All Questions
Tagged with real-numbers abstract-algebra
11
questions with no upvoted or accepted answers
3
votes
0
answers
107
views
Proof that for all nonzero real numbers $a$, $\frac{1}{a}$ is nonzero
I was wondering if someone could check my proof that "For all $a\in\mathbb{R}$, if $a\neq 0$ then $\frac{1}{a}\neq 0$". The definitions/assumptions I am basing the proof off of come from &...
- 31
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
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
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
1
vote
0
answers
90
views
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
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
0
votes
0
answers
61
views
Is subtraction on the reals isomorphic to division on the positive reals?
I know that the magma $(\mathbb{R};+)$ of addition on the real numbers is isomorphic to the magma $(\mathbb{R}^+;\times)$ of multiplication on the strictly positive real numbers. I wonder, is it the ...
- 21.3k
0
votes
0
answers
93
views
Is every formally real field isomorphic to a subfield of the reals?
A formally real field is a field $K$ such that $-1$ is not a sum of squares in $K$. Clearly subfields of $\mathbb{R}$ are formally real. I also know finite fields and algebraically closed fields are ...
- 2,185
0
votes
0
answers
53
views
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
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