For questions about $\mathbb{R}$, the field of real numbers. Often used in conjunction with the real-analysis tag.

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1answer
20 views

The terminology for particular subsets of the power set of R

The set $X = \{\{x\in\Bbb R\mid x<a\}\mid a\in\Bbb R\}$ , which is a subset of the power set of R. Is there a terminology for the set $X$? My intent is to search the related literature about the ...
0
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2answers
33 views

Is there such a thing as complex rational numbers and does it have the same properties as the usual complex numbers as extension of the real numbers?

I've been wondering if there is any use to defining a set that is isomorphic to $\mathbb{Q}^2$ (in the same way that $\mathbb{C}$ is isomorphic to $\mathbb{R}^2$). I immediately see a problem with ...
0
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0answers
25 views

On the rearrangement of an infinite series of real numbers. [duplicate]

A chapter of a text book ended with. If we rearrange infinitely terms of a series that converges only conditionally, we may get results that are far different from the original series ...
1
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5answers
47 views

How is “smaller than” defined on $\mathbb{R}$?

According to http://en.wikipedia.org/wiki/Binary_relation Binary relations are used in many branches of mathematics to model concepts like "is greater than", "is equal to", and "divides" in ...
1
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1answer
53 views

An awkward Functional Equation

Find all $f:\mathbb{R} \rightarrow \mathbb{R}$ satisfying $$f(f(y)+x^2+1)+2x=y+(f(x+1))^2$$ for all $x,y \in \mathbb{R}.$ I proved that $f$ is bijective, but I am stuck there. Any help please?
-1
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0answers
49 views

An inequality with a real number represented by a Dedekind cut [closed]

Theorem 3: Let $r$ be a real number $(A,B)$ (meaning a Dedekind cut). Then $r \ge a$ for all $a$ in $A$, and $r < b$ for all $b$ in $B$. If $r$ is irrational, then $r > a$ for all $a$ in $A$. ...
-1
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1answer
40 views

How do you prove the commutativity of multiplication of all real numbers?

How do you prove the commutativity of multiplication of all real numbers? I have researched a lot before, and I found out that defining real numbers is an important step towards answering this ...
1
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1answer
31 views

Closeness of infinite union of closed sets

Is the set $\bigcup_{x \geq 0} \left\{\frac{1}{x+1} \right\} $ closed? For all $x \geq 0$, the set $\left\{\frac{1}{x+1}\right\}$ is a single point, therefore it is closed. But I am not sure about ...
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0answers
44 views

Partitioning real numbers with sum $1$ to sets

If the sum of a finite number of positive real numbers is $1$ and each of them is less than $x$, then those real numbers can be partitioned into $50$ sets (some of which may be empty) such that the ...
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4answers
34 views

Prove that for $\forall x \in (0,1]$, $\exists n \in \mathbb{N}$ such that $\frac{1}{n+1} \leq x \leq \frac{1}{n}$

Prove that $\forall x \in (0,1]$, $\exists n \in \mathbb{N}$ such that $\frac{1}{n+1} \leq x \leq \frac{1}{n}$ This is a completely obvious statement (it was taken for granted in another proof) ...
1
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1answer
31 views

Which of the following are subspaces

I was working on subspace and found a problem that check following are subspaces of $\Bbb{R}^3(\Bbb{R})$ or not. 1) W = {($a^5$,0,0) : a∈$\Bbb{R}$} 2) U = {($a^2$,0,0) : a∈$\Bbb{R}$} My Attempt : I ...
1
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1answer
36 views

Dedekind cuts: proof every interval of $\mathbb{R}$ contains both rational and irrational numbers

I'm trying to follow the proof that every interval $(a,b)$ of $\mathbb{R}$ contains both rational and irrational numbers given on page 19 of Real Mathematical Analysis by Pugh. Word for word, he ...
1
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0answers
24 views

Is the computable numbers equal to the set of all the limits of finite length algebraic expressions?

Let's call $C$ the set of computable real numbers and $L$ the set of all the (existing) limits of finite length algebraic expressions. By $L$ I mean the set of all converging limits $\lim_{x_1 \to ...
14
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3answers
796 views

Transcendental number

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? ...
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2answers
26 views

Extracting real and imaginary numbers from a complex number

How can I get the real number and the imaginary number from: $$\frac{3+i}{5-12i}$$
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2answers
65 views

How do I show convergence of this sequence?

Given a sequence $\{a_n\}$ of positive real numbers such that $\sum\limits_{n=1}^\infty a_n<\infty$. Suppose that there exists $k\in \Bbb N$ such that $a_{n+k}\leq a_n, \,\,\,\forall n.$ Question: ...
2
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0answers
26 views

Subsets of real numbers that aren't complete

If all bounded subsets of $\mathbb{R}$ have an infimum and a supremum, then why does a bounded subset consisting of only rational numbers (such as $\{x\in\mathbb{Q}:x^2<2\}$) break the property? ...
4
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3answers
753 views

What do we call the front part of a decimal number? [duplicate]

I have the following number. 23.45 There are two parts of this number. 23 and 45. What is the mathematical name of the 23 part?
3
votes
1answer
74 views

Is the set of real numbers really uncountably infinite?

