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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Let $a,b \in \mathbb{R^+}$; prove that if $ab \geqslant 1$, then $a \geqslant 1$ or $b \geqslant 1$

Is this a valid proof: We take the contrapositive statement, that if $a < 1$ and $b < 1$, then $ab < 1$, and suppose, to the contrary, that $ab\geqslant 1$. It would follow that \begin{align*} ab && \geqslant && 1\\ a &&…
Izzy M.
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Extended Euclidean Division and Series

We know that the extended Euclidean division of 8 by 3 results in the periodic decimal number 2,666 ... We can also see that the series indicated by the decimal number, this is, 2 + 6/10 + 6/100 + 6/1000 + ... converges to 8/3. Let a and b be…
Paulo Argolo
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Supremum and infimum of a set of reals

Find the supremum and infimum of the following set $$ A:= \left\{1 + \frac{(-1)^n}{n} \mid n\in \mathbb{N}\right\}.$$ Here if $n$ is near infinity then the fraction part is $0$, so the value of $A$ is $1$. So any of the terminal values are is $1$.…
Sami
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What's between an irrational and rational number?

There is a rational number between two irrational numbers, and an irrational number between two rational numbers. So what's between an irrational and rational number? I know about rational numbers being found between 2 real numbers but I don't know…
fiat.lucy
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What is the smallest number with any mathematical value of property?

I know that Graham's number is the biggest number ever used in a mathematical demonstration. Does a similarly unimaginably small number, with any worth of note mathematical property, exist? Please note that I'm talking about the smallest absolute…
Mauro F.
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Is $\{(m+ka,n+kb) : m,n,k\in\mathbb Z\}$ dense in $\mathbb R^2$ if $a,b\notin\mathbb Q$?

Can anyone prove here that for $a,b\in\mathbb R\setminus\mathbb Q$ we have that the set $$ \{(m+ka,n+kb) : m,n,k\in\mathbb Z\} $$ is dense in $\mathbb R^2$? I know that the projections onto the coordinate axes both are dense in $\mathbb R$, but I…
Friedrich Philipp
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A full gross number

I noticed in my Steam inventory the following card: Half Gross A full gross is greater than 99, but we can do with half. (the image shows the number $ 72 $) What is a full gross number? Google results just show me the gross bill amount/gross…
hjpotter92
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Laws of Indices with real exponents

Let x be any positive real number, and m, n be real numbers. Is it true that $x^{m+n} = x^m \times x^n$? If so, how do you argue this? What other definitions are required save for $x^0 = 1$ and $x^{m+1} = x \times x^m$, with m being a real number?
Anirban Mandal
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Real Numbers-Write in Interval Form

$|5 - x^{-1}| \lt 1$ $-1 \lt 5 -\frac{1}{x} \lt 1$. $-1-5 \lt -\frac{1}{x} \lt 1-5$. $-6 < -\frac{1}{x}< -4$. $6 > \frac{1}{x} > 4$. $6x > 1 > 4x$. I get stuck after this
user385779
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Help with proving that all ζ ∈ ℚ have a periodic decimal expansion

I've been trying to prove that all ζ ∈ ℚ have periodic decimal expansion. I assume that a terminating decimal expansion counts as a periodic decimal expansion as well since the zeros are periodic as well. I managed to prove that for all ζ ∈ ℝ with…
David
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Choosing a number in 0 to 1

I can pick a number in the interval from all real numbers from 0 to 1. Say this number is 0.42. Now the probability for drawing this number most be zero since we have infinity many numbers in the interval 0 to 1. Is this true and is it not a paradox…
Ole Petersen
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How to prove this is a vectorspace?

my question asks the following: Prove the following are members of the set of all positive real numbers $$x+y=xy$$ $$kx=x^k$$
LetsDoCode
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Simplifying surds

I got a question in my textbook that was to simplify the expression: $6\sqrt{162c^7d^5}$, and the answer says it is $54c^3d^2\sqrt{2cd}$, but I'm not sure how this happened as $5^2\cdot162$ doesn't equal $54^2\cdot2$. ($c$ and $d$ are positive real…
Kiwi
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Proof about infinite sum of fractions

STATEMENT: $t \in R$, $0 < t < 1$, $$\sum_{i=1}^\infty t^i = \frac{t}{1-t}$$ EXAMPLE: let t = $\frac{1}{2}$ then, $\frac{1}{2}$ + $\frac{1}{4}$ + $\frac{1}{8}$ + ... = 1 let n=$\frac{2}{3}$ then, $\frac{2}{3}$ + $\frac{4}{9}$ +…
Kuta
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