34

My brother and I have been discussing whether it would be possible to have a "smallest positive number" or not and we have concluded that it's impossible.

Here's our reasoning: firstly, my brother discussed how you can always halve something, $(1, 0.5, 0.25, \dotsc)$. I myself believe that it is impossible because of something I managed to come up with. You can put an infinite amount of zeroes in the decimal place before a number, $(0.1, 0.01, 0.001, \dotsc)$. I am not entirely sure if our reasoning is correct though. I have been told that there is a smallest number possible but I decided to see for myself.

Xander Henderson
  • 24,007
  • 25
  • 53
  • 82
  • 1
    Also note that your brother says you can always divide by $2$, while you're actually saying that you can always divide by $10$. – Frenzy Li Aug 14 '16 at 14:09
  • 20
    Your brother suggested to divide by $2$, you suggested to divide by $10$. Pretty much the same argument. Works perfectly for positive values, but keep in mind that there are negative values as well... which brings me to the main point - what do you consider by "smaller" (i.e., what is the context in which this question is asked - the set of positive numbers, the known universe, etc)? – barak manos Aug 14 '16 at 14:09
  • There is no smallest *positive* real number – user251257 Aug 14 '16 at 14:11
  • 7
    Assuming you're considering real numbers, there is no smallest positive number. Both you and your brother provided proofs of this. – Git Gud Aug 14 '16 at 14:12
  • Thanks Git Gud! true hero to my self confidence –  Aug 14 '16 at 14:13
  • 21
    @PixelFallHD Mathematics is not physics. Do not mix the two. There is no smallest positive real number and than tells you nothing about parasites or the physical world you perceive. – Git Gud Aug 14 '16 at 14:13
  • Maths is fun. I'm 12, is being a mathematician plausible for me? –  Aug 14 '16 at 14:14
  • @PixelFallHD Unless someone here knows you, no one here can tell you that. – Git Gud Aug 14 '16 at 14:14
  • 1
    There is, in a sense, a smallest distance in physical space. It is the Planck length, the distance at which you can no longer distinguish between points (https://en.wikipedia.org/wiki/Planck_length) – Carser Aug 14 '16 at 14:14
  • 13
    @Carser: our cousins at Physics SE [don't agree](http://physics.stackexchange.com/questions/185939/is-the-planck-length-the-smallest-length-that-exists-in-the-universe-or-is-it-th). – Martin Argerami Aug 14 '16 at 14:57
  • @MartinArgerami If you mean they don't agree with each other, then I agree with you! ;) – Carser Aug 14 '16 at 15:09
  • 1
    @ carser "There is, in a sense, a smallest distance in physical space. It is the Planck length, the distance at which you can no longer distinguish between points" but it does not mean there is no length which is lesser that planck length .it just mean YOU CAN NO LONGER DISTINGUISH BETWEEN POINTS.ie if we consider a length which is less than Planck length, you can't see any differences in the answer.since their difference is very very small which is "NEGLIGIBLE".DONT SAY ANYTHING WHITHOUT KNOWING IT COMPLETELY! – Sathasivam K Aug 14 '16 at 16:23
  • @T.Bongers May be but idk exactly whether it is a duplicate question but he want to clarify his doubt. – Sathasivam K Aug 14 '16 at 16:38
  • Of course it depends on which numbers you consider. If you consider the natural numbers, there's definitely a smallest positive one, namely $1$. – celtschk Aug 14 '16 at 18:57
  • 4
    Yet another time when I am amazed by how such a simple question can become so popular... – Wojowu Aug 14 '16 at 20:25
  • 1
    Even in extended number systems, such as the surreals, where the "number" $\frac{1}∞$ is greater than zero, you can still divide it by 2, or 10, etc. – Neil Aug 14 '16 at 21:08
  • As far as you and your brother are concerned, "number" means "decimal"? Mathematicians do not use the term that way... – GEdgar Aug 14 '16 at 22:19
  • 8
    @Wojowu the popular questions on this site are virtually always simple. They're the most accessible to voters, and there are more users qualified to give high quality answers. – Matt Samuel Aug 14 '16 at 23:41
  • 1
    @SathasivamK I don't think Carser was disagreeing. Note that they only said "in a sense" and then qualified that "sense" with exactly the same disclaimer as you did. Anyway, there's REALLY no need to SHOUT ABOUT IT. If you want to emphasise a point, try using `_italics_`, `*italics again*`, or `**bold**`. – underscore_d Aug 15 '16 at 13:58
  • @underscore_d. OK – Sathasivam K Aug 15 '16 at 15:04
  • 3
    Your logic is fine. But instead of saying "You can put an infinite amount of zeroes in the decimal place before a number", it is more accurate to say "You can put an _unlimited_ amount of zeroes in the decimal place before a number." – TonyK Aug 15 '16 at 20:58

