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.

4498 questions
6
votes
3 answers

Symbol for set of strictly positive real numbers?

Is there any standard symbol for the set $\{x\in\mathbb{R} : x > 0\}$? I think $\mathbb{R}^{+}$ usually includes zero. Some sources say I should use $\mathbb{R}^{*}_{+}$ but it looks slightly bizarre to me. Suggestions?
user255451
  • 79
  • 1
  • 2
5
votes
6 answers

How does one partition real numbers in two equinumerous sets (non-trivial)?

I better explain my goal with simplified example. I look for a function $f:(0,1)\rightarrow\{0,1\}$ such that $\forall \varepsilon>0 \,\,\exists \text{ bijection }\varphi:\{x\in(0,\varepsilon): f(x) = 0\}\rightarrow\{ x\in(0,\varepsilon):f(x)=1\}$.…
nakajuice
  • 2,549
5
votes
0 answers

Solve the equation $\sqrt{x^2-3x+18}+\sqrt{x^2+5x+2} = 2\sqrt{x^4+2x^3+5x^2+84x+36}.$

Solve the equation $$\sqrt{x^2-3x+18}+\sqrt{x^2+5x+2} = 2\sqrt{x^4+2x^3+5x^2+84x+36}.$$ I don't know where to start, but what I know is that $$(x^2-3x+18)(x^2+5x+2) = x^4+2x^3+5x^2+84x+36.$$ Any suggestion would be appreciate.
Longirsu
  • 51
5
votes
2 answers

Prove that $\max\{a+b,c+d\} \leq \max\{a,c\} + \max\{b,d\}$

I've got the number $\max\{a,b\}$ and $a,b\in \mathbb R$ $\max\{a,b\}=a, a\geq b$ or $\max\{a,b\}=b, a
George K.
  • 129
5
votes
3 answers

Simplifying an Expression further

I have trouble doing this and I'm not sure why are they the same and what are the steps I need to do to reach the simplified answer. For example ... $\frac{-8}{\sqrt{128}}$ This is the same as $\frac{-1}{\sqrt{2}} $ I'm not sure how to reach…
user307640
  • 2,632
  • 2
  • 17
  • 27
5
votes
4 answers

What makes negative numbers different from positive numbers other than their being (almost) opposite?

To quote from Wikipedia's article on negative numbers Negative numbers represent opposites. If positive represents movement to the right, negative represents movement to the left. If positive represents above sea level, then negative represents…
user311559
4
votes
1 answer

$A_\sqrt{2}$ isn't a Dedekind cut?

In a problem we consider a cut of $\mathbb{Q}$ a subset $A\subset\mathbb{Q}$ that fulfills: $A\neq\emptyset$ $\forall (q,q')\in A\times\mathbb{Q},\,q'
Scientifica
  • 8,781
4
votes
6 answers

Quick and painless definition of the set of real numbers

I am looking for a simple way to describe the underlying set of the real numbers without getting into cauchy sequences or dedekind cuts. Furthermore, I want the description to not rely on some notion of equivalence (like how one can use the notion…
aaron
  • 149
4
votes
3 answers

Can every uncountable subset of $\mathbb R$ be split up this way?

For me this question is like a fish that anytime when I (seem to) catch it, manages to slip out of my hands again. If $U$ is an uncountable subset of $\mathbb R$ then can it be shown that some $x\in\mathbb R$ exists such that $U\cap(-\infty,x)$ and…
drhab
  • 151,093
4
votes
1 answer

Dividing real number into two sets

I wonder the following question: Is there a partition of $\mathbb{R}$ into two disjoint subsets $A$ and $B$ such that $B$ satisfies $B=A+A$, namely $B=$ {$ x+y|x,y \in A $}?. Here, "partition into disjoint sets" means $A\cup B=\mathbb{R}, A\cap…
Wembley Inter
  • 632
4
votes
2 answers

Does there exist a nonzero rational number $x$ for every irrational number $y$ such that the product of $x$ and $y$ is rational?

I came across this statement in my math textbook: $$\exists x\in\mathbb{Q}\ :\ \forall y\in\mathbb{R}\setminus\mathbb{Q},\ xy\in\mathbb{Q}$$ The only number $x$ for which it's true (that I can think of) is $0,$ as in: $$xy=0\times y=0$$ I know that…
Zikta
  • 51
4
votes
1 answer

Is every centered set of real numbers of the form A-A?

A centered set of real numbers is a set $S$ of real numbers such that $0 \in S$ and $x \in S \rightarrow -x \in S$. Also, let $A$ and $B$ be sets of real numbers. The difference $A - B$ is the set $\{a - b | a \in A, b \in B\}$ It is easy to prove…
user107952
  • 20,508
4
votes
1 answer

Is it possible to construct $\Bbb R$ directly from $\mathcal{P}(\Bbb N)$

We usually construct $\Bbb R$ from Dedekind cuts. It is surprising to me that $|\mathcal{P}(\Bbb N)|=|\Bbb R|$. Is it possible to construct $\Bbb R$ directly from $\mathcal{P}(\Bbb N)$ rather than $\Bbb N \to \Bbb Z \to \Bbb Q \to \Bbb R$? In that…
Akira
  • 17,367
4
votes
1 answer

Prove the existence of a real number satisfying a property

Let $x_1, x_2, . . , x_n$ real numbers from $[0, 1]$. Prove there is $x \in [0, 1]$ so that $|x - x_1| + |x - x_2| + . . . + |x - x_n| =\frac n 2$ My attempt Let $f:[0,1] \rightarrow R, f(x)=|x - x_1| + |x - x_2| + . . . + |x - x_n|$. Because…
user261263
4
votes
1 answer

Is it true that $\Bbb R \subset \Bbb R^2$?

My teacher said that $\Bbb R \subset \Bbb R^2$, but I don't understand how this is possible since the elemnts of $\Bbb R$ are single numbers and the elements of $\Bbb R^2$ are pairs of numbers. Help please.
user288682
1
2
3
…
20 21