All Questions
Tagged with real-numbers abstract-algebra
98
questions
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90
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Proof that $\mathbb Q$ and $\mathbb R$ are Archimedean ordered fields
I searched for "archimedean ordered field" on this website and Google but didn't find much.
Exercises:
(pages 90 and 101 of Analysis I by Amann and Escher)
My attempt:
These exercises seem ...
- 4,252
1
vote
1
answer
91
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Alternative to the proof on Wolfram Mathworld that $\mathbb Q$ is the smallest subfield of $\mathbb R$
Exercise 2 (b) on page 100 of Analysis I by Amann and Escher asks me to show that $\mathbb Q$ is the smallest subfield of $\mathbb R$.
Wolfram MathWorld gives the following reasoning:
I don't know ...
- 4,252
0
votes
2
answers
117
views
Nontrivial subring of $\mathbb{R}$ not containing $1$
Are there examples of nontrivial subrings of $\mathbb{R}$ that do not contain $1$? If not, how can we prove they don't exist? The definition of "ring" here is really "rng"; rings ...
- 1,751
-3
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1
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106
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Easy looking hard homogenious inequality [closed]
For non negative $a,b,c$ $27{(a+b)}^{2}{(b+c)}^{2}{(c+a)}^{2}\ge64abc{(a+b+c)}^3$
I've tried to open the brackets but I didn't see how to proceed.
3
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1
answer
136
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Is there an ordered field with distinct subfields isomorphic to the reals?
Is there an ordered field with distinct subfields isomorphic to the field $\mathbb R$ of real numbers?
- 19.9k
13
votes
2
answers
533
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Is there an "algebraic" way to construct the reals?
It's possible to construct $\mathbb{Q}$ from $\mathbb{Z}$ by constructing $\mathbb{Z}$'s field of fractions, and it's possible to construct $\mathbb{C}$ from $\mathbb{R}$ by adjoining $\sqrt{-1}$ to $\...
- 694
6
votes
2
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118
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Two "different" definitions of $\sqrt{2}$
In Walter Rudin's Principles of Mathematical Analysis (3rd edition) (page 10), it is proved that
for every $x>0$ and every integer $n>0$ there is one and only one positive real $y$ such that ...
user765539
1
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2
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104
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Proof that $\frac{1}{2} + \frac{1}{2} = 1$ using just the algebraic properties of $\mathbb R$
Like the title says, can you prove rigorously that $\frac{1}{2} + \frac{1}{2} = 1$ using only the nine field properties of $\mathbb R$? I don't know if addition and multiplication are supposed to be ...
- 817
2
votes
1
answer
99
views
If A is a square matrix of size n with real entries, with $A = A^{p+1}$, then $rank(A) + rank (I_n - A^p) = n$
Let A be a square matrix of size n with real entries, $n \geq 2$, with $A = A^{p+1}$, $p \geq 2 $, then $$rank(A) + rank (I_n - A^p) = n$$
If p is prime, in addition, $$rank (I_n - A)=rank (I_n - A^...
4
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1
answer
183
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Two uncountable subsets of real numbers without any interval and two relations
Are there two uncountable subsets $A, B$ of real numbers such that:
(1) $(A-A)\cap (B-B)=\{ 0\}$,
(2) $(A-A)+B=\mathbb{R}$ or $(B-B)+A=\mathbb{R}$ ?
We know that if one of them contains an interval,...
- 2,491
2
votes
1
answer
635
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Is an automorphism of the field of real numbers without ordering the identity map? [duplicate]
We know an automorphism on $\mathbb{R}$ must fix $\mathbb{Q}$. If we assume the usual order structure and topology on $\mathbb{R}$, then we can use the density of $\mathbb{Q}$ to show an automorphism ...
- 501
0
votes
0
answers
53
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solve for function which is the written in terms of itself and its inverse
Let, $f:\rm I\!R \rightarrow \rm I\!R$ be some function that is equal to a linear combination of itself and its inverse. Is is possible write an explicit formula for $f(x)$?
$$
f(x) = af(x) + bf^{-1}(...
- 85
-2
votes
1
answer
48
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Construct a sequence in A that converges to the supremum of A [closed]
It is similar to this question that I learned quite a bit from:
Showing the set with a $\sup$ has a convergent sequence
But I want to ask how can I construct an example of (Sn).
i.e.
If A is a ...
- 3
0
votes
1
answer
151
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Showing a set is an ideal in a ring of real-valued functions
If $F$ is a ring of all real-valued functions defined on $\mathbb{R}$, is $S = \{f ∈ F | f(0) = 1\}$ an ideal?
What I'm thinking is $(f+g)(0) = f(0)+g(0) = 1+1 = 2$
and hence $f + g$ is in $S$? Is ...
- 819
1
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1
answer
140
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On the existence of an algebraically closed field containing other fields
This question arose while I was reading a paper I found in the web.
It might be very simple, but I don't know the answer.
Let $\mathbb{R}$ be the set of real numbers and $\mathbb{Q}_p$ the set of all $...
