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

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1answer
46 views

Which algebraic intuition can be used in fields

I wonder what basic laws of arithmetic of reals e.g. $x^n y^m = (xy)^{m+n}$ holds for fields. Every time I take some book on abstract algebra it proves very abstract and unpractical properties. So I ...
1
vote
1answer
47 views

Is there something wrong with this proof of the cancellation rule?

I was helping a friend of mine with math homework. We were dealing with real numbers, and had to prove the cancellation rule; that is $$a,b,c, \in \mathbb{R} \land a+c=b+c \implies a=c$$ He had a ...
0
votes
3answers
99 views

Why is the Absolute Value of $3-π$ equal to $π - 3$

Why if the absolute value is the distance from $0$ on the number line, is $\lvert 3-\pi \rvert = \pi - 3$ and $\left\lvert\sqrt{2}-1\right\rvert= \sqrt{2}-1$
0
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0answers
30 views

real vs complex numbers

Can someone write REAL numbers in rectangular form as well? And if so, is it useful? For example: On the complex plane, (x + yi) is x units on the Real x axis and y units on the Imaginary y axis. If ...
0
votes
1answer
65 views

Definable real numbers

Reading this Wikipedia page I found this definition: A real number $a$ is first-order definable in the language of set theory, without parameters, if there is a formula $\phi$ in the language ...
2
votes
1answer
31 views

Monotone Convergence Theorem (for real sequences) equivalent to the Least Upper Bound Property?

Some days ago I have asked this question to which André Nicolas gave a link to this paper which contained a proof of the Least Upper Bound Axiom from Monotone Convergence Theorem via Archimedian ...
2
votes
1answer
36 views

Sequence in an uncountable set of real numbers

Let $A$ be an uncountable subset of the real numbers, I think the following is true: There is an injective sequence $a:N\to A$ such that $\sum_{n=1}^\infty a_n$ diverges. This might also be true ...
0
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0answers
23 views

Standard Euclidean Metric on $\mathbb{R}^{k^2}$.

Does the standard euclidean metric on $\mathbb{R}^{k^2}$ refer to $$ \left(\sqrt{\sum_{i=1}^{k}x_i^2}\right)^2 = \sum_{i=1}^{k}x_i^2 $$ or does $\mathbb{R}^{k^2}$ have some other meaning, like ...
1
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1answer
33 views

Courant. Real numbers determined by nested sequences of rational intervals.

In his book Introduction to Calculus and Analysis vol.1, page 95 Courant writes: Every nested sequence of intervals with real end points contains a real number. To prove this, consider closed ...
0
votes
3answers
47 views

Prove the following.

Let $a,b \in R$. Prove that if $a \gt 2$ and $b=1+ \sqrt{a-1}$ then $2 \lt b \lt a$. Explanation: I know I start by breaking the conclusion into two parts when I'm thinking about it. So, when I'm ...
-3
votes
2answers
34 views

Prove and disprove the following inequality.

Prove: $ 0 \le a \lt b$ implies $ 0 \le a^2 \lt b^2 $ and $0 \le \sqrt{a^3} \lt \sqrt{b^3}$. Now show that the statement is false if the hypothesis $a \ge 0$ or $a \lt 0$ is removed. EDIT: Someone ...
1
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1answer
30 views

Define $x^3$ = $x \times x^2$. Prove that if $x_1, x_2$, . . . represents $x$, then $x_1^3$, $x_2^3$, . . . represents $x^3$ [closed]

I'm a little bit lost on where to start this problem. My initial thought is to work backwards. Say $x^3$ is a Cauchy sequence. Then for some $j, k \geq m$ contingent on $n$, we have |${x^3}_j - ...
0
votes
1answer
31 views

Prove the associative law for the addition of real numbers

The problem asks us to prove the commutative and associative laws for the addition of real numbers. The commutative proof seems straightforward. I am wondering how to approach the proof of the ...
0
votes
1answer
31 views

How to tell the difference between interval and coordinate notation from context?

I am working on a practice problem with sets. (the answer key) At first I was confused by the notation Ai = (0,i), i is a natural number. I looked up the use of paranthesis and saw that they could ...
0
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0answers
20 views

Numerical system that includes the limit targets such as $0^+$, $0^-$, $1^+$ etc

I wonder what is the name of a mathematical system extending the real numbers that includes signed zero along with unsigned zero as well as other "limit targets", such as $1^+=1+0^+$, $5^-$ etc, so ...
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0answers
27 views

Inequation with fractional part [closed]

Solve the inequality $x-x\{x\} < 1$, where $x$ is a real number, and $\{x\}$ is it fractional part.
3
votes
1answer
157 views

Solve an inequality using Cauchy-Schwarz Inequality

Le $a,b,c,d \in \mathbb{R^{+}}$. Using Cauchy-Schwarz Inequality prove that the following inequality holds: $$\frac{1}{\frac{1}{a+c} + \frac{1}{b+d}} \ge \frac{1}{\frac 1a + \frac 1b} + ...
21
votes
6answers
2k views

Can a complex number ever be considered 'bigger' or 'smaller' than a real number, or vice versa?

