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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Multiples of triangular numbers

I know that $\;2T_2=T_3\;$ is a triangular number. Can anyone suggest me other triangular numbers whose two multiple is again a triangular number?
Anu
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Is $0.\sqrt9$ a valid number?

Is $0.\sqrt9$ valid number? Are such numbers allowed? First, I thought the value of the above number can be 0.3 but then it occurred how I would interpret this number: $0.65\sqrt2$ or $0.65\sqrt229$ Are such numbers valid?
user33786
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Definition of modulus function

Good day. I'm an A-Level student and I've recently learnt about the modulus operation. Based on both wikipedia and my A-Level textbook, for $x\lt 0, |x|= -x$. However, since $0 = -0, |0| = -0$ and thus when $x = 0, |x| = -x$. Therefore, shouldn't…
Robin Ting
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Where is my mistake...?

Let $$x=0.99999\ldots.$$ Clearly $x$ is a rational number. I want to find $a,b$ such that $$x=\frac{a}{b}.$$ Clearly $10x-x=9$ and thus $x=1$. So $$1=0.99999\ldots$$ Where is my mistake?
Schüler
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The numbers between two numbers, and an infinitely small positive number

This question sounded a bit foolish to me, but at the same time, a bit intriguing if nothing else. Given any arbitrary number $a$ and $c$ where $a≠c$, the set containing all numbers on the interval $[a, a+c]$ contains infinitely many numbers. Can…
Descartes Before the Horse
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Real numbers axioms proof, is this correct?

What do you think is my proof correct? To prove x.0=0 x.0 = x.0 + 0 (0 is additive identity) = x.0 + (x + (-x)) (x must have an additive inverse) = (x.0 + x) + (-x) (by associativity) = (x.0 + x.1) + (-x) (1 is multiplicative identity) = x.(0+1) +…
Adrián Juárez Carrasquedo
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If $x\in [0,1)$ then there exists $n\in \mathbb N$, $x\leq 1-\dfrac{1}{n}$ ?

I'm trying to show that $\bigcup_{n\in \mathbb N^*}[0,1-\dfrac{1}{n}]=[0,1)$ and I'm stuck at the following step: If $x\in [0,1)$ how to justify that there exists $n\in \mathbb N$ such that $x\leq 1-\dfrac{1}{n}$ ? It seems like an Archimedean…
palio
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Bounded and unbounded interval

According to Hoy et al., an interval is bounded if it is impossible to go off to infinity while remaining inside of it and unbounded otherwise. Using this definition, how do we show that [a,$\infty$) is unbounded? I do understand that this is an…
PGupta
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What is the probability of $\lfloor a+b \rfloor < \lfloor a \rfloor + \lfloor b \rfloor$

Given $a,b \in \mathbb{R}, a\ge0, b\ge0$, what is the probability of: $$ \lfloor a+b \rfloor < \lfloor a \rfloor + \lfloor b \rfloor $$
marolafm
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Trivial (?) Proof

Take, $a$, $b$, $c$, $d$ to be real numbers. If $a>b$ and $c>b$ and $d>b$, can we prove that TECHNICALLY, $a+c>d$, or $a+d>c$, or $c+d>a$. Kind of a brain teaser but I am looking for some honest proof. Thanks.
John
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$E$ and $E^c$ dense in $\mathbb{R}$ with measure $>0$

Is it a way to construct such a space $E$ (the measure is intended as the Lebesgue measure), with this additional property : $$ \forall u,t\in \mathbb{R}, \ (u\neq v), \qquad \lambda(E\cap [u,v] )=\lambda(E^c\cap [u,v])>0 $$ For exemple, is it a…
Netchaiev
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Concept of bounded and well ordered sets

Why we can't say $]1,2]$ has a Minimum? Is it because we use this assertion only in the context of a partial order or total order which are not strict? The Well-Ordering-Theorem says that every set can be well-ordered. A set X is well-ordered by a…
Hölderlin
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Meaning of a statement

What does the following statement mean ? “The expression $((a^b)^c)^d$ has five interpretations. The simplest of all is $a^{bcd}$”
Aditi
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how to prove $\operatorname{lub} A = \operatorname{glb} B$ where $\emptyset \neq A \subset \mathbb R$ be bounded above in $\mathbb R$.

Let $\emptyset \neq A \subset \mathbb R$ be bounded above in $\mathbb R$. Let $B$ be the set of upper bounds of $A$. Show that B is bounded below and $\operatorname{lub} A = \operatorname{glb} B$
Abhishek Pal
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Substitute s for cos u and ds for -sin u

Can someone explain why when going from this step: $$\frac23\int \frac{\sin u}{\cos u}du$$ where substituting $s=\cos u$ and $ds = -\sin u\,du$ produces $$\frac23\int -\frac1s ds$$ My work shows it to be this from the substitution of $s$ for $\cos…
user04445
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