All Questions
Tagged with real-numbers abstract-algebra
98
questions
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195
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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
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1
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131
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Which algebraic intuition can be used in fields
I wonder what basic laws of arithmetic of reals e.g. $x^n x^m = x^{m+n}$ holds for fields. Every time I take some book on abstract algebra it proves very abstract and unpractical properties. So I ...
- 2,420
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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 ...
- 1,732
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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 $\bbR$-...
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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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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}...
- 4,322
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269
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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
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332
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What properties do you lose when you extend your number set? [duplicate]
So in $\mathbb{R}$ and $\mathbb{C}$ you have both associative and commutative property, but as you extend to $\mathbb{H}$ you lose the commutative property, and $\mathbb{O}$ loses the associativity. ...
- 5,339
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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
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90
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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
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1
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31
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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
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2
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100
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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
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3
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121
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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
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119
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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
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117
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Nontrivial subring of $\mathbb{R}$ not containing $1$
Are there examples of nontrivial subrings of $\mathbb{R}$ that do not contain $1$? If not, how can we prove they don't exist? The definition of "ring" here is really "rng"; rings ...
- 1,751