# Questions tagged [real-numbers]

For questions about $\mathbb{R}$, the field of real numbers. Often used in conjunction with the real-analysis tag.

2,552 questions
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### What is the use of extension of the real and complex field?

What is the use of the extension of the real and complex field to extended real and complex field (including $\infty$)? If the extended real and complex sets are no more field then what is the use of ...
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### Why is a < 0 the only solution to the following inequality?

I have been given the following equation, semi-derived from the quadratic equation: $\frac{+\sqrt{b^2-a}}{a}<\frac{-\sqrt{b^2-a}}{a}$ I need to prove that ${a}<0$ is a possible real solution ...
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### Prove $\forall x,y \in \mathbb{R} \$ $x^2+y^2 \geq x^2-y^2$

I know this may sound obvious, but I was wondering if both $x, y$ are real numbers, then why is it that $$x^2+y^2\geq x^2-y^2.$$
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### Exercise regarding density of rational numbers in $\mathbb{R}$ [on hold]

Let $a,b\in\mathbb{R}$ be such that $a<b$. Show that $$\exists\ n\in\mathbb{N}\ \ \{p/2^n : p\in\mathbb{Z}\} \cap (a,b) ≠ \emptyset.$$ I suppose two cases : $a>0$ and $a<0$. First, ...
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### Why $x=x$? Why $x=y$ and $y=z$ imply $x=z$? (Assume, $x$, $y$ and $z$ are reals.)

I recently have been studying the axiomatic construction of the set of real numbers through the Peano axioms for the natural numbers. It seems to me, the only things needed to proceed with this body ...
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### Group automorphism of multiplicative group of real number field

Let $\mathbb{R}$ be the real number field and $\mathbb{R}^{\times}$ be the multiplicative group of it. $\mathrm{Aut}(\mathbb{R}^{\times})$ denotes the group automorphism of $\mathbb{R}^{\times}$. [...
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### If $x$ and $y$ are irrational, then $x^y$ is irrational

I thought it was true, however my textbook claims it to be false. I need a counter example but I can't really think of one.
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### Why is it necessary to show subsequence convergence in the extreme value theorem?

I’m probably making this more complicated than it needs to be, but I’m trying to figure out why it is necessary to prove the convergence of a subsequence in proving the extreme value theorem (EVT). I ...
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### Can one sample rational number from $[0, 1]$ in $\Bbb R$?

I learned from real analysis course that when we devide [0,1] into irrational part $I$ and rational part $R$, we have $m(R) = 0, m(I) = 1$. So my question is, can I say we have zero probability that ...
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### Can linear combinations of real numbers with integer coefficients produce every real number?

Can there exist a finite set $S$ such that the set $$S' = \{ \sum a_i \beta_i, a_i \in \mathbb Z, \beta_i \in S\}$$ Is the set of all real numbers? If no, prove the same
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### Jordan's theorem implication

In class we proved we can write a function in bounded variation as the difference of two non-decreasing functions. (Jordan's theorem) My prof then said this suggests that this could be a good source ...
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### Proving $∀ε>0:x<y+\epsilon ⇒ x<y$

If $x, y \in \Bbb R$ and $x<y+\epsilon$ for every $\epsilon >0$, then $x<y$. Okay so I went about this by proving the contrapositive. Proof: Let $x,y\in\Bbb R$ and let $\epsilon>0$. ...
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### Do cubics always have one real root? [closed]

I've seen a few conflicting pieces of information online. So far, I know that with real coefficients there will always be one real root. But how about with complex coefficients? At very least ...
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### Mapping $\mathbb{R}^n$ “close” to integer points

For $\epsilon>0$ and a given $n$, define $$S_\epsilon=\bigcup_{x\in\mathbb{Z}^n}B_\epsilon(x)\subset\mathbb{R}^n$$ where $B_\epsilon(x)$ is the open ball of radius $\epsilon$ around the point $x$ ...
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### Bijection between points on a line and the real numbers

I need some help. Quoted from Elliott Mendelson's book, "Real numbers will have to be defined in such a way that, not only are the ordinary arithmetic operations of addition, subtraction, ...
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### Why the universe of neural network is a ring K?

I wish to find an algebraic or category theory approach to describe neural network in particular to have use algebraic method to extract the concept of 'synaptic weight'. I find this for the moment, ...
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### On some inequality about positive numbers [closed]

I was wondering whether you can help me with the following question: Let $a,b\geq0$. Is there some $c\geq 1$ such that \begin{equation} (ab)^c\left|(a^4+1)^{\frac{-3}{4}}-(b^4+1)^{\frac{-3}{4}}\...
The addition of Real numbers is commutative, so instead of saying we can find the sum of a sequence $\{a_1,...,a_n\}$ of real numbers that are pairwise not equal, we can say that there is a sum of a ...
### Prove that if $a < b$ and $c < d$ then $a + c < b + d$
I'm having a tough time with this questions. Could someone perhaps give me a hint? I'm only allowed to use the axioms of real numbers: A1. $a + b = b + a$ A2. $a + (b + c) = (a+b) + c$ A3. \$a + 0 = ...