- 1,345
-1
votes
1
answer
430
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how can we define real number to form of a (first-order) structure of type $\mathcal L$?
We know that :
Let $\mathbb{R}$ denote the set of all real numbers. Then:
1-) The set $\mathbb{R}$ is a field, meaning that addition and multiplication are defined and have the usual properties.
2-)...
- 4,772
0
votes
1
answer
165
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The properties of real numbers field [closed]
I know, that the multiplicative group of $\mathbb{R}$ is create on the set $\mathbb{R}\setminus \{0\}$. But how we can multiply real numbers on the $0$ after this?
This point was unswered, I think. ...
- 595
0
votes
2
answers
71
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Proving $x^{r} \cdot x^{s} = x^{r + s}$ provided two facts
I have the following two facts:
For positive numbers $a$ and $b$ and natural numbers $n$ and $m$, we have $a = b$ if and only if $a^{n} = b^{n}$ if and only if $a^{1/m} = b^{1/m}$.
For a ...
user381493
10
votes
2
answers
1k
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Are the real numbers the unique Dedekind-complete ordered set?
A totally ordered set is Dedekind-complete if any subset which has an upper bound also has a least upper bound. Now any two ordered fields which are Dedekind-complete are order-isomorphic as well as ...
- 10.3k
3
votes
1
answer
206
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Abstracting Magnitude Measurement Systems (i.e. subsets of ${\mathbb R}^{\ge 0}$) via Archimedean Semirings.
I did some googling but could not find any easily accessible theory so I am going to lay out my ideas and ask if they hold water.
Definition: A PM-Semiring $M$ satisfies the following six axioms:
(1)...
- 11.5k
14
votes
2
answers
2k
views
Finite dimensional division algebras over the reals other than $\mathbb{R},\mathbb{C},\mathbb{H},$ or $\mathbb{O}$
Have all the finite-dimensional division algebras over the reals been discovered/classified?
The are many layman accessible sources on the web describing different properties of such algebras, but ...
- 950
3
votes
1
answer
487
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Homomorphism from $\mathbb R^2\to \mathbb C$
Is it possible to define a surjective ring homomorphism from $\mathbb R^2$ onto $\mathbb C$? The multiplication defined on $\mathbb R^2$ is as follows:
$(a,b)(c,d)=(ac,bd)$
- 4,928
6
votes
4
answers
207
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Is $\mathbb Q$ a quotient of $\mathbb R[X]$?
Is there some ideal $I \subseteq \mathbb R[X]$ such that $\mathbb R[X]/I \cong \mathbb Q$?
$I$ is clearly not a principal ideal.
- 10.5k
8
votes
1
answer
1k
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Is this a field?
Let $S$ be the set of all the ordered pairs in the cartesian plane. That is:
$$S=\{(x,y)|\ \ x, y \in \Bbb{R}\}$$
Then, If $a=(a_1, a_2)$ and $b=(b_1, b_2)$ are two arbitrary elements of $S$, the ...
- 1,598
3
votes
1
answer
243
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Find all $\mathbb{Q}$ subspaces of $\mathbb{Q} \times \mathbb{R}$
While working to Apply Lam's theorem to determine all the left ideals of $\begin{pmatrix}\mathbb{Q}&\mathbb{R}\\0&\mathbb{R}\end{pmatrix}$ I have encountered the problem of determining all ...
- 285
1
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1
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195
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Proof on Rational Numbers
I am trying to determine whether the following structure forms a Ring under the Real Number Definition of Addition and Multiplication
Consider the set of Real Numbers of the form:
$A = \{a + bp \:|\:...
user150203
1
vote
1
answer
85
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Are these two mathematical objects the same from a practical standpoint, or literally identical mathematical objects? [closed]
This question is derived from another question that I recently asked.
Take the two mathematical objects $\{ \mathbf{x} \in \mathbb{R}^n \mid x_1, x_2, \ldots, x_n \in \mathbb{Z} \}$ and $\{ \mathbf{x}...
- 4,322
4
votes
1
answer
715
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$n$-dimensional integer space? Or $\{ \mathbf{x} \in \mathbb{R}^n | x_1, x_2, ..., x_n \in \mathbb{Z} \}$?
If $\mathbf{x} \in \mathbb{R}^n$, then we would have $x_1, x_2, ..., x_n \in \mathbb{R}$, right? This is commonly known as $n$-dimensional space.
My question is, could we also have such a thing as $\...
- 4,322
2
votes
0
answers
88
views
Finding equivalent submodular function
Let $V$ be a finite set of points in $\mathbb{R}^n$ with $d(x,y)$ denoting the usual Euclidean distance between two points $x$ and $y$. I am trying to order points from the set $V$ iteratively, at ...
- 2,340
0
votes
1
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139
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How construct $\mathbb R$ algebraically ?
We can construct $\mathbb Q$ from $\mathbb Z$ using a quotient. Topologically, $\mathbb R$ is construct as the space where all Cauchy-sequence of $\mathbb Q$ converge (and we can also construct $\...
- 56k