Questions tagged [natural-numbers]

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

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1answer
84 views

Estimation of a polynomial

I'm currently reading the following paper: http://arxiv.org/abs/1209.0612 and got stuck on Proposition 3.1 (2). The claim translated to polynomials is the following: Assume $n\geq 3, c\geq 1, d\...
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4answers
1k views

Of all the possible combinations of positive numbers that sum to 10, which has the largest multiplication?

Of all the possible combinations of positive numbers that sum to 10, which has the largest multiplication? I had also got a clue: it's related to e. Please help! (...
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2answers
950 views

How to define a bijection between natural numbers and the set of all polynomials with natural coefficients and finite variables?

Is there an explicit algorithm which establish a bijection between polynomials with finite variables and natural coefficients and natural numbers. Does anyone have one of these?Thanks.
3
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2answers
637 views

Interesting or non-obvious finite subsets of the natural numbers

I was recently explaining to someone how to prove that there are infinitely many prime numbers, and I mentioned to them that it's not immediately obvious, upon first encountering the natural numbers, ...
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3answers
3k views

Field with natural numbers

To make sure that we are talking about the same, I would like to post the relevant definitions I know first. Definitions: A pair $(G, +)$ where $G$ is a set and $+: G \times G \rightarrow G$ is ...
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2answers
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Consistency of Peano axioms (Hilbert's second problem)?

(Putting aside for the moment that Wikipedia might not be the best source of knowledge.) I just came across this Wikipedia paragraph on the Peano-Axioms: The vast majority of contemporary ...
14
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2answers
270 views

Prove that $\left (\frac{a^2 + b^2 +c^2}{a+b+c} \right) ^ {(a+b+c)} > a^a b^b c^c$

Prove that $\left (\dfrac{a^2 + b^2 +c^2}{a+b+c} \right) ^ {(a+b+c)} > a^a b^b c^c$ if $a$, $b$ and $c$ are distinct natural numbers. Is it possible using induction?
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2answers
3k views

Proof that $1$ is the least natural number using the Principle of Well Ordering

I'm doing a question from a PDF on Abstract Algebra.(The pdf can be found here http://abstract.pugetsound.edu/ 2012 edition). I have to show that the Principle of Well-Ordering implies that $1$ is the ...
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4answers
364 views

Euler's theorem for powers of 2

According to Euler's theorem, $$x^{\varphi({2^k})} \equiv 1 \mod 2^k$$ for each $k>0$ and each odd $x$. Obviously, number of positive integers less than or equal to $2^k$ that are relatively ...
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14answers
2k views

How can we produce another geek clock with a different pair of numbers?

So I found this geek clock and I think that it's pretty cool. I'm just wondering if it is possible to achieve the same but with another number. So here is the problem: We want to find a number $n \...
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3answers
1k views

Perfect square with digit-sum 15

Prove that there is not a single natural number $N$ with sum of digits equal to 15 that is the square of an integer.
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3answers
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Proof that binomial coefficient is a natural number [duplicate]

Possible Duplicate: Proof that a Combination is an integer What is the proof that the binomial coefficient is a natural number? $$k\ge0,n\ge k \implies {n \choose k} \in N,$$ I guess it's a ...
3
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1answer
137 views

Prove that $\mathbb{N}$ is nonwhere dense in $\mathbb{R}$

Prove that the set $ \displaystyle{\mathbb{N} =\{1,2,3, \cdots \} }$ is nonwhere dense in metric space $ \displaystyle{ \left( \mathbb{R} ,|\cdot| \right)}$ . I have found a solution in two steps: I ...
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6answers
4k 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'...
2
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1answer
2k views

Cardinality of the power set of natural numbers

I was reading an article on infinite sets and I came across a proof about how the power set of natural numbers has a greater cardinality than the set of natural numbers. I know that both given that ...
7
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1answer
956 views

Can the natural numbers be uncountable?

Definition of a countable set, from Stanford, as I didn't want to quote Wikipedia: Definition. A set S is countable if |S| = |N|. Thus a set S is countable if there is a one-to-one mapping of ...
2
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1answer
69 views

Maximising the product of exponents, but minimising the product of the base raised to its respective exponent

Given the following sequences: let value = $(b_0^{p_0})(b_1^{p_1})\cdots(b_n^{p_n})$ let productOfExponents = $p_0 \cdot p_1 \cdots p_n$ Where $p_i\geq 0$ and $p_i$ an element of $\mathbb{N}$ for ...
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2answers
256 views

find the minimum value of $a+b+c$

There are natural numbers: $a$, $b$, $c$. $$\begin{cases} ab+bc+ca+\frac32(a+b+c)=5015,\\ 2abc-a-b-c=6366 \end{cases} $$ I need to find the minimum value of $a+b+c$. To my mind there's ...
1
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3answers
834 views

Find the least natural number, for which the statement is true

could anyone help me with a small problem? I need to find the least natural number $n$, $n>1$ for which the statement is true. Statement: For any $n$ natural numbers, we can find two, $a$ and $b$,...
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2answers
645 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, ...
3
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2answers
446 views

Axiom schema and the definition of natural numbers

An axiom schema is used to generate the axioms, which inductively define the natrual numbers using the empty set and the successor function $S$. I don't understand why you have to define this set as ...
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3answers
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Is there a natural number between $0$ and $1$?

Is there a natural number between $0$ and $1$? A proof, s'il vous plaît, not your personal opinion. (Assume the Peano Postulates.)
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3answers
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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&...
2
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1answer
131 views

Is the estimation of number's name's length and comma-grouping feasible?

I am thinking in a mathematical problem that probably is already formulated and even solved. It is about big integers and someting else. Let n be an integer positive number. For n := 1,000 we have ...
2
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1answer
133 views

Show that there is some $p$ in $\omega$ for which $m+p^+=n$.

This is an old exercise I wrote up, but I'm unhappy with my solution. I assume only the basic properties of addition and multiplication for natural numbers as sets. Assume that $m$ and $n$ are ...
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3answers
393 views

Is 3/2 undefined when only considering the natural numbers?

If we only consider the natural numbers, is 3/2 undefined? If not what is the answer?
5
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1answer
954 views

Set theoretic definition of a Natural Number

I am unable to understand the motivation behind the set theoretic definition of a natural number. The definition given in the book by Goldrei is as follows: First he defines an inductive set: A set $...
0
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1answer
811 views

“Encode” a single digit in a multi-digit number in the smallest way possible

This may have more to do with computing than Mathematics in its application, however this has been giving me a headache for some time and I see no other recourse than to ask... Given a natural ...
64
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7answers
33k 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 ...
1
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3answers
335 views

Strict ordering on natural numbers

I'm studying on K. Hrbacek and T. Jech, Introduction to Set Theory. In the third chapter, they prove the usual properties of the strict ordering on natural numbers in the following way: They prove ...
91
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8answers
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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 ...