All Questions
Tagged with real-numbers abstract-algebra
98
questions
5
votes
1
answer
121
views
Is there a binary operation over the nonnegative reals satisfying the metric and group axioms?
Is there a binary operation over the nonnegative reals which satisfies the metric axioms and the group axioms? I.e., find an $f : S \times S \to S$ such that $(f,S)$ follows the group and metric ...
user275490
2
votes
1
answer
129
views
If $a,b$ and $b,c$ are algebraic, can we effectively say $a,c$ are algebraic?
Lets say for two transcendental complex numbers x,y in $\Bbb C$ that they are algebraically independent, if there doesn't exists a non-zero polynomial $p(X,Y) \in \Bbb Q[X,Y]$, such that:
$$p(x,y)=0$$...
user4414
2
votes
2
answers
187
views
Homomorphism Between a Geometric Algebra and its Field of Scalars?
Given a geometric algebra defined over a real vector space, is is possible to construct a homomorphism between the elements of the geometric algebra and the reals?
I was pondering an example of this: ...
- 23
1
vote
2
answers
338
views
If $a$ and $b$ are positive rational numbers, why might we want to call the $√a$+$√b$ and $√a$-$√b$ are conjugates?
If $a$ and $b$ are positive rational numbers, why might we want to call the $√a$+$√b$ and $√a$-$√b$ are conjugate?
I know the definition of conjugates. Using that I can see this. What is the purpose ...
user436595
14
votes
2
answers
556
views
Can $\mathbb{R}^{+}$ be divided into two disjoint sets so that each set is closed under both addition and multiplication?
Can $\mathbb{R}^{+}$ be divided into two disjoint nonempty sets so that each set is closed under both addition and multiplication?
I know if we only require both sets to be closed under addition then ...
- 169
4
votes
1
answer
177
views
Abstract concept tying real numbers to elementary functions?
Real numbers can be broken into two categories: rational vs. irrational. Irrational numbers can be approximated, but never fully represented by rational numbers.
Analytic functions have Taylor ...
- 247
1
vote
2
answers
269
views
real numbers of the form $\frac{m}{10^n} $ with $m,n \in \mathbb{Z} $ and $n \geq 0$ is dense in $\mathbb{R}$ . [duplicate]
Problem :
Verify if the statement if true of false -
The set $S$ of all real numbers of the form $\frac{m}{10^n} $ with $m,n \in \mathbb{Z} $ and $n \geq 0$ is dense in $\mathbb{R}$ .
I think this ...
- 2,208
0
votes
2
answers
41
views
Can someone help me this polar form question?
The red bubble means I was wrong, but I don't know why. The second picture is the way I did this question.
Sorry for the inconvenience, I am still learning LaTex.
- 143
2
votes
3
answers
245
views
Inverse of isomorphism between two fields is isomorphism
I'm using this definition of isomorphism between two systems satisfying the axioms of Dedekind-complete totally ordered fields to show that its inverse is an isomorphism too. I can be easily proved ...
- 779
1
vote
2
answers
452
views
Must an automorphism on the group of real numbers under multiplication maintain sign?
Suppose we have an automorphism $\phi$ under the group $(\mathbb{R}^{\#},\,\cdot)$. I need to show that $\phi$ preserves the sign of the numbers, or that $\phi(\mathbb{R}^+)=\mathbb{R}^+$ and $\phi(\...
0
votes
0
answers
45
views
Properties of $\mathbb{R}$
Introductory analysis classes make it a point to observe some of the critical properties of the reals that have allowed so much to be done with them.
My takeaways are that the most important ...
- 2,233
2
votes
1
answer
449
views
Question on the archimedean property
Let $a,b \in \Bbb ℝ$. Suppose that $a>0$. Prove that there is some $n\in \Bbb N$ such that $b\in[-na, na]$.
I understand how the Archimedean Property can be used to prove this statement if $b$ is ...
- 107
2
votes
0
answers
48
views
Basic proof that every polynomial over $\mathbb R$ factorizes into at most quadratic ones
Is there a proof without using that in $\mathbb C$ every polynomial factorizes into linear ones that every polynomial over $\mathbb R$ factorizes into linear or quadratic ones?
- 18.2k
2
votes
1
answer
195
views
Is this an isomorphism possible?
I am working on the following homework problem:
Let $\phi$ be an isomorphism from $\mathbb{R}^*$ to $\mathbb{R}^*$ (nonzero reals under multiplication). Show that if $r>0$, then $\phi(r) > 0$.
...
- 1,862
5
votes
3
answers
843
views
Is $\mathbb{Q}$ isomorphic to $\mathbb{Z^2}$?
Most of us are aware of the fact that $\mathbb{C}$ is isomorphic to $\mathbb{R^2}$, as we can define $\mathbb{C}$ as follows :
$$\mathbb{C} := \left\{z : z=x+iy \ \ \ \text{where} \ \ \langle x,y \...
