I'm just confused by the question, please don't answer it. Logically it makes sense that if n_0...
12. ### Finding number of natural numbers N such that............

Homework Statement Find the number of natural numbers n such that (n2-900)/ (n-100) is an integer.Homework EquationsThe Attempt at a Solution I have done the following: (n2-900-9100+9100)/ (n-100) or,{ (n2 - (100)2) + 9100 }/ (n-100) or, (n+100) + (9100)/(n-100) I hope you can understand this...
13. ### Proof of an inequality with natural numbers

Homework Statement Prove that ##\forall n \in \mathbb{N}## $$\frac{n}{2} < 1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{2^n - 1} \leq n \text{ .}$$ Homework Equations Peano axioms and field axioms for real numbers. The Attempt at a Solution Okay so my first assumption was that this part...
14. ### I Proof that 1 is an odd number using Peano Axioms of naturals

So I was just writing a proof that every natural number is either even or odd. I went in two directions and both require that 1 is odd, in fact I think that 1 must always be odd for every such proof as the nature of naturals is inductive from 1. I am using the version where 1 is the smallest...
15. ### Are natural numbers mental abstractions?

Say you have an orange and a banana. You can say that they are two fruits. But this pertains to the categorization of fruit, which could be considered a mental construct of a category. You cannot say that you have two yellow objects, because you really don't. Relative to the category of color...
16. ### MHB Natural Numbers with Repeating Digits: Solving for Perfect Squares

Find all natural numbers $n$, for which: $\sqrt{1 \underbrace{4...4}_{n \: times}} \in \Bbb{N}$
17. ### I The Set of Positive Integers - a Copy of the Natural Numbers

I am reading Ethan D. Bloch's book: The Real Numbers and Real Analysis ... I am currently focused on Chapter 1: Construction of the Real Numbers ... I need help/clarification with an aspect of Theorem 1.3.7 ... Theorem 1.3.7 and the start of the proof reads as follows: n the above proof we...
18. ### MHB The Set of Positive Integers as a Copy of the Natural Numbers ....

I am reading Ethan D. Bloch's book: The Real Numbers and Real Analysis ... I am currently focused on Chapter 1: Construction of the Real Numbers ... I need help/clarification with an aspect of Theorem 1.3.7 ... Theorem 1.3.7 and the start of the proof reads as...
19. ### MHB The Well-Ordering Principle for the Natural Numbers

I am reading Ethan D. Bloch's book: The Real Numbers and Real Analysis ... I am currently focused on Chapter 1: Construction of the Real Numbers ... I need help/clarification with an aspect of Theorem 1.2.10 ... Theorem 1.2.10 reads as follows: Towards the end (second last line) of the...
20. ### I The Well-Ordering Principle for the Natural Numbers ....

I am reading Ethan D. Bloch's book: The Real Numbers and Real Analysis ... I am currently focused on Chapter 1: Construction of the Real Numbers ... I need help/clarification with an aspect of Theorem 1.2.10 ... Theorem 1.2.10 reads as follows: Towards the end (second last line) of the...
21. ### MHB Addition and Multiplication of Natural Numbers - Bloch Th. 1.2.7 .... ....

I am reading Ethan D. Bloch's book: The Real Numbers and Real Analysis ... I am currently focused on Chapter 1: Construction of the Real Numbers ... I need help/clarification with an aspect of Theorem 1.2.7 (1) ... Theorem 1.2.7 reads as follows: https://www.physicsforums.com/attachments/6976...
22. ### I Addition/Multiplication of Natural Numbers - Bloch Th. 1.2.7

I am reading Ethan D. Bloch's book: The Real Numbers and Real Analysis ... I am currently focused on Chapter 1: Construction of the Real Numbers ... I need help/clarification with an aspect of Theorem 1.2.7 (1) ... Theorem 1.2.7 reads as follows: In the above proof of (1) we read the...
23. ### B Natural Numbers and Odd Numbers

Given any finite set of natural numbers, it seems evident that the odd numbers form a subset of the natural numbers. But what happens "at infinity"? I mean, if we account for all infinitely many natural numbers, there would be also infinitely many odd numbers. In such case, is it still true that...
24. ### A What if the (semi) field characteristic of N is not zero?

By definition, the characteristic of a field is the smallest number of times one must use the ring's multiplicative identity element (1) in a sum to get the additive identity element (0). Can we use the same rule for the set of natural numbers? If yes, I found a problem, that has something to...
25. ### B Cardinality and Natural Numbers

Hi, I hoping someone might be kind enough to possibly tell me where I have made an error :) I'm more of a recreational maths person, lol - and I'm trying to make a scheme that 'maps' any decimal number to a natural one. The method I have come up with is a bit odd, I'm hoping it works but still...
26. ### Disproving Statement: No Natural Numbers Satisfy y^5 + 1 = (x^7-1)/(x-1)

Homework Statement Prove that there doesn't exist natural numbers x and y such that the statement holds true. y^5 + 1 = (x^7-1)/(x-1)The Attempt at a Solution I was able to simplify the term down to y^5 / x = x^5 + (x^5-1)/(x-1) Not sure what to do with it
27. ### MHB Finding 3 natural numbers by the rule of Sarrus

Here is a problem I am working on: Using the Rule of Sarrus: $$\begin{vmatrix} x & y & z \\ z & x & y \\ y & z & x \\ \end{vmatrix} =x^3+y^3+z^3-3xyz,$$ find $x, y, z$ such that $x^3+y^3+z^3-3xyz = 315.$ And here is what I have gotten so far: By row and column operations and by factoring out...
28. ### I What does N^2 mean in the case of natural numbers?