I've always had this doubt. It's perfectly reasonable to say that, for example, 9 is bigger than 2. But does it ever make sense to compare a real number and a complex/imaginary one? For example, ...
0
votes
1answer
22 views

Equivalence in the montone convergence theorem

If a monotone sequence is bounded it converges. Does convergence implies monotonicity and boundedness?
0
votes
0answers
39 views

$\lim \frac{x_1+ \dots +x_n}{y_n+\dots +y_n}=a$ [duplicate]

Suppose $y_n>0$ for all $n\in \Bbb{N}$, with $\sum y_n=+\infty$. If $\lim \frac{x_n}{y_n}=a$, prove that $\lim \frac{x_1+ \dots x_n}{y_n+\dots +y_n}=a$.
0
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1answer
53 views

Dedekind Cuts in Construction of the real line [closed]

Is each Dedekind cut a unique real number? or when we apply the process(Dedekind cut), do we get a bunch of real numbers instead of a unique one. If we get a unique real number, is the unique real ...
0
votes
0answers
26 views

Is the representation of $0$ unique?

So all integers $n$ can be represented by either an integer $\pm|n|$ or $\pm|n-1|$ follow by an infinitely long string of $9$'s, as $\cdots\,-2 = 1.\underline{9},\quad -1 = 0.\underline{9},\quad ...
1
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3answers
86 views

Which number is higher $2^{600}$ or $3^{400}$?

Which number is higher $2^{600}$ or $3^{400}$ ? I know that the solution is $3^{400}>2^{600}$ bot how to explain that. without using a calculator.
0
votes
1answer
23 views

Show that if $a, b \in \mathbb{R}$ and $a \neq b$, the intersection of the $\epsilon$ neighborhoods of a and b is $\emptyset$

I know the $\epsilon$-neighbordhood for a $\in \mathbb{R}$ is defined as the set $V_\epsilon(a)$ of $x \in \mathbb{R}$ such that |x-a| < $\epsilon$. To prove that the intersection of two ...
4
votes
1answer
54 views

Difficult to prove $\lim_ {n \to \infty} x_n - \liminf y_n = \limsup (x_n - y_n)$

I wanted to prove something here about $\liminf$ and $\limsup$, but it is not coming out, I need some help. Let $\{x_n\}$ and $\{y_n\}$ two sequences of real numbers such that $0\leq y_n\leq x_n$ for ...
1
vote
2answers
32 views

My proof of: $|x - y| < \varepsilon \Leftrightarrow y - \varepsilon < x < y + \varepsilon$

Is it reasonable to prove the following (trivial) theorem? If yes, is there a better way to do it? Let $x, y \in \mathbb{R}$. Let $\varepsilon \in \mathbb{R}$ with $\varepsilon > 0$. ...
0
votes
3answers
59 views

Complex number problem- separating into real and imaginary parts!

Please help with a question that I am working on just now...:) If $z=2e^{i\theta}$ where $0<\theta<\pi$, how can I find the real and imaginary parts of $w=(z-2)/(z+2)$? Hence, how can I ...
0
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1answer
34 views

Clarification on intuition behind one to one correspondence?

My book - Discrete Mathematics and its Applications This is my book's definition on if an infinite set is countable And the example it gave The "infinite set is countable if and only if it is ...
3
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0answers
60 views

Questions about decimal expansion being able to represent all real numbers

I read this in several books, and there's a Wikipedia article unquestionably stating that reals must be representable by means of regular language generated from finite alphabet. My questions are: ...
2
votes
1answer
86 views

Why is this not a function?

This problem is from Discrete Mathematics and its Applications This is the definition that the book gave of function Here is my work so far It's pretty clear to me that 1b and 1c are not ...
2
votes
1answer
44 views

Well ordering of $\mathbb{N}$ using inductive sets

In this book (Elementary Real Analysis by Thomson-Bruckner p.22), $\mathbb{N}=\left\{ 1,2,...\right\}$ (In some, $0\in\mathbb{N}$). In an exercise, a set $S\subset\mathbb{R}$ is inductive if $1\in S$ ...
0
votes
0answers
25 views

Proof of equivalence of the different forms of the completeness axiom

How can it be proven that the different forms of completeness axiom are equivalent and that if an ordered field satisfies one form of it, it satisfies all of them? I am referring to this article ...
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votes
1answer
49 views

Unitary Method : Hair Color and the Developer [closed]

The direction on the hair color bottle says use 1 oz of hair color to 1.5 oz of developer I need to use 1.5 oz of hair color How much developer would I need?
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3answers
111 views

Handling division by zero axiomatically

Suppose we define the multiplicative inverse function on real numbers as follows: $\forall{x \in \mathbb{R}}(x \neq 0 \implies x \times \frac{1}{x} = 1) $. Consider this truth table. \begin{array} ...
2
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1answer
31 views

Can we find a relation between the three integrers $m$, $j$ and $k$?

