Questions tagged [natural-numbers]
For question about natural numbers $\Bbb N$, their properties and applications
158
questions
127
votes
9
answers
76k
views
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 ...
- 4,251
184
votes
10
answers
124k
views
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 ...
- 9,804
56
votes
8
answers
8k
views
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'...
- 1,225
17
votes
2
answers
9k
views
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 ...
- 337
8
votes
7
answers
5k
views
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 ...
- 203
70
votes
9
answers
10k
views
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. ...
- 1,109
121
votes
12
answers
11k
views
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 ...
user1324
11
votes
2
answers
1k
views
The history of set-theoretic definitions of $\mathbb N$
What representations of the natural numbers have been used, historically, and who invented them? Are there any notable advantages or disadvantages?
I read about Frege's definition not long ago, which ...
- 32.2k
8
votes
4
answers
592
views
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 ...
- 47.3k
23
votes
3
answers
14k
views
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 ...
- 15.5k
6
votes
1
answer
341
views
Prove that $n!=\prod_{k=1}^n \operatorname{lcm}(1,2,...,\lfloor n/k \rfloor)$ for any $n \in \mathbb N$
I try to prove the following formula:
$$n!=\prod_{k=1}^n \operatorname{lcm}(1,2,...,\lfloor n/k \rfloor)$$
I noticed that
$\upsilon_{p}(\operatorname{lcm}(1,2,...,\lfloor n/k \rfloor)) = s$ iff $\...
- 119
6
votes
4
answers
925
views
Proof by induction that if $n \in \mathbb N$ then it can be written as sum of different Fibonacci numbers [duplicate]
Proof that every natural number $n\in \mathbb N$, can be written as the sum of different Fibonacci numbers between $F_2,F_3,\ldots,F_k,\ldots$.
For example: $32 = 21 + 8+3 = F_8+F_6+F_4$
Research ...
- 1,324
3
votes
4
answers
10k
views
$\sqrt{17}$ is irrational: the Well-ordering Principle
Prove that $\sqrt{17}$ is irrational by using the Well-ordering property of the natural numbers.
I've been trying to figure out how to go about doing this but I haven't been able to.
- 41
3
votes
3
answers
137
views
Solve in $\mathbb N^{2}$ the following equation : $5^{2x}-3\cdot2^{2y}+5^{x}2^{y-1}-2^{y-1}-2\cdot5^{x}+1=0$
Question :
Solve for natural number the equation :
$5^{2x}-3\cdot2^{2y}+5^{x}2^{y-1}-2^{y-1}-2\cdot5^{x}+1=0$
My try :
Let : $X=5^{x}$ and $Y=2^{y}$ so above equation
equivalent :
$2X^{2}+(...
- 2,323
10
votes
4
answers
759
views
Should $\mathbb{N}$ contain $0$? [closed]
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 ...
- 12.6k