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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
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Set Theory (Real Numbers)

I have seen in a book that a number whose square is nonnegative is called real number. How can we explain what a real number is?
Waqar Ali Shah
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Find all values of $n$ for which $n^2+96$ is a perfect square

There can be infinite values of $n$. The statement is true for $n=2,5$. How to find out others? Please tell if there is any formula.
virat
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Extracting real and imaginary numbers from a complex number

How can I get the real number and the imaginary number from: $$\frac{3+i}{5-12i}$$
D_R
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Wilson's theorem states that if n is a prime number, it will divide (n-1)! + 1, using this find the smallest divisor of 12!+6! +12!×6! + 1?

Wilson's theorem states that if $n$ is a prime number, it will divide $(n-1)! + 1$, using this find the smallest divisor of $12!+6! +12!×6! + 1$? I checked yahoo answers and there someone gave the answer as $7$(which is wrong). The answer is $91$.…
Kousik
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$\Bbb R^n \times (0,\infty)$ what does this mean?

Just began to read about PDEs. There are a whole list of notations which I don't understand and the book isn't expecting a reader who is as inexperienced as me. Also don't understand what this means: $U_T = U \times (0,T]$ I know the bit in the…
PDElearner
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Which of this two numbers is the bigger?

How can I approach this problem of comparison between these two numbers. Any hints please. $A = 1000^{1000}$ or $B = 1\times 3\times 5\times \dots \times 1997$
mohamez
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Density of unbounded set modulo t

Let $A$ be an unbounded set in $\mathbb{R}$. Then consider the set $M(t)=\{a\mod t|a \in A \}$ in an interval $[0,t]$,for given $t>0$. Then some of $t$ will make $M(t)$dense in $[0,t]$. (This is my conjecture, so not maybe...) But my question is :…
curiosity
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Is there two integral numbers and a real number which satisfy $np=mr+q$?

$p$ and $r$ are given real numbers, both are positive and p is greater than r ($p,r \in R$, $p>r>0$) I need to prove or give a counter-example with the following conditions: Exist $n,m \in Z$, where $n>0$ and $m>0$, and exist $q \in R$, where…
User20141219
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Prove if $x ∈ \mathbb{R}$, such that $0 ≤ x ≤ 1$, and $m,n ∈\mathbb{ N}$, with $m ≥ n$. Then $x^m ≤ x^n$

How to prove the following prop. Let $x \in \mathbb{R}$, such that $0 \le x \le 1$, and $m,n \in\mathbb{ N}$, with $m \ge n$. Then $x^m \le x^n$. I don't exactly know where to begin with this proof, any guidance is appreciated.
raza arshad
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Prove there exists a real number $x$ such that $xy=y$ for all real $y$

Prove: There exists a real number $x$ such that for every real number $y$, we have $xy=y.$ In class I learned that I can prove a statement by: proving the contrapositive, proof by contradiction, or proof by cases. Can I do something along the…
Math Major
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Prove using a proof by contradiction: There is no smallest positive real number

Prove using a proof by contradiction: There is no smallest positive real number Let us assume the contradiction: There is a smallest positive real number. How do I continue?
user178415
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Proof about real numbers

Question is from Apostol's Vol. 1 One-variable calculus with introduction to linear algebra textbook. Page 28. Exercise 1. If $x$ and $y$ are arbitrary real numbers with $x
George Apriashvili
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Cardinality of an intersection

Is it true that given two real intervals $[a, b] $ and $[c, d] $, the cardinality of their intersection is either $0$ (when they're disjoint), $1 $ (when either $ b=c $ or $ d=a$) or $\mathfrak{c}$?
user132181
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How to prove the following numbers odd and even respectively?

Question:If a and b are two odd positive integers such that a>b, then prove that one of the two numbers (a+b)/2 and (a-b)/2 is odd and the other is even. My answer: since a and b are odd positive integers they have to be in the form 2q + 1; so a =…
Abhishek Mhatre
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Construct a bijection from $(0,1]$ to $(0,1)$ and $[0,1]$ to $(0,1)$

I have proved this but my teacher wants me to put more but I have no idea what to add. He says he wants a proof that they are explicitly in fact a bijection. For the first one this is what I did $g(x)=x$ if $x$ is not contained in $A$ Otherwise…
Ayoshna
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