All Questions
Tagged with real-numbers abstract-algebra
98
questions
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61
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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
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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
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1
answer
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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
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1
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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 ...
- 2,979
2
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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
95
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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
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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
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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
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1
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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
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443
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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
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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
12
votes
3
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460
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Is there a minimal generating set of reals which additively generate all the reals?
Is there a set $S$ of real numbers such that the submagma generated by $S$ under addition is the entire set of real numbers, but such that no proper subset of $S$ generates the entire set of real ...
- 21.3k
8
votes
1
answer
262
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$A,B$ such that $A\cap B=\emptyset$ and $A\cup B=\mathbb{R}$ and $B=\{x+y : x,y\in A\}$?
If set $A,B$ satisfy $A\cap B=\emptyset,A\cup B=I$, and $B=\{x+y : x,y\in A\}$, can $I$ be real number set $\mathbb{R}$?
I think the answer is yes, but I can't construct it. If $A$ is odd number set, ...
- 3,247
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93
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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
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vote
2
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72
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Deriving the addition and product on $\mathbb C$
I am studying Stillwell's Elements of Algebra. In Chapter 3, Section 3.6, he writes
There is a unique extension of $+$, $-$, $\times$, $\div$ to $\mathbb C$ satisfying the field properties since if ...
- 4,057
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votes
1
answer
59
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$H_q(X) = 0$ if $X \subset \mathbb{R}^n$
Is it true that if $X \subset \mathbb{R}^n$ then $H_q(X) = 0$ if $q \geq n$. I have this statement (unproved) in my algebraic topology notes and I'd like to know whether this is true just for the sake ...
- 5,164
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1
answer
222
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What is $\operatorname{Hom}_{\mathbb{Z}}(\mathbb{R},\mathbb{R})$?
We already know that $\operatorname{Hom}_{\mathbb{Z}}(\mathbb{Q},\mathbb{Q}) = \mathbb{Q}$. Why? Because each homomorphism $f$ is uniquely determined by the value $f(1)$ and we can then calculate for ...
- 2,767
3
votes
1
answer
433
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Why is $\mathbb{R}/\mathbb{Z}$ not an $\mathbb{R}$-vector space?
This is an embarrassing question which might seem elementary and possibly silly, but its suddenly confusing me. Clearly I'm missing something very obvious.
Take the structure $\mathbb{R}/\mathbb{Z}$. ...
- 2,517
0
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1
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748
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Prove that real multiplication distributes over addition
The distributive property of real numbers states that $“$for all $a, b, c \in \mathbb{R}$, we've $a⋅(b + c) = a⋅b + a⋅c$ and $(b + c)⋅a = b⋅a + c⋅a”$. How to prove this field property of real numbers? ...
user783493
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2
answers
44
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Proving a set X is dense in [0,1] equivalence relation [duplicate]
Let the relation in $\mathbb{R}: x \equiv y \ \mbox{mod} \ \mathbb{Z}$, when $x-y \in \mathbb{Z}$. For each $n \in \mathbb{N}$, let $x_n \in [0,1)$ such that $x_n \equiv \sqrt{n} \ \mbox{mod} \ \...
4
votes
1
answer
109
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Elements of $\operatorname{Hom}_{\mathbb{Z}} (\mathbb{Q}, \mathbb{Q}/\mathbb{Z}) $ as Cauchy sequences
There is an isomorphism of abelian groups $\operatorname{Hom}_{\mathbb{Z}} (\mathbb{Q}, \mathbb{Q}/\mathbb{Z}) \cong \mathbb{R}$, a proof is based on three observations:
$\operatorname{Hom}_{\mathbb{...
- 6,375
1
vote
1
answer
78
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Axiom Q in Fischer's Intermediate Real Analysis
In Intermediate Real Analysis by Emanuel Fischer page 6, the author states an axiom that says
(Axiom Q) If $x$ and $y$ are real numbers, where $z+y\neq z$ holds for some real $z$, then there exists a ...
- 573
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1
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117
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Irreducible polynomial with degree 2 over $\mathbb{R}$
Let $f(x) \in \mathbb{R}[x]$ with $\deg f=2$. Show that $f(x)$ is irreducible over $\mathbb{R}$ if and only if $f(x)=(x-a)^2 +b^2$, where $a,b \in \mathbb{R}$ and $b \neq 0$.
I don't know how to do ...
2
votes
2
answers
144
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Is there a reasonable limit to how far you can generalise complex numbers? [duplicate]
Real numbers satisfy a(bc) = (ab)c as well as ab = ba. They are also comparable.
Generalising to complex numbers, everything stays the same, except the numbers lose their comparibility.
Generalising ...
- 195
-2
votes
2
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258
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For any numbers $a, b,$ and $c,$ $a + b = a + c$ if and only if $b = c$ [duplicate]
I was reading about the field of real numbers $\mathbb{R},$ and a basic question arose in my mind.
How one should prove that, for any numbers $a, b,$ and $c,$ $a + b = a + c$ if and only if $b = c?$
...
- 3,794
2
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1
answer
51
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Can we define a new multiplication on $\Bbb R$ ( addition and identity element remain the same) while its properties of being a field persist? [closed]
If the identity element of $\Bbb R$ were permitted to be redefined, then it could be redefined as any nonzero number while keeping $\Bbb R$ a field;
By induction, we can prove that only the product ...
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