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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Can two distinct irrational numbers be final segments of each other?

Let $x$ and $y$ be real numbers. I define the relation "$y$ is a final segment of $x$" if the decimal expansion of $x$, considered as an infinite string, has the decimal expansion of $y$ as a suffix. We omit any leading zeroes in the decimal…
user107952
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Find all $ a,b,c \in \mathbb{R} $ that satisfies the equations, $ \bullet $ $ a+b+c=63$ and $ \bullet $ $ab+bc+ac=2021$?

Find all $ a,b,c \in \mathbb{R} $ that satisfies the equations, $ \bullet $ $ a+b+c=63$ $ \bullet $ $ab+bc+ac=2021$ ? I try to solve this problem but going to the result that this problem has no solutions at all ... My attempt about the solution is…
Atique Ahmed
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How to prove $\sqrt{x} - \sqrt{x-1}>\sqrt{x+1} - \sqrt{x}$ for $x\geq 1$?

Intuitively when $x$ gets bigger, $\sqrt{x+1}$ will get closer to $\sqrt{x}$, so their difference will get smaller. However, I just cannot get a proper proof.
djsg
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What is the difference between $\mathbb{R}^+$ and $\mathbb{R}^*$?

I know that both of them contain all positive numbers from $\mathbb{R}$ but one notation contains $0$ too. I don't know which one. Thanks in advance.
Ge To
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Construct a set such that $a, b \in X \implies a + b \not\in X$

I'm looking for an uncountable set $X \subset \mathbb{R}_{\geq 0}$ such that for all $a, b \in X$, $a \neq b \implies a + b \not\in X$. Two points about this question: first of all, I'm not sure which area of mathematics this falls under so I used…
user883107
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Find the maximum between the minima

For every $n\in\mathbb Z^+ = \{1, 2,\dots\}$ set of positive integers, let $r_n$ be the minimum value of $|c−d\sqrt3|$ for all nonnegative integers $c$ and $d$ with $c+d=n$. Find, with proof, the smallest positive real number $g$ with $rn\leq g$…
Gabriela Da Silva
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Is $(-2)^{\frac{7}{5}}=((-2)^7)^{\frac{1}{5}}$ or not?

In Wolfram Alpha this statement is false. But how? Because $(a)^{\frac{b}{c}}=(a^b)^{\frac{1}{c}}$. Is there any condition. Please tell me.
Mathematics
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Find the least upper bound of the set $A=\{\frac{1}{y+x};x >1\}$; y>0 is fixed

I am looking for nice ways of proving that 1 is the least upper bound of the set $A=\{\frac{1}{y+x};x>1\}$ where y>0 is fixed. One (not so nice and unachieved) way is to first prove that 1 is a upper bound of any element of A and then use the…
Yagami
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Proving that the set $E=\left\{\frac{xy}{x^{2} +3y} ;x >0;y >0\right\}$ does not have an upper bound

It has been 2 days that i am trying to prove that the set E defined by $$E=\left\{\frac{xy}{x^{2} +3y} ;x >0;y >0\right\}$$ `doesn't have a upper bound,but without success I first supposed that E has a upper bound M and i tried to find x and y…
Yagami
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Greatest Lower Bounds

Let $Y = \{1/(1 + |x|): x \in \mathbb{R}\}$. This means that $Y$ is the collection of all the numbers of form $1/(1 + |x|)$, where $x$ is real. Does $Y$ have a maximum element (i.e. is there a $y_0$ in $Y$ such that $y \leq y_0$ for every $y$ in…
Sam Connell
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Are there any properties of real numbers that we know of which are not derivable from the axioms of a complete ordered field?

Of course there may be properties of the real numbers not derivable from the axioms (some Gödelian hand-waving here, I have only studied up to multivariable calc. and only dipped my feet in DEs + linear algebra), but is there anything we know to be…
Pineapple Fish
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Distance between two numbers less than every real number greater than 0

If I had two numbers $a,b \in \mathbb{R}$, is it true that $$(\forall \epsilon \in \mathbb{R}, \epsilon > 0), |a-b| < \epsilon \implies a = b$$ This is not part of any homework or assignment, but it is something I was wondering about. If it is the…
Kookie
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What does it mean to write a number in decimal form?

This is going to sound really elementary but I had this question about (this is a problem in Number Theory) 'find all four digit numbers such that, when written in decimal,...' Now the question itself is not that bad but what does it mean…
UnsinkableSam
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How to find x from the equation $\sqrt{x + \sqrt{x}} - \sqrt{x - \sqrt{x}} = m \sqrt{\frac{x}{x+\sqrt{x}}}$

For m are real number, Find x from the equation $$\sqrt{x + \sqrt{x}} - \sqrt{x - \sqrt{x}} = m \sqrt{\frac{x}{x+\sqrt{x}}}$$ I tried to multiply $\sqrt{x + \sqrt{x}}$ to the both sides and I get $$x + \sqrt{x} - \sqrt{x^{2} - x} = m \sqrt{x}$$ What…
Anas Ghazi
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Why multiplication of real numbers is commutative??

In real analysis books, it is taken as axiom. But I want to know the logic behind it. As for example, 32*75 means adding 32 for 75 times and 75*32 means adding 75 for 32 times. How to be sure these two quantities are equal? Please mention any book…
DREAM ISI-CMI-IIT
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