All Questions
Tagged with real-numbers abstract-algebra
98
questions
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55
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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
3
votes
0
answers
107
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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
3
votes
1
answer
85
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Historically, when have the the real numbers been constructed via the "positive" (non-negative) reals first, and then usual real numbers second?
There has been something that has been bugging me for the longest time, at least since grad school.
In the teaching of mathematics, during the construction of the "usual" real numbers, why ...
- 1,622
1
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1
answer
44
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What does Artin mean by "real numbers are the *only* ones needed for the usual for the usual algebraic operations?"
In page 81 of the 2nd edition Michael Artin's Algebra, he introduces fields and presents $\mathbb{R}$ as a familiar example, but goes on to say that "the fact that they are the only ones needed ...
16
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1
answer
168
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What is the "higher cohomology" version of the Eudoxus reals?
The "Eudoxus reals" are one way to construct $\mathbb{R}$ directly from the integers. A full account is given by Arthan; here is the short version: A function $f: \mathbb{Z} \to \mathbb{Z}$ ...
- 1,247
-4
votes
2
answers
124
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How can the reals be the set of all points on a number line when there exist non-constructible reals? [closed]
We are given the intuition that the reals form all the numbers on the numberline. However, this intuition wasn't working for me as the existence of non-constructible reals seems to me to imply that ...
- 2,947
2
votes
1
answer
73
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$(H,*)$ group with some properties $\Rightarrow H$ not an interval.
Let $(H, \ast)$ be a group, where $H \subseteq (0, \infty)$, which has these properties:
$x \in H \Rightarrow \frac{1}{x} \in H$
$2023 \in H$, and
$x \ast y = \frac{1}{x} \ast \frac{1}{y}$ for any $x,...
- 167
3
votes
1
answer
94
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Why there would be no additive inverse in real numbers if we changed the definition of a cut?
We define a cut to be a proper subset of rationals such that:
1- It is not the empty set $\emptyset$,
2- It is closed to the left, meaning that if $p \in \alpha, q<p \Rightarrow q \in \alpha.$
So, ...
- 138
3
votes
1
answer
76
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Sets of real numbers which are anti-closed under addition
Let $(M,*)$ be a magma, that is, a set with a binary operation. I define a subset $S$ of $M$ to be anti-closed under $*$ iff for all $x,y$ in $S$, $x*y \notin S$. For example, the set of negative real ...
- 21.3k
0
votes
2
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118
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Positive definite (inner product)
In my linear algebra course, we defined the positive definite of the inner product where $\langle z,z\rangle \ge 0$. My professor stated that because of this $\langle z,z\rangle \notin\mathbb{C}$?
...
- 782
1
vote
1
answer
144
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A vector space contains $\mathbb{R}$ but have scalar product defined differently than vector product
Suppose we have a vector space with the underlying field being $\mathbb{R}$. Just out of curiosity, what are some examples of vector space $(V,+,\cdot)$, where $\mathbb{R} \subsetneq V$, but these ...
- 1,690
1
vote
1
answer
78
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Is every ring homomorphism between real algebras also real-linear?
$\def\bbR{\mathbb{R}}
\def\bbQ{\mathbb{Q}}$The comment from Vladimir Sotirov in March 2022 in this answer could be interpreted as suggesting the possibility that every ring homomorphism between $\bbR$-...
0
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3
answers
441
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What is the smallest infinite field?
The real numbers and the rational numbers are both fields, but what is the smallest field. Is the set of rational numbers smaller than the set of reals, and if so is there a 'smaller' infinite set ...
0
votes
1
answer
80
views
Characterizations of the reals
I know that one characterization of the reals is that it is the only Dedekind-complete ordered field. Are there any other characterizations of the reals as a field?
- 4,057
1
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0
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67
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What's the proof that the only Dedekind-complete field is the reals? [duplicate]
I know that the field of the rational numbers is ordered but not Dedekind-complete. What's the proof that the only Dedekind-complete field is the reals?
- 4,057