Let $r>4$ and $n>1$ positive integers and let $α$ be a positive real number. Let us define the following three positive integers: $$ \begin{align*} m &= \lfloor r^{(n+1)^2} \alpha \rfloor ...
0
votes
2answers
13 views

Find the number of digits of the number $k$ in function of $r$ and $n$

Let $α∈(0,1)$ be an irrational number with infinitely digits after the decimal point. Let $r>4$ and $n>1$ be positive integers. Let $$k=⌊r^{n²}α⌋$$ where $⌊.⌋$ is the floor function. My ...
0
votes
1answer
17 views

Question about Weierstrass approximation theorem

I hope that someone can answer this question. Can the Weierstrass approximation theorem be generalized to the interval $(-\infty,\infty)$? If for example one covers $(-\infty,\infty)$ with ...
1
vote
1answer
42 views

Can we deduce that $⌊r^{n}α⌋≃r^{n}α$ when $r→∞$?

Let $α∈(0,1)$ be an irrational number and let $n≥1$ be a fixed positive integer. For any $r>4$ we define the positive integer $$k=⌊r^{n}α⌋$$ where $⌊.⌋$ denotes the floor function. My question is: ...
0
votes
2answers
83 views

How do we make sense of angles which take irrational measures such as $\sqrt 2 ^\circ$?

If you were asked to draw such an angle how would you do so? Would you take it to a limit? Can the degree measure take the value of all real numbers?
0
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0answers
36 views

Baby Rudin: Chapter 1, Problem 6(d). How to complete this proof?

I've searched for a question, and it has been already asked here Baby Rudin: Chapter 1, Problem 6{d}. How to complete this proof?, but it hasn't yet a satisfactory answer. Can you help? If you go ...
0
votes
1answer
37 views

$r+s \leq x+y$: How to prove it?

If the following were true, I could complete an exercise. Is it really true? If it is, has anybody some hint? If it is not, what the counter-example? Need some help! Thanks Let $t\in\Bbb{Q}$ and ...
1
vote
1answer
44 views

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. ...
1
vote
1answer
48 views

$a^n-b^n=(a-b) (\sum_{i=0}^{n-1}a^ib^{n-1-i})$ [duplicate]

How do I use finite induction to prove that $$a^n-b^n=(a-b) (\sum_{i=0}^{n-1}a^ib^{n-1-i}), \forall a,b\in \Bbb{R}\space \text{and} \space \forall n \in \Bbb{N}?$$ Ok, for $n=2$ it's fine. ...
0
votes
1answer
34 views

Determine all $a\in\mathbb{R}$ so that a series converges

How do I determine all $a\in\mathbb{R}$ for a series $\sum \limits_{n=1}^\infty (-1)^n \cdot \frac{a^n}{n}$ so that the series converges? I know that the series converges for $a=1$ And I ...
3
votes
1answer
60 views

Dedekind's Cuts Lemma

I'm studying Dedekind's Cuts and his construction of Real numbers from the Rational ones. Here we are allowed to use $\Bbb{Q}$ as an ordered field and all all its properties (Archimedean Property, his ...
90
votes
11answers
6k views

Is there a domain “larger” than (i.e., a supserset of) the complex number domain?

I've been teaching my 10yo son some (for me, anyway) pretty advanced mathematics recently and he stumped me with a question. The background is this. In the domain of natural numbers, addition and ...
2
votes
1answer
42 views

How I can prove the following equivalence: $f(x)= 2/(2^{1/x}-1)⇔x=1/k$

Let us consider the function (see http://mathworld.wolfram.com/DevilsStaircase.html): $$f(x)=∑_{n=1}^{∞}⌊nx⌋/2ⁿ$$ for $x∈(0,1)$, where $⌊nx⌋$ is the floor function. This function is monotone ...
6
votes
4answers
120 views

Are logarithms the only continuous function on $(0, \infty)$ that has this property?

Are logarithms the only continuous function on $(0, \infty)$ that has this property? $$ f(xy) = f(x) + f(y) $$ If so, how would we show that? If not, what else would we need to show that a function ...
0
votes
1answer
37 views

$xy \le xz$ if both $y \le z$ and $0 \le x$. (very easy proof exercise)

As an exercise, I tried to prove the following theorem. Please share your thoughts about what I wrote. (The proof only uses the utensils which are listed below.) Theorem Let $x,y,z \in ...
0
votes
1answer
47 views

Does every real number have a decimal expansion?

Can every real number be written in decimal expansion? I mean, can every real number $a$ be expressed as follows: $$\text{For }\, a \in \mathbb {R}^{+},\quad ...