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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Division question

Let Z be the number of 8-digit numbers with 8 different digits, none of which is 0. How many 8 digit numbers exist that are divisible by 9, that have 8 different digits, none of which is 0. Answer in terms of Z. I tried putting in 9, also tried…
user140161
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Prove this statement?

I am having trouble with the following proof: Prove that for every three positive real numbers a, b, and c that $(a+b+c)*(\frac{1}{a} + \frac{1}{b} + \frac{1}{c}) \ge 9$. I have tried to directly prove this but all I get are dead ends.
Ansdai
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Prove consequence of Archimedean property

I have to proof that for $x,y\in \mathbb R$ with $y>0$, it exists $n\in\mathbb N$ such that $x
taue2pi
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Can a number be positive and negative at the same time?

Last week someone asked me if I could solve $3x+5 = 3x-5$. I think he just looked up unsolved problems or something like that, but as far as I can tell it has no solution... other than if $x$ was positive and negative $5/3$. So I told him it was…
RegularHumanBeing
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Adding the same real to itself finitely many times

Take any positive real number $a$. Is there always some natural number $n$ such that $a\cdot n \geq 1$? And, I guess more generally, is there always some natural number $n$ such that $a\cdot n\geq b$ for any $b\in\mathbb{R}$?
Questionbeggar
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Proving $x^a \cdot x^b = x^{a+b}$ for $a,b,x \in \mathbb R$ using dedekind cuts

How simple is it to prove the statement that for each $a,b,x \in \mathbb R$, we have $x^a \cdot x^b = x^{a+b}$, using Dedekind cuts? The reason I'm curious about this question is that someone was trying to solve $(\sqrt 2)^2 = 2$ using Dedekind…
Snared
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deduce that $\mathbb{Q}$ and $\mathbb{D}$ are dense

i have a question let $b \geq 2$ and for $n$ a natural number $Q_n = \Bigg\{ \frac{k}{b^n} ,k\in \mathbb{N} , 0 \leq k < b^n \Bigg\} $ i have proved for all real number in $[0,1]$. there exists $r_n \in Q_n$ such that $r_n \leq x < r_n +…
ana nadi lwa3r
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a natural map to reals?

If one encodes the real numbers as the surreal numbers with countable birthdays, it seems that the tree representation can be mapped to the naturals by a simple breadth first traversal. What am I missing?
JPER
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Why should a and b be co-primes in a÷b to be a rational number

A rational number can be represented in the form of a/b where a and b are integers and b≠0. a and b should be co-prime. My question is why do a and b need to be co-prime?
Zehran Bashir
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Between any 2 distinct real numbers, does there exist a real number such that its decimal expansion terminates in base 10?

Between any 2 distinct real numbers, does there exist a real number such that its decimal expansion terminates in base 10? Also, does this result hold in any natural number base? Thank you.
Brothersquid
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Number of points in an interval with quadratic spacing

Suppose I want to distribute unknown points $\{\lambda_1,\lambda_2,\dots,\lambda_n\}$ on an interval $[\lambda_\text{min}, \lambda_\text{max}]$ (on the real line) with spacing $\delta\lambda_i:=\lambda_{i+1}-\lambda_i$ given…
FizzleDizzle
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Why does there exist no real number such that it is equal to an integer multiple of any other number?

Was reading about waves in my Physics textbook and a mathematical fact was invoked which I was curious about: If we combine an infinitely large number of sinusoidal component waves, each with infinitesimally different reciprocal wavelength drawn…
physBa
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Prove that $\forall a,b {\in} \mathbb{Q} \:\:\exists x {\in}\mathbb{R}{\setminus} \mathbb{Q}\:\:a \lt x \lt b$

I'm trying to use cardinalities to prove that $$\forall a,b {\in} \mathbb{Q} \:\:\exists x {\in}\mathbb{R}{\setminus} \mathbb{Q}\:\:a \lt x \lt b.$$ Is the following proof of correct? Let $ I = \mathbb{R} \setminus \mathbb{Q} $ Assume $ \exists a,b…
Itay Hagadi
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I’m using what I want to demonstrate? Inverse of the product equals product of inverses

I want to prove that the inverse of a product equals the product of inverses, but I’m not quite sure if to I am using what I want to demonstrate. If so, can you explain me a way to avoid this kind of mistakes? Thanks for the advice! Let $a$ and $b$…
Aya Monge
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Definitions of subtraction and division

I'm reading some lecture notes, but I don't understand their definitions of subtraction and division. It states that subtraction is the inverse operation of addition, which I believe, but states that $a - b = c$ means that $a = b + c$. Similarly,…
Cardinality
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