What does the N^2 mean in this case? (Image below) Does it mean, for all two pairs of natural numbers, a and b? How would I represent non pair numbers, i.e. how would I write "For integers k,l, and m such that k>1, l>2, m>k+l" all in one line?
29. M

### I Cancellation law multiplication natural numbers

Hello everyone. I wanted to prove the following theorem, using the axioms of Peano. Let ##a,b,c \in \mathbb{N}##. If ##ac = bc##, then ##a = b##. I thought, this was a pretty straightforward proof, but I think I might be doing something wrong. Proof: Let ##G := \{c \in \mathbb{N}|## if ##a,b...
30. ### MHB Have I gone wrong in putting irrationals in one to one correspondence with natural numbers?

Hi, New member here and have been dabbling with some aspects of George Cantor's work. I think I have found a way to put the irrationals in one to one correspondence with natural numbers but thousands of mathematicians over the years might disagree. Is there a subtle error ( or even a blatant...
31. ### What is the term for all numbers except the natural numbers?

Is there a term used to describe all number except the natural numbers?
32. ### MHB Solving Equation In The Set Of Natural Numbers

Find all the natural numbers $a$ and $b$ such that $(a^2+1)(b^2+1)+45=2(2a+1)(3b+1)$.
33. ### MHB Find solutions in natural numbers

Find the solutions in natural numbers for the following equation: \frac{10}{x+10}+\frac{10\cdot 9}{(x+10)(x+9)}+\cdots+\frac{10\cdot 9\cdot 8 \cdots\cdot 3 \cdot 2 \cdot 1}{(x+10)(x+9)(x+8)\cdots(x+3)(x+2)(x+1)}=5
34. ### Proof of |2^N x 2^N| = |2^N| with N the natural numbers

Hello, At my exam I had to proof the title of this topic. I now know that it can easily be done by making a bijection between the two, but I still want to know why I didn't receive any points for my answer, or better stated, if there is still a way to proof the statement from my work. My work...
35. ### Understanding the Closed Set of Natural Numbers

Hi, How and why set of natural numbers is closed?
36. ### Proving that the natural numbers are countable, stuck.

Homework Statement [/B] I'm working through a problem in Abott's Understanding Analysis, second edition, the statement of the problem being: "Fix a member n of the natural numbers and let An be the algebraic numbers obtained as roots of polynomials with integer coefficients that have degree n...
37. ### Proving combination is a natural number by induction

Hi, I've seen on on several sites that you can prove that nCr, where r<=n, is a natural number. I'm not sure how to do this by induction. So I need help on this proof. How do I write this as a mathematical statement at the start of the induction proof? Thank you
38. ### How to construct the natural numbers with hypersets

I'm not a logician or mathematician but a philosopher (with dyscalculia) so please forgive me for skipping the technicalities... My question is this: Is there in the theory of non-well-founded sets (hypersets) something analogous to the set-theoretic construction of the natural numbers in ZF...
39. ### Closed/Open Sets and Natural Numbers

Homework Statement I am trying to understand why ℕ the set of natural numbers is considered a Closed Set. 2. Relevant definition A Set S in Rm is closed iff its complement, Sc = Rm - S is open. The Attempt at a Solution I believe I understand why it is not an Open Set: Given that it...
40. ### MHB Each subset of the natural numbers is finite or countable

Hello! (Smile) Proposition: Each subset of the natural numbers is finite or countable. Proof: Let $X \subset \omega$. First case: $X$ is bounded. That means that $(\exists k \in \omega)(\forall y \in X) y \leq k$. Then $X \subset k+1$ and $X$, as a subset of a finite subset, is finite ...
41. ### MHB Can Recursion Define Natural Number Addition?

Hello! (Wave) For each pair of natural numbers $m \in \omega, n \in \omega$ we define: $$m+0=m\\m+n'=(m+n)'$$ We fix a $m$ and recursively the operation $m+n$ is defined for any $n \in \omega$. Knowing for example that $m+0=m$ we can conclude what $m+1$ means. $$m+1=m+0'=(m+0)'=m'$$ and...

50. ### MHB Prime numbers vs consecutive natural numbers.

An easy question. All "odd" number can be expressed as a sum of consecutive natural numbers. Example: 35=17+18 35=5+6+7+8+9 35=2+3+4+5+6+7+8Question: Demonstrate that prime numbers (except for the "2"), can only be expressed as the sum of two consecutive natural numbers.