Questions tagged [real-numbers]

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

Filter by
Sorted by
Tagged with
275
votes
27answers
42k views

Is it true that $0.999999999\dots=1$?

I'm told by smart people that $$0.999999999\dots=1$$ and I believe them, but is there a proof that explains why this is?
43
votes
6answers
16k views

Is an automorphism of the field of real numbers the identity map?

Is an automorphism of the field of real numbers $\mathbb{R}$ the identity map? If yes, how can we prove it? Remark An automorphism of $\mathbb{R}$ may not be continuous.
135
votes
6answers
19k views

Induction on Real Numbers

One of my Fellows asked me whether total induction is applicable to real numbers, too ( or at least all real numbers ≥ 0) . We only used that for natural numbers so far. Of course you have to change ...
67
votes
5answers
12k views

Completion of rational numbers via Cauchy sequences

Can anyone recommend a good self-contained reference for completion of rationals to get reals using Cauchy sequences?
41
votes
9answers
5k views

Does $1.0000000000\cdots 1$ with an infinite number of $0$ in it exist?

Does $1.0000000000\cdots 1$ (with an infinite number of $0$ in it) exist?
12
votes
9answers
632 views

Providing that: $\lim\limits_{n\to\infty} \left(\frac{a^{\frac{1}{n}}+b^{\frac{1}{n}}}{2}\right)^n =\sqrt{ab}$

Let $a$ and $b$ be positive reals. Show that $$\lim\limits_{n\to\infty} \left(\frac{a^{\frac{1}{n}}+b^{\frac{1}{n}}}{2}\right)^n =\sqrt{ab}$$
51
votes
13answers
6k views

Do we really need reals?

It seems to me that the set of all numbers really used by mathematics and physics is countable, because they are defined by means of a finite set of symbols and, eventually, by computable functions. ...
7
votes
1answer
952 views

What is $\operatorname{Aut}(\mathbb{R},+)$?

I was solving some exercises about automorphisms. I was able to show that $\operatorname{Aut}(\mathbb{Q},+)$ is isomorphic to $\mathbb{Q}^{\times}$. The isomorphism is given by $\Psi(f)=f(1)$, but ...
6
votes
3answers
426 views

After removing any part the rest can be split evenly. Consequences?

Let $S$ be a finite collection of real numbers not necessarily distinct . If any element of $S$ is removed then the remaining real numbers can be divided into two collections with same size and same ...
4
votes
1answer
821 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 ...
48
votes
10answers
7k views

Why are integers subset of reals?

In most programming languages, integer and real (or float, rational, whatever) types are usually disjoint; 2 is not the same as 2.0 (although most languages do an automatic conversion when necessary). ...
49
votes
10answers
5k views

Is “$a + 0i$” in every way equal to just “$a$”?

I'm having a little argument with my friend. He says that "$a + 0i$" is, in every way, absolutely equal to "$a$" (e.g.: $2 + 0i = 2$). I say this is practically the case, so in every calculation you ...
44
votes
8answers
29k 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, ...
7
votes
3answers
935 views

Using Zorn's lemma show that $\mathbb R^+$ is the disjoint union of two sets closed under addition.

Let $\Bbb R^+$ be the set of positive real numbers. Use Zorn's Lemma to show that $\Bbb R^+$ is the union of two disjoint, non-empty subsets, each closed under addition.
9
votes
5answers
3k views

What is a natural number?

According to page 25 of the book A First Course in Real Analysis, an inductive set is a set of real numbers such that $0$ is in the set and for every real number $x$ in the set, $x + 1$ is also in the ...
37
votes
7answers
24k views

Are there real numbers that are neither rational nor irrational?

I wouldn't have asked this question if I hadn't seen this image: From this image it seems like there are reals that are neither rational nor irrational (dark blue), but is it so or is that ...
3
votes
2answers
915 views

Find Number Of Roots of Equation $11^x + 13^x + 17^x =19^x $

The Equation $11^x + 13^x + 17^x =19^x $ Has No Real Roots Only One Real Roots Exactly Two Real Roots More than Two Real Roots What I have done is The function $f(x)=11^x + 13^x + 17^x -19^x ...
141
votes
14answers
17k views

Why does an argument similiar to 0.999…=1 show 999…=-1?

I accept that two numbers can have the same supremum depending on how you generate a decimal representation. So $2.4999\ldots = 2.5$ etc. Can anyone point me to resources that would explain what the ...
37
votes
7answers
4k views

Is the real number structure unique?

