Questions tagged [natural-numbers]

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

581 questions
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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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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 ...
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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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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'...
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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 ...
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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 ...
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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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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 ...
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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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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, ...
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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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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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“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 ...
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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 ...
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 ...