The proof that the set of real numbers is uncountably infinite is often concluded with a contradiction. In the following argument I use a similar proof by contradiction to show that the set of ...
0
votes
1answer
33 views

Wilson's theorem states that if n is a prime number, it will divide (n-1)! + 1, using this find the smallest divisor of 12!+6! +12!×6! + 1?

Wilson's theorem states that if $n$ is a prime number, it will divide $(n-1)! + 1$, using this find the smallest divisor of $12!+6! +12!×6! + 1$? I checked yahoo answers and there someone gave ...
1
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1answer
33 views

How to show $(1+a)^n<1+2^n a$ for all $n\in\mathbb N$ and $a\in (0, 1)$?

There is an induction problem which is baffling me. I'm supposed to use induction to show the inequality $$(1+a)^n< 1+2^n a,$$ for all $n\in\mathbb N$ and $a\in (0, 1)$. I guess there must be some ...
2
votes
2answers
63 views

Function whose limit does not exist at all points

There are functions which are discontinuous everywhere and there are functions which are not differentiable anywhere, but are there functions with domain $\mathbb{R}$ (or "most" of it) whose limit ...
0
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1answer
30 views

Real logarithm of a real matrix?

What is the real logarithm of \begin{equation} \begin{pmatrix} -1 & 1 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 1 \\ 0 & 0 & 0 & -1 \end{pmatrix}? ...
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0answers
8 views

On the matter of order completion (Dedekind completion)

So I was introduced recently to this idea of "order completion" of partially ordered set. A more special case is when this poset is in fact a completely ordered set, such as rationals. Completing the ...
2
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3answers
54 views

Find the inverse of $f(x,y) = (x+3y,3x+y)$

Given the function $f : \mathbb{R}^2 \to \mathbb{R}^2$ as $f(x,y) = (x+3y,3x+y)$. Find $f^{-1}$ .( Assume $f$ is a bijection) I know how to find $f^{-1} (x) = (3x+2)$ or anything with one ...
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2answers
82 views

Does $y = -1^x$ where $x∈ℝ$, change exponentially?

Is $y = -1^x$ an exponential curve, or just a sinusoidal one, can it be said to change exponentially as with positive exponents? I'm sure W/A showed this as being sinusoidal with an integer period. ...
0
votes
1answer
27 views

Two brands of chocolate are available in packs of 24 and 15 respectively. If I need to buy

Two brands of chocolate are available in packs of 24 and 15 respectively. If I need to buy an equal number of chocolates of both kinds, what is the least number of boxes of each kind I would need to ...
1
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2answers
27 views

Cardinality of $E=\left\{\left(x,y\right):x,y>0\text{ and }x+y,xy\in\mathbb{Q}\right\}\subseteq\mathbb{R}^2$

I want to find the cardinality of $$E=\left\{\left(x,y\right):x,y>0\text{ and }x+y,xy\in\mathbb{Q}\right\}\subseteq\mathbb{R}^2.$$ This problem came from a recent real analysis comprehensive exam ...
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2answers
29 views

How do I interpret this operation?

This question has to do with operations and exploring their characteristics. I have just learned how to extract info from an operations table (what is the identity, inverse, etc.), but this question ...
1
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1answer
23 views

Is this inequality of real numbers true?

Let $\alpha\in (0,1/2)$ be a parameter. Is it true thet for every $x>y>0$ real numbers we have $$y^{-\alpha} - x^{-\alpha} \leq C y^{-\alpha -\frac{1}{2}} (x-y)^{1/2}$$ for some constant ...
2
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0answers
22 views

Obtain an inequality of real numbers

Let $x,y> 0$ be real numbers such that $x>y$. Let $\alpha \in (0,1/2)$ be a parameter then I obtained the following inequality: $$y^{-\alpha} - x^{-\alpha} \leq C ...
20
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7answers
2k views

Example of uncomputable but definable number

Every computable number is definable. However, the converse is not true. What is an example of a real number that is definable but that is NOT computable? I guess if it is there, we can "define" ...
15
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8answers
1k views

Can we have a one-one function from [0,1] to the set of irrational numbers?

Since both of them are uncountable sets, we should be able to construct such a map. Am I correct? If so, then what is the map?
0
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1answer
7 views

Conversion to predicates including number

How can i convert these sentences into predicates, I'm a little bit confused since it includes numbers in the sentences? How can i represent the count operation? $f_1$ : There are $500$ employees in ...
5
votes
1answer
74 views

How can the axioms (and primitives) of Tarski's axiomatization of $\Bbb R$ be independent?

