Questions tagged [real-numbers]

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

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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?
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 ...
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 ...
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 ...
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 ...
103
votes
24answers
17k views

Why do we still do symbolic math?

I just read that most practical problems (algebraic equations, differential equations) do not have a symbolic solution, but only a numerical one. Numerical computations, to my understanding, never ...
88
votes
4answers
6k views

Cover of “Gödel, Escher, Bach”

Consider the cover image of the book "Gödel, Escher, Bach", depicted below. The interesting feature is that it shows the existence of a subset of $\mathbb{R}^3$ which projects onto $\mathbb{R}^2$ in ...
67
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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?
66
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16answers
12k views

Why can a real number be defined as a Dedekind cut, that is, as a set of rational numbers?

I don't know if my textbook is written poorly or I'm dumb. But I can't bring myself to understand the following definition. A real number is a cut, which parts the rational numbers into two classes....
65
votes
3answers
4k views

Does multiplying all a number's roots together give a product of infinity?

This is a recreational mathematics question that I thought up, and I can't see if the answer has been addressed either. Take a positive, real number greater than 1, and multiply all its roots ...
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 ...
51
votes
13answers
7k 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. ...
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 ...
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). ...
44
votes
8answers
30k 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, ...
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.
42
votes
10answers
11k views

Picking two random real numbers between 0 and 1, why isn't the probability that the first is greater than the second exactly 50%?

I attempted to answer this question on Quora, and was told that I am thinking about the problem incorrectly. The question was: Two distinct real numbers between 0 and 1 are written on two sheets of ...
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?
40
votes
7answers
3k views

Category-theoretic description of the real numbers

The familiar number sets $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$ all have "natural constructions", which indicate, why they are mathematically interesting. For example, equipping $\mathbb{N}$ with ...
37
votes
7answers
25k 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 ...
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 ...
33
votes
5answers
2k views

How many pairs of numbers are there so they are the inverse of each other and they have the same decimal part?

I was wondering... $1$, $\phi$ and $\frac{1}{\phi}$, they have something in common: they share the same decimal part with their inverse. And here it comes the question: Are these numbers unique? How ...
32
votes
6answers
4k views

Are there many fewer rational numbers than reals?

Today my professor asked me to figure out the probability of getting a rational number from $[0,1]$. His answer was that the probability is $0$. Why is this?
32
votes
4answers
1k views

$p_n(x)=p_{n-1}(x)+p_{n-1}^{\prime}(x)$, then all the roots of $p_k(x)$ are real

$p_0(x)=a_mx^m+a_{m-1}x^{m-1}+\dotsb+a_1x+a_0(a_m,\dotsc,a_1,a_0\in\Bbb R)$ is a polynomial, and $$p_n(x)=p_{n-1}(x)+p_{n-1}^{\prime}(x),\qquad n=1,2,\dotsc$$ then, there exist $N\in\Bbb N$, such ...
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" (...
29
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6answers
3k views

Confusion about Tao's construction of reals

Background: I am currently studying real analysis using Tao's Analysis Volume One and so far I am really enjoying myself though I seem to have run into some confusion regarding professor Tao's ...
27
votes
4answers
2k views

Is there a website like OEIS for real constants?

I'm hoping an online service exists where I can type in say 3.14159 and it then shows a bunch of 'interesting' (however one would define that) numbers. Naturally in that instance it would bring up $\...
26
votes
9answers
6k views

Prove that any real number can be expressed as the sum of two irrational numbers

Prove that any real number $r$ can be expressed as the sum of two irrational numbers $x$ and $y$. Progress: I have a specific example for any rational number $r$: $x = r-\pi$ and $y = \pi$ (or ...
25
votes
5answers
2k views

Were “real numbers” used before things like Dedekind cuts, Cauchy sequences, etc. appeared?

Just the question in the title, I'm trying to understand how something like analysis could be developed without formal constructions of the real numbers. I'm also very interested, if the answer is "...
24
votes
4answers
3k views

Is it irrational?

Suppose I generate a number $0 < x < 1$. In general, after the decimal point, the first digit is $1$, the second is $0$, the third is $1$, etc. However, every digit in position $n$ has a $1/n$ ...
24
votes
0answers
292 views

Smallest Subset of $\mathbb{R}_{>0}$ Closed under Typical Operations

Let $S$ denote the smallest subset of $\mathbb{R}_{>0}$ which includes $1$, and is closed under addition, multiplication, reciprocation, and the function $x,y \in \mathbb{R}_{>0} \mapsto x^y.$ ...
23
votes
1answer
654 views

Is $\sin(e)$ rational or irrational?

