All Questions
Tagged with real-numbers abstract-algebra
98
questions
-1
votes
1
answer
430
views
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-)...
- 4,782
0
votes
1
answer
165
views
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
views
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
views
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
views
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 ...
- 940
3
votes
1
answer
485
views
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
views
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
views
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
views
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
vote
1
answer
195
views
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
views
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}...
- 4,312
4
votes
1
answer
710
views
$n$-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 $\...
- 4,312
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
answer
139
views
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