- 13.2k
0
votes
3
answers
121
views
Prove $\mathbb{R}$ does not contain a subring isomorphic to $\mathbb{C}$
I'm trying to prove that the quaternions ring $\mathbb{H}$ is not a $\mathbb{C}$-algebra, so I assume $\mathbb{H}$ actually is a complex algebra and that implies that there exists an injective ring ...
- 1,261
1
vote
3
answers
2k
views
Arithmetic Operations with Infinities in Real Analysis
Infinity is not a number , thus we cannot perform the usual arithmetic operations that we do with real numbers
This is the usual reason given when asked why we can't perform the usual arithmetic ...
- 13.2k
0
votes
2
answers
963
views
$R/\Bbb Z$ isomorphic to $R/(2\pi \Bbb Z)$
I was told that $\mathbb{R}$$/$$\mathbb{Z}$ is isomorphic to $\mathbb{R}/2\pi \mathbb{Z}$ when these groups are taken under addition. Is this always true? I do not specifically see why this has to be ...
- 263
9
votes
2
answers
1k
views
Proving (without using complex numbers) that a real polynomial has a quadratic factor
The Fundamental Theorem of Algebra tells us that any polynomial with real coefficients can be written as a product of linear factors over $\mathbb{C}$. If we don't want to use $\mathbb{C}$, the best ...
- 23.7k
1
vote
1
answer
31
views
Dividing with imaginary numbers, simplifying
Alright, so I have $8-\frac{6i}{3i}$.
I multiplied by the conjugate of $3i$, and got $-18-\frac{24i}{9}$.
This is the part that confuses me, because I don't know how to divide this. Can I divide ...
- 11
1
vote
2
answers
100
views
Is the element $(0,0,0)\in\mathbb{R}^3$ a divisor of zero?
(I'm assuming that $\mathbb{R}^3=\mathbb{R}\times\mathbb{R}\times\mathbb{R}$)
In my assignment, I'm told to prove that exactly one of the following can be true for an element $(x,y,z)\in\mathbb{R}^3$
...
- 2,429
6
votes
2
answers
392
views
Path From Positive Dedekind Cuts to Reals?
Don't spend a lot of time on this. I'm certain I could bang it out myself; but maybe there's an answer out there that someone already knows.
Say we use Dedekind cuts to construct the reals. Addition ...
- 90.9k
2
votes
0
answers
133
views
Extended real numbers as algebraic structure
I need to work with real numbers, but extended to have an additional element. This element, I denote by $\odot$ and my set is: $\mathbb{R}_{\odot}=\mathbb{R}\cup\{\odot\}$. This element should behave ...
- 165
4
votes
1
answer
3k
views
Existence of identity element for binary operation on the real numbers.
We define a new operation
$$x*y= x+y+xy,$$
on the set of real numbers with the usual addition and multiplicaton. Has this operation got an identity element?
It seems clear for me that there is one ...
- 203
0
votes
1
answer
42
views
find all subgroups of G where: $0 \ne r \in \Re$ $G = <r>$
I need to find all subgroups of G where:
$G \lt \Re$
$0 \ne r \in \Re$
$G = <r>$
$\Re$ is the group of real numbers and G is a subgroup.
Edit :
the operation is +
I tried thinking about ...
- 947
4
votes
1
answer
456
views
Is there any proper subring of $\mathbb{R}$ with field of fractions equal to $\mathbb{R}$? [duplicate]
Is there any proper subring of $\mathbb{R}$ with field of fractions equal to $\mathbb{R}$? Can we construct that proper subring? Is it necessarily an integral domain?
Updated: Is there an example to ...
- 420
5
votes
2
answers
1k
views
How to define the operation of division apart from the inverse of multiplication?
Sorry if this question is too far out there, but I'm looking for a rigorous definition of the division operation. As I have seen it before, $a/b$ is the solution to the equation $a=xb$. While I am ...
- 2,049
2
votes
1
answer
213
views
What would be interesting maps to use on that Eudoxus reals?
I'm trying to understand Eudoxus Reals. From wikipedia:
Let an almost homomorphism be a map $f:\mathbb{Z}\to\mathbb{Z}$ such that the set $\{f(n+m)-f(m)-f(n): n,m\in\mathbb{Z}\}$ is finite. We say ...
- 24.2k
3
votes
2
answers
923
views
Why is the observation that proof of the Fundamental Theorem of Algebra requires some topology not tautological?
I have heard it mentioned as an interesting fact that the Fundamental Theorem of Algebra cannot be proven without some results from topology. I think I first heard this in my middle school math ...
- 1,387
18
votes
3
answers
2k
views
Why is it so hard to prove a number is transcendental?
While reading on Wikipedia about transcendental numbers, i asked myself:
Why is it so hard and difficult to prove that $e +\pi, \pi - e, \pi e,
\frac{\pi}{e}$ etc. are transcendental numbers?
...
- 1,154