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

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0
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0answers
17 views

When $\cos x$ is transcendental?

About the transcendence of trigonometric functions I know that: 1) if $x$ is an algebraic number than $\cos x$ is transcendental. 2) if $p=\dfrac{m}{2^n}$ with $m,n \in \mathbb{Z}$ than $\cos ...
0
votes
0answers
27 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
44 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
votes
0answers
29 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
votes
2answers
62 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
votes
3answers
100 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: ...
0
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2answers
29 views

Help prove that $\alpha=\{y\in\mathbb{Q}|(y\le0)\vee(y^{2}\le2)$} is a non-rational Dedekind cut [on hold]

I need to prove that $\alpha=\{y\in\mathbb{Q}|(y\le0)\vee(y^{2}\le2)$} is a non-rational Dedekind cut. My professor defines a Dedekind cut as: $\{\alpha,\beta\}$ is a partition of $\mathbb{Q}$, ...
-2
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2answers
40 views

Prove that ($\mathbb{R}$, $\le$) is a partial order

I was told that the relation $\le$ is a total order on R, it is dense, and it has a least upper bound property. I actually have don't understand those 3 properties... :/
5
votes
4answers
114 views

Need help with proof of existence of $\sqrt{2}$

I am working my way through the proofs on this page. I am stuck on "4. The real number $\sqrt{2}$ exists." It begins: We will get $\sqrt{2}$ as the least upper bound of the set $A = \{q\in Q | q^2 ...
0
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0answers
26 views

Order Axioms: '<' or '$\leq$'?

In the axioms given for the real numbers, I see that the order axioms are sometimes given for the '<' relation and sometimes for '$\leq$'. Which is more commonly used these days?
5
votes
1answer
93 views
+50

Prove (or disprove): $a\times b=c\times d$, the solutions for $x$ in the equation $\frac {a^x+b^x}{c^x+d^x} = \frac{a+b}{c+d}$ is only $\pm 1$.

Prove (or disprove): If $a,b,c,d$ are positive real numbers with $a\times b=c\times d$, then the only solutions for $x$ in the equation $$\frac {a^x+b^x}{c^x+d^x} = \frac{a+b}{c+d}$$ are $x = \pm 1$. ...
0
votes
5answers
45 views

Cubic Equation. (Factorisation)

I'm given this question, factorise $4x^3-7x-3$. Is this answer acceptable? $(x+\frac{1}{2})(x-\frac{3}{2})(x+1)$.
0
votes
0answers
21 views

Proving initial segment in Q

Let $A = \{\alpha_\lambda \mid \lambda \in \Lambda\}$ be a non-empty family of initial segments in $\Bbb Q$. Prove that: If $\bigcap A \ne \varnothing$, then $\bigcap A$ is an initial segment in ...
0
votes
1answer
31 views

Proving that a rational cut is a Dedekind cut

I'm new here, so I don't know how to do the fancy symbols. Sorry... This is for my intro. to adv. math class, and I've been struggling this entire semester. I kinda understand the concept being ...
0
votes
3answers
65 views

Which of this two numbers is the bigger?

How can I approach this problem of comparison between these two numbers. Any hints please. $A = 1000^{1000}$ or $B = 1\times 3\times 5\times \dots \times 1997$
1
vote
1answer
52 views

About subspaces of $\mathbb{R}$ as vector space over $\mathbb{Q}$.

In many texts is noted the analogy between the transcendence degree of a field extension and the dimension of a vector space, so I'm tempting to use such analogy to better understand the structure of ...
3
votes
3answers
549 views

Question about a solution of a system of three non linear equations in three unknowns

Let $a$, $b$ and $c$ be positive real numbers such that $$ a + \frac{1}{b} = 3$$ $$b + \frac{1}{c} = 4$$ $$ c + \frac{1}{a} = \frac{9}{11} $$ then $$ a \times b \times c =?$$ I tried doing this ...
1
vote
0answers
22 views

Density of unbounded set modulo t

Let $A$ be an unbounded set in $\mathbb{R}$. Then consider the set $M(t)=\{a\mod t|a \in A \}$ in an interval $[0,t]$,for given $t>0$. Then some of $t$ will make $M(t)$dense in $[0,t]$. (This is my ...
-5
votes
1answer
37 views

How many $9$'s are there in $0.999\dots$ in the first $n$ places after the decimal point? [closed]

I am counting the number of $9$'s that will come in the first $n$ digits after the decimal point in $0.999\dots$ How many $9$'s will be there? For example, if I take $0.333\dots$ we will have $n$ ...
1
vote
2answers
68 views

At what points of r in real number is f continuous?