For a frame of reference, I'm an undergraduate in mathematics who has taken the introductory analysis series and the graduate level algebra sequence at my university. In my analysis class, our book ...
13
votes
2answers
453 views

A series with only rational terms for $\ln \ln 2$

We all know that $$ \ln 2 = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n}. $$ Do you know a series with only rational terms for $$\ln \ln 2 = ?$$ Let's exclude base expansions with non explicit ...
9
votes
1answer
143 views

Remove any number and the remaining numbers can be partitioned into two subsets of equal sum; prove all numbers are equal. [duplicate]

Supposed I have a list of $n$ real numbers, where $n$ is odd. The list is constructed such that I can remove any arbitrary number from the list, and the remaining numbers can be partitioned into two ...
13
votes
1answer
537 views

Constructing the reals from the integers

A map $f\colon\mathbb{Z}\longrightarrow\mathbb Z$ is called a quasi-homomorphism if the set$$\{f(m+n)-f(m)-f(n)\,|\,m,n\in\mathbb{Z}\}$$is bounded. Let $R$ be the set of these functions. Let's ...
5
votes
4answers
624 views

Is any real-valued function in physics somehow continuous?

Consider the following well-known function: $$ \operatorname{sinc}(x) = \begin{cases} \sin(x)/x & \text{for } x \ne 0 \\ 1 & \text{for } x =0 \end{cases} $$ In physics, the sinc function has ...
3
votes
1answer
431 views

Enough Dedekind cuts to define all irrationals?

Assuming that there are uncountably infinitely many irrationals between any two consecutive rationals, how can the Dedekind cuts (defined on the countably infinite rationals) define all the ...
111
votes
11answers
8k 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 ...
30
votes
7answers
4k 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" (...
17
votes
2answers
5k views

Is there a bijection between the reals and naturals?

I found this pop math article saying that there was a paper published last year that proved that the cardinalities of the reals and naturals are equal. Is this true or is it a misinterpretation of the ...
17
votes
3answers
1k views

A topology on the set of lines?

Of course any set $X$ can have a topology, but are there more natural topologies, metrics or similar on the set of straight lines in $\mathbb R^2$?
16
votes
3answers
606 views

Could Euclid have proven that multiplication of real numbers distributes over addition?

In Euclid's day, the modern notion of real number did not exist; Euclid did not believe that the length of a line segment was a quantity measurable by number. But he did think it made sense to talk ...
13
votes
4answers
2k views

Enumeration of rationals from Stein-Shakarchi's Real Analysis (Chapter 1, Exercise 24)

The exercise is from Stein-Shakarchi's Real Analysis (Chapter 1, ex. 24). Does there exist an enumeration $\{r_{n}\}_{n=1}^\infty$ of the rationals such that the complement of $\bigcup_{n=1}^{\...
6
votes
1answer
2k views

Is the extended real line a metric space?

I've got a question reading the demonstration of the Theorem 3.2 in POMA of Rudin. Indeed, he says that every convergent sequence in a metric space is bounded. My question is: Is $\bar{\mathbb{R}}$ ...
10
votes
2answers
504 views

Real Induction Over Multiple Variables?

I've seen in several different places* that one can use normal mathematical induction to prove the truth of a statement that relies not on just one variable (say, $x$,) but multiple variables (for ...
4
votes
4answers
379 views

What makes negative numbers different from positive numbers other than their being (almost) opposite?

To quote from Wikipedia's article on negative numbers Negative numbers represent opposites. If positive represents movement to the right, negative represents movement to the left. If positive ...
0
votes
9answers
2k views

Does it make any sense to prove $0.999\ldots=1$?

I have read this post which contains many proofs of $0.999\ldots=1$. Background The main motivation of the question was philosophical and not mathematical. If you read the next section of the post ...
7
votes
1answer
2k views

Every ordered field has a subfield isomorphic to $\mathbb Q$?

I'm going through the first chapter in a text on real analysis, which contains preliminaries on ordered fields, the real numbers, etc. Supposedly I had learned about such things already, in calculus, ...
7
votes
3answers
6k views

Does same cardinality imply a bijection?

This came up today when people showed that there is no linear transformation $\mathbb{R}^4\to \mathbb{R}^3$. However, we know that these sets have the same cardinality. I was under the impression ...
14
votes
2answers
13k views

The set of real numbers is a subset of the set of complex numbers?

