Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [real-numbers]

For questions about $\mathbb{R}$, the field of real numbers. Often used in conjunction with the real-analysis tag.

The field of real numbers, usually denoted by $\mathbb{R}$ or $\mathbf{R}$ is a field equipped with an order, which is complete with respect to that order. Moreover, it is the only ordered field which is complete (up to isomorphism). The real numbers are used as basis for measuring "length".

The real numbers can be classified in various ways: rational and irrational numbers; algebraic and transcendental numbers; computable and non-computable numbers; etc.

The real numbers carry a natural topology, which is generated by the order. The topology can be induced by a naturally arising complete metric. See more on Wikipedia.

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Standard Euclidean Metric on $\mathbb{R}^{k^2}$.

Does the standard euclidean metric on $\mathbb{R}^{k^2}$ refer to $$ \left(\sqrt{\sum_{i=1}^{k}x_i^2}\right)^2 = \sum_{i=1}^{k}x_i^2 $$ or does $\mathbb{R}^{k^2}$ have some other meaning, like $\mathbb{R}^k \times \mathbb{R}^k$?
Mr.Young
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Dedekind Cuts in Construction of the real line

Is each Dedekind cut a unique real number? or when we apply the process(Dedekind cut), do we get a bunch of real numbers instead of a unique one. If we get a unique real number, is the unique real number then plotted as a line segment between…
novice
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Is something wrong with this counting technique of reals?

I know the topic of countability of reals has been discussed a lot, but I still don't understand the proofs, including the well-kown diagonal approach. So, please forgive my dilettantism if it has place. First of all, the main subject: is there…
user129516
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Is 0.00…001 a real number?

Is $0.\overline{0}1 \in \mathbb{R}$? If so, is $0.\overline{0}2>0.\overline{0}1$ and $0.\overline{0}1 \neq 0$? If not, is there a way to define the smallest Number that is not $0$?
LostReminder
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Collection list of un-real numbers

I am sorry if this question is off-topic here. This is not really a question, but more a request to provide examples of un-real numbers. So far I know only one example that I will put in the list below. However, to justify this post, I will yet ask…
brilliant
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Number Groupings for Coding

On a coding assignment, I want my program to detect 3 groups of any number going from 1 to 6. eg. 1 3 4 5 5 3 5 There is 1 group of three 5's but how would I specify that there are three fives? Also, how may pairs of 3 are there in the numbers going…
Luis Briceno
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If I have two consecutive Integers and I have the following formula $n(m+1)^2$ is it even of odd?

I am helping my sister study for the praxis exam of this study book, and I reviewed a question based on number theory. I see it involves constant integers my question is: If $m$ and $n$ are consecutive integers, which can never be even? Choose all…
Jose M Serra
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Define number by relation to $0$?

I've been trying to define what a number is and I've come forth with the definition that a number is a mathematical object with certain properties, such as value and that this value is determined by how much greater or less than a number is than 0.…
user756857
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How many digits has $\sqrt{2}$?

How many digits has $\sqrt{2}$? Countable or uncountable many?
nonuser
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How can I prove by contradiction that there is no real number such that $q^2 = -1$

How can I prove by contradiction that there is no real number such that $q^2 = -1$. You would have to assume that there exists a $q$ that satisfies $q^2 = -1$. But I can´t understand how I am supposed to prove this. Do I have to first assume that…
Bob Pen
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Use Wilson Theorem for the divisor

Use Wilson Theorem to find the smallest possible number which completely divides (12! + 6! + 12! × 6! + 1!). Wilson Theorem → Wilson Theorem states that if n is a prime number then n divides [(n-1)!+1] completely. ↓↓ Answer Below…
Aditya Shrivastava
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How can I prove that $\inf B=\sup A$?

Let $B$ be the set of all the upper bounds of the non-empty bounded subset $A\subseteq\Bbb R$. Prove that $\inf B=\sup A$. I divided it into two areas ($\inf B>\sup A$, $\inf B<\sup A$) and tried to show a contradiction. Is it the right approach?…
user656291
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Can we say for any given interval in $\mathbb R$ that every point in this interval is accumulation point?

Accumulation point means briefly that if $x_0$ is acc. point in set $S$ then for any given $\epsilon>0$, $((x_0-\epsilon, x_0+\epsilon ) \cap S )\setminus\{x_0\} \not = \emptyset$ My reasoning is, since every (open) interval is open set for every…
Micheal Brain Hurts
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representing Order pairs

I was reading Swokowski's book of calculus I noted that say [The Set of all order pairs will be denote by $\mathbb{R}*\mathbb{R}$ ] Does that mean if I have $(a,b)$ and $(c,d)$ then $(a,b)$ and $(c,d)$ = $\mathbb{R}*\mathbb{R}$ ?
Ammar
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How to prove that $x \in \mathbb{Q}$ such that $s^2 < x^2 < 5$.

$S$ is a supremum of a set $ E = \{x \in \mathbb{Q} : x > 0, x^2 < 5 \}$ And I want to prove that $ s^2 = 5 $. So I was trying to show that $ s^2 \geq 5 $ and $ s^2 \leq 5 $. It follows that $s^2 = 5$ I proved that $ s^2 \leq 5 $ But I couldn't…
alryosha
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