Please have a look at the picture above. Do we really have such points? If that's the case, how do I prove it? Many thanks.
0
votes
2answers
43 views

Can a Subset be considered an Element for Field Axioms

I have the subset $L\subset \Bbb Q$ that is Dedekind cut. I want to prove that $L+(-L)=0$ I want to do this using the field axiom of Additive Inverse, but Additive Inverse specifically deals with ...
-1
votes
0answers
37 views

Convergent series and real numbers [closed]

Prove that every decimal representing a positive real number can be expressed as a convergent series. Any ideas?
0
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0answers
55 views

Inverse and named fixed values, with ↑↑?

The inverse of $+$ is $-$, of $\times$ is $/$ and of $\text{^}$ is Log. Continuing upwards hyperoperationally, what is the inverse of $↑↑$? Whats more somtimes values that are fixed are given ...
0
votes
2answers
15 views

behavior of function between two bounds

Let $f, U, L : [0,1] \rightarrow \mathbb{R}$ be three functions with the property that (1) U and L are continuous functions (2) $\forall x \in [0,1]$, $L(x) \leq f(x) \leq U(x)$ (3) ...
0
votes
0answers
14 views

Multiplicative inverse in terms of dedekind cuts.

Let $(A|B)>0$ be a Dedekind cut. Suppose its multiplicative inverse is $(C|D)$. I am trying to show that $C=\mathbb Q^{\le 0}\cup\{c>0:\frac{1}{c}<r\text{ for some } r\in \mathbb Q\setminus ...
0
votes
1answer
26 views

Decimals and equivalence relations

I am told that decimals set up an equivalence relation on the Reals and that decimal numbers and the Reals are not the same thing. I believe this also clarifys the famous $.\bar{9}=1$. That $1$ and ...
0
votes
1answer
19 views

If a function is derivable in a point then there exists an open interval around the point in which the function is continuous

Let $f:\ \mathbb{R} \rightarrow \mathbb{R}$ be a function that is differentiable in $a\in \mathbb{R}$. Prove or give a counterexample: There exists a $\delta \in \mathbb{R}_0^+$ as follows: $$ ...
0
votes
2answers
58 views

Sequences and real numbers

Based on the answers so far I restate the question: on p. 63 of his volume on Analysis (http://math.univ-lyon1.fr/~okra/2011-MathIV/Zorich1.pdf), Zorich says: “We now answer the question whether some ...
0
votes
0answers
19 views

Explanation of successor set

What exactly is a successor set? Could someone provide a simple explanation as to why the real number is a successor set?
1
vote
1answer
41 views

Proof for reverse triangle inequality $|x - y| \ge |x| - |y|$. [duplicate]

Only step by step hint leading to the final proof. How should I take it from here?
1
vote
2answers
24 views

Proof for triangle inequality for case $x+y<0$

Here's the attempt in case $x+y<0$: In case $$x+y<0,$$ it must be true that $$(x<0\wedge y<0)$$ $$x\in \mathbb{R}\text{ is negative IFF }-x\in \mathbb{R}^+$$ $$x + y < 0 < -(x + y) ...
4
votes
6answers
110 views

Proof that $a+\frac{1}{a}\in\mathbb{Z}$ iff $a=\pm1$

I showed someone how to prove by induction that if $a+\frac{1}{a}\in\mathbb{Z}$ then also $a^n+\frac{1}{a^n}\in\mathbb{Z}$. He noted that there was no need for induction since obviously ...
1
vote
1answer
57 views

Proving $(\frac{1}{y})(\frac{1}{z}) = (\frac{1}{yz})$

Of course for $y,z \neq 0$. I want to prove this simply by using the Algebraic properties of the real numbers. The way I approached this is as follows: 1) By definition of the reciprocal $y \cdot ...
2
votes
1answer
45 views

Negative Zero in the set of real numbers

So according to the laws of algebra for a given $x$, one denotes by $-x$ the number such that $x+y = 0$ and is called negative of x. The subtraction operation is given by $z - x = z + (-x)$. It is ...
0
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0answers
29 views

My try at the completeness axiom proof

By the completeness axiom: (1) Every non-empty subset S of $\mathbb{R}$ that has an upper bound has a lowest upper bound. (2) Every non-empty subset S of $\mathbb{R}$ that has a lower bound has a ...
1
vote
1answer
23 views