So, I was taught that $\mathbb{Z}\subseteq\mathbb{Q}\subseteq\mathbb{R}$. But since the set of complex numbers is by definition $$\mathbb{C}=\{a+bi:a,b\in\mathbb{R}\},$$ doesn't this mean $\mathbb{R}\...
11
votes
1answer
407 views

How to prove that: $19.999<e^\pi-\pi<20$?

I would like to know how to prove $$e^\pi-\pi\sim 20.$$ More precisely, I want to show by using only mathematical tools that, $$19.999<e^\pi-\pi<20$$ I have checked with online ...
7
votes
5answers
2k views

Is this direct proof of an inequality wrong?

My professor graded my proof as a zero, and I'm having a hard time seeing why it would be graded as such. Either he made a mistake while grading or I'm lacking in my understanding. Hopefully someone ...
3
votes
2answers
242 views

$x,y$ are real, $x<y+\varepsilon$ with $\varepsilon>0$. How to prove $x \le y$? [closed]

$x,y \in \mathbb R$ are such that $x \lt y + \epsilon$ for any $\epsilon \gt 0$ Then prove $x \le y$.
2
votes
1answer
169 views

Change of coordinate codomain from $[-1,1]$ to $[0,1]$

How does one translate coordinates from $[-1,1]$ to $[0,1]$? That is, suppose we have an ordered pair $(x,y)$ which lies between $[-1,1]$ and want to push into the range delimited by $[0,1]$. A lot ...
2
votes
1answer
204 views

Definitions of supremum

This question is for me to better understand the beginning of a real analysis course. We are provided with two definitions of supremum as follows: Def 1 : Let $S$ be a set in $\mathbb{R}$ be bounded ...
2
votes
3answers
928 views

Prove that $\lfloor 2x \rfloor + \lfloor 2y \rfloor \geq \lfloor x \rfloor + \lfloor y \rfloor + \lfloor x+y \rfloor$ for all real $x$ and $y$. [closed]

Prove that $\lfloor 2x \rfloor + \lfloor 2y \rfloor \geq \lfloor x \rfloor + \lfloor y \rfloor + \lfloor x+y \rfloor$ for all real $x$ and $y$. How should I solve this? I can't think of a way with ...
1
vote
6answers
428 views

Solutions of $f(x)\cdot f(y)=f(x\cdot y)$ [duplicate]

Can anyone give me a classification of the real functions of one variable such that $f(x)f(y)=f(xy)$? I have searched the web, but I haven't found any text that discusses my question. Answers and/or ...
3
votes
2answers
299 views

Why do new real numbers show up in Gödel's constructible hierarchy

In the "fat" cumulative hierarchy of sets, all the real numbers appear on level $\omega+1$. In Gödel's constructible hierarchy, some real numbers appear at that level, but (if I'm not mistaken) more ...
3
votes
2answers
308 views

Well-orderings of $\mathbb R$ without Choice

The question is about well-ordering $\mathbb R$ in ZF. Without the Axiom of Choice (AC) there exists a set that is not well-ordered. This could occur two ways: a) there are models of ZF in which $\...
2
votes
3answers
6k views

If $a<b$, $c<d$ then $a+c<b+d$?

This is how I managed to prove it: I know $a \lt b$ and $c \lt d$ thus, $b-a$ and $d-c$ are real positive numbers. Then, $b-a + d-c \gt 0$ and because of this $a+c \lt b+d$ Did I prove it right?
1
vote
2answers
1k views

Bijection between an infinite set and its union of a countably infinite set

I have $A$ as an infinite set and $S$ as a countably infinite set, (so that means there exists a one-to-one correspondence between $S$ and $\mathbb{N}$). How do I show that there always exists a ...
107
votes
3answers
2k views

All real numbers in $[0,2]$ can be represented as $\sqrt{2 \pm \sqrt{2 \pm \sqrt{2 \pm \dots}}}$

I would like some reference about this infinitely nested radical expansion for all real numbers between $0$ and $2$. I'll use a shorthand for this expansion, as a string of signs, $+$ or $-$, with ...
56
votes
16answers
7k views

Why are real numbers useful?

A question (by a fellow CS student taking a first course in calculus, presumably after the lecture in which continuity was introduced: was as follows. In the real, physical world, we deal with ...