While reading through this Wikipedia page about Tarski's axiomatization of the reals, a particular bit of text jumped out at me: Tarski proved these 8 axioms and 4 primitive notions independent. ...
0
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2answers
25 views

Integral of strictly real function has imaginary component

Intuitively and informally speaking, $\int_{a}^{b}f(x)dx$ is summing all of the values $f(x)$ yields for $x\in [a,b]$. So it would make sense that if $f(x)$ is strictly real over $[a,b]$, then ...
1
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3answers
57 views

Uncountable infinity

The "number" of real numbers in $[0,1]$ is uncountably infinite, just as the "number" of real numbers in $[0,10]$ is uncountably infinite. However, my intuition would tell me the second interval has ...
2
votes
1answer
22 views

Describe the union and intersection of $n$ open neighborhoods of the same point

Given real numbers $x, \delta_1,..., \delta_n, $ I am asked to describe $$ \bigcap_{i=1}^n N(x, \delta_i ) \; \; \; and \; \; \; \bigcup_{i=1}^n N(x, \delta_i ) $$ where $N(x, \delta_i ) = \{ y: ...
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0answers
19 views

Showing an element is not in a cover of $\mathbb Q$

Given the Calkin-Wilf enumeration of $\mathbb Q$, $\{q_n\}_{n=1}^\infty$, one can define a cover of the positive rationals: $\left\{ \left(q_n-\frac{1}{2^n}, q_n+\frac{1}{2^n}\right) ...
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2answers
29 views

$\Bbb R^n \times (0,\infty)$ what does this mean?

Just began to read about PDEs. There are a whole list of notations which I don't understand and the book isn't expecting a reader who is as inexperienced as me. Also don't understand what this ...
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1answer
21 views

Linear Algebra Orthogonality Help

I am struggling with this one exercise from self-learning. I simply do not understand what it is asking. If someone could walk me through this problem I would be very grateful.
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1answer
55 views

Find all the automorphisms of $(\mathbb{R},<)$, the real numbers with the usual ordering

Find all the automorphisms of $(\mathbb{R},<)$, the real numbers with the usual ordering Obviously the identity mapping, $\iota : \mathbb{R} \to \mathbb{R}, \iota(r) = r$ and the mapping of ...
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3answers
56 views

Proving uncountability of $\mathbb R$ only using the complete ordered field axioms

If we define the real numbers abstractly as a complete ordered field (like described in the Wikipedia page), how can we prove that they are uncountable? In other words, using just the axioms of a ...
6
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1answer
81 views

Could the real numbers have been invented without the natural numbers

The real numbers are constructed from the rational numbers which are constructed from the integers which, in turn, are constructed from the natural numbers. But if we had no notion of the natural ...
3
votes
1answer
89 views

When $\cos x$ is transcendental?

About the transcendence of trigonometric functions I know that: 1) if $x$ is an algebraic number $\ne 0$ than $\cos x$ is transcendental. 2) if $p=\dfrac{m}{2^n}$ with $m,n \in \mathbb{Z}$ than ...
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0answers
34 views

Continuity of exponentiation (Terence Tao's book-Lemma 6.7.1)

How to use the sequence $(x^{1/k})_{k=1}$ convergent at 1 (Cauchy sequence) for testing whether $eventually\ $ $\epsilon\cdot x^{-M}-close$ to 1 ? How to get the next result: ...
2
votes
1answer
51 views

Is there any condition while applying law of exponents?

${[(-3)^2]}^\frac{1}{2}$ = ${(-3)^2}^\frac{1}{2}$ = $-3^1$ = $-3$ But counted other way it is $9^\frac{1}{2} = \surd{9} = 3$ where I went wrong?
0
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0answers
31 views

Does rule of three make sense for other than real numbers?

I'm currently working on a software tool which can make calculations based on the rule of three. I can make it more simple and just support real numbers, or I can make it "universal" to support ...
0
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2answers
64 views

Calculate $\sqrt{7+5\sqrt{2}}-\sqrt{3-2\sqrt{2}}$

I need help with calculating the following,$$\sqrt{7+5\sqrt{2}}-\sqrt{3-2\sqrt{2}}$$ i have tried to solve it as $$\sqrt{\left(\sqrt{7+5\sqrt{2}}-\sqrt{3-2\sqrt{2}}\right)^2}$$ but i've come nowhere.
4
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3answers
106 views

Proving $\left(a+\frac{2}{a}\right)^2+\left(b+\frac{2}{b}\right)^2\ge \frac{81}{2}$ for all positive real $a,b$ such that $a+b=1$

I approached this problem in two different ways, but only one was successful. I'll post the latter as an answer, while here follows the first approach: I expanded the squares: ...