Relating proof of triangle inequality to properties of order relation

I'm using the book "Mathematical Analysis" by Brend S.W Schroder. On page $7$, he tried to relate the proof for $|x+y| \leq |x| + |y|$ to the properties of order relation, specifically, "If $x \leq ...
0
votes
1answer
30 views

Switching limits counterexample

Let $(f_n)_n$ be a row of functions that converges non-uniformly to a function $f$. $$ f_n: A \subseteq \mathbb{R} \rightarrow \mathbb{R} $$ Let $a$ be an accumulation point in $\mathbb{R} \cup ...
0
votes
1answer
38 views

Prove that if $0<x\le y$ then $1/y \le 1/x$ using ordered field axioms

$$\text{Suppose that }x,y,z \in \mathbb{R}^+$$ $$\text{ Then, either the case $x = y$ or $x \leq y$}$$ $$\text{If $x = y$, then }x^{-1} = y^{-1}$$ $$\text{If $x < y$ then via proof by contradiction ...
1
vote
1answer
45 views

Prove that if $x\leq y$ then $x+z\leq y+z$

Does the below proof looks correct for the above question? $$\text{Either x = y or (y - x) $\in $ }\mathbb{R}^+$$ $$\text{Case 1: x=y $\forall $z$\in $}\mathbb{R}^+$$ $$\text{x+z=y+z}$$ $$\text{Case ...
0
votes
1answer
58 views

Prove the number x is positive IFF x>0

I'm trying to work on an exercise from a book which asked to prove the above question. It seems fairly trivial although I cannot be sure if my attempt suffices. To prove that the number x is ...
0
votes
0answers
23 views

What's the conjectured optimal running time for an exponential function algorithm restricted to [0, 1]?

If such an algorithm were used, for each positive integer ''n'', what's the upper bound on the computation time for the ''n''th digit after the decimal place. The Schönhage–Strassen algorithm runs on ...
-1
votes
1answer
34 views

$\sup(A\cup B)=\max\{\sup A,\sup B\}$ proof [closed]

Suppose $M$ is an upper bound for $A\cup B$ then, that implies $M\geqslant \sup(A)$, or, $M\geqslant\sup(B)$ But how does this implies that $\sup(A\cup B)\geqslant \max\{\sup A,\sup B\}$?
1
vote
1answer
28 views

Closed Subsets of the Real Line that are Uncountable

If a subset of the real line is uncountable and closed, does it have to contain a closed interval? Is there any theorem related to this?
2
votes
1answer
45 views

Dedekind cuts in Rudin's PMA

I'm working on Appendix to chapter I of Rudin's Principles of mathematical analysis and I have the following problem: Given a positive cut $\alpha$ and a rational $x>1,$ how can I prove that there ...
4
votes
2answers
52 views

How can “$y = \sqrt{-x}$ ” be sketched on the x-y plane?

I was reading James Stewart's book Calculus 5th edition (international student edition) and I came across an example that seemed wrong to me. In chapter 1 section 3, he talks about transformations of ...
0
votes
3answers
26 views

Density of Rational Numbers in Lowest Terms in R

I have a question about the density of the rational numbers when they're strictly in lowest terms in $\mathbb{R}$. I'm looking at all rational numbers in lowest terms, $\frac{p}{q} \in [0,1]$, and ...
0
votes
1answer
18 views

Is the set of elements of sequence closed?

Let $(x_n)_n$ be a convergent sequence in $\mathbb{R}$. Let $A$ be the following set: (suppose $A$ is infinite) $$ A= \{x_n \mid n \in \mathbb{n} \} $$ Prove or give a counterexample: ...
1
vote
1answer
30 views

Metric and Absolute value function on $\mathbb R$ [closed]

I'm contemplating the notion of the absolute value function on $\mathbb R$ as well as of the usual metric on $\mathbb R$. It seems to me that each one of those can be seen in light of the other. ...
1
vote
2answers
56 views

How is $\mathbb{R}_{+}^{\ast }$ defined?

I don't know what set $\mathbb{R}_{+}^{\ast }$ usually denote? How is it defined? Does it contain zero ? Does it contain infinity?
2
votes
1answer
72 views

Does $\sin(n)$ hit all real numbers between $-1$ and $1$ if $n$ is an integer?

More concisely, is this statement true: $$\forall\; y \in \left(\left[-1, 1\right] \subset \mathbb R\right) \;\exists\; n \in \mathbb N \;\vert \sin n = y$$ At first I thought that since $\sin$ is ...