We know that $\pi$ and $e$ are transcendental numbers. Here $\sin(x)$ is a real trigonometric function. We know that $\sin(\pi)=0$ which is rational. Now I am wondering to know that whether $\sin(e)$ ...
23
votes
2answers
5k views

Showing that $\sqrt[3]{9+9\sqrt[3]{9+9\sqrt[3]{9+\cdots}}} - \sqrt{8-\sqrt{8-\sqrt{8+\sqrt{8-\sqrt{8-\sqrt{8+\cdots}}}}}} = 1$?

$$\sqrt[3]{9+9\sqrt[3]{9+9\sqrt[3]{9+\cdots}}} - \sqrt{8-\sqrt{8-\sqrt{8+\sqrt{8-\sqrt{8-\sqrt{8+\cdots}}}}}} = 1$$ In the second nested radical, the repeating pattern is $(-,-,+)$. I approached this ...
21
votes
7answers
4k views

Does $0 < x < 0$ imply $x =0$?

In Real Analysis class, a professor told us 4 claims: let x be a real number, then: 1) $0\leq x \leq 0$ implies $x = 0$ 2) $0 \leq x < 0$ implies $x = 0$ 3) $0 < x \leq 0$ implies $...
20
votes
5answers
3k views

Dedekind cuts for $\pi$ and $e$

I tried to search in the internet about this but did not get any exciting answers. So my question is: How is construction of transcendental numbers like $\pi$ and $e$ explained via Dedekind cuts?
19
votes
10answers
5k views

What is so wrong with thinking of real numbers as infinite decimals?

Timothy Gowers asks What is so wrong with thinking of real numbers as infinite decimals? One of the early objectives of almost any university mathematics course is to teach people to stop thinking ...
19
votes
2answers
477 views

How can you show by hand that $ e^{-1/e} < \ln(2) $?

By chance I noticed that $e^{-1/e}$ and $\ln(2)$ both have decimal representation $0.69\!\ldots$, and it got me wondering how one could possibly determine which was larger without using a calculator. ...
19
votes
1answer
432 views

An interesting way of expressing any real number using the harmonic series.

I recently saw the identity $$ \frac{1}{1} - {1 \over 2} +{1 \over 3} - {1 \over 4} + {1 \over 5} - {1 \over 6} \dotsb = \log(2) $$ which I found rather interesting. I was intrigued by the way a ...
18
votes
6answers
4k views

when product of irrational numbers = rational number?

let $a$ and $b$ be irrational numbers. when do we have $ a \cdot b $ = rational number? for example $\sqrt{2} \cdot \sqrt{2}=2$. I was wondering if there some conditions for the product to be a ...
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
7answers
3k views

What is the purpose of showing some numbers exist?

For example in my Analysis class the professor showed $\sqrt{2}$ exists using Archimedean properties of $\mathbb{R}$ and we showed $e$ exists. I want to know why it's important to show their existence?...
17
votes
8answers
3k 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?
17
votes
3answers
1k views

Does the concept of permutation make sense for a set indexed by the real numbers?

I know that the concept of permutation makes sense for sequences, which are sets indexed by the natural numbers (if the sequence is infinite) or indexed by the first $n$ natural numbers (if the ...
17
votes
4answers
1k views

Why does $ a_n = \frac {a_{n-1} + \frac {2}{a_{n-1}}}{2}$ converge to an irrational number?

There is a problem in my textbook that goes like this $$ a_n = \frac {a_{n-1} + \frac {2}{a_{n-1}}}{2}$$ and $$a_0 =1$$ for all $n\ge1$. It is monotonically decreasing sequence of rational ...
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
1k 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? ...
16
votes
3answers
607 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 ...
15
votes
15answers
7k views

Can the product of three complex numbers ever be real?

Say I have three numbers, $a,b,c\in\mathbb C$. I know that if $a$ were complex, for $abc$ to be real, $bc=\overline a$. Is it possible for $b,c$ to both be complex, or is it only possible for one to ...
15
votes
8answers
1k views

When trying to learn analysis from bottom up, what numbers should I first construct?

I am interested in studying more analysis and related topics. However, I want to make sure I do so well and without making too many broad jumps in my learning. In some books I have seen, the author ...
15
votes
5answers
450 views

A curiosity: how do we prove $\mathbb{R}$ is closed under addition and multiplication?

So I tried looking around for this question, but I didn't find much of anything - mostly unrelated-but-similarly-worded stuff. So either I suck at Googling or whatever but I'll get to the point. So ...