6 Answers6

53

A simple proof by contradiction works here.

  • Suppose that $a$ is the smallest positive real number.
  • Next, divide it by $n$ (where $n>1$) to get $\displaystyle\frac a n$.
  • This new number is smaller than $a$.

Your brother choose $n=2$, while you chose $n=10$.

So we can deny the existence of a smallest positive real number since

... there is a "smallest" number and yet there is a number smaller than it.

Same argument works with positive rational numbers.

Frenzy Li
  • 3,543
  • 2
  • 31
  • 52
  • 9
    I think it is $0.\overline01$. – EKons Aug 14 '16 at 15:52
  • 33
    @ΈρικΚωνσταντόπουλος You can refer to quid's answer. $0.\bar01$ is not a valid decimal expansion because, to quote, "one cannot have infinitely many $0$ and then the first... non-zero digit, in a decimal expansion." Should you insist that it is a valid representation of any number, then it would be interpreted by the following limit:$$0.\bar01=\lim_{n\to\infty} 0.\underbrace{000\cdots0}_{n\text{ digits}}1 = \lim_{n\to\infty} \frac{1}{10^{n+1}} = 0,$$ which is zero, and not a positive number. – Frenzy Li Aug 14 '16 at 15:59
  • @ΈρικΚωνσταντόπουλος I believe it is $0.\overline02$. $0.\overline01$ would be $0$ with $0.\overline01$ added to it. – bb216b3acfd8f72cbc8f899d4d6963 Aug 14 '16 at 17:05
  • 3
    @Peanut $$0.\bar02=\lim_{n\to\infty} 0.\underbrace{000\cdots0}_{n\text{ digits}}2 = \lim_{n\to\infty} \frac{2}{10^{n+1}} \leqslant \lim_{n\to\infty} \frac{1}{10^n} = 0,$$ which is zero as well. Same if you insist so with $0.\bar03$ to $0.\bar09$ or even $0.\bar0$ followed by $\bar9$. They are all invalid decimal writing, and should they be equal to any value, they'd be zero. – Frenzy Li Aug 14 '16 at 17:13
  • 12
    I think I think it is actually $0.\overline{00}1$ :) – Ovi Aug 14 '16 at 22:18
  • 7
    To all the folks in the comments, http://math.stackexchange.com/questions/979177/does-1-0000000000-cdots-1-with-an-infinite-number-of-0-in-it-exist/979217#979217 – Asaf Karagila Aug 14 '16 at 23:14
  • I always hope for some mention of nonstandard analysis in these questions, but don't see one here – Nate 8 Aug 15 '16 at 04:23
  • 3
    Meanwhile, I always hope to avoid mention of nonstandard analysis, usually in vain, when the question is obviously in the context of real numbers (even if it's from a person who doesn't have the sophistication to use that technical term). – Daniel R. Collins Aug 15 '16 at 04:59
  • I think it is 1-0.999... ;) – Bradley Thomas Aug 15 '16 at 13:33
  • @BradThomas Assuming you aren't making a stale joke, that would be 0. Infinitely repeatings 9s is just another representation of 1. – chepner Aug 15 '16 at 13:36
  • 2
    Note: This is only true among the real numbers. In the surreal numbers there are many such numbers that are smaller than all positive real numbers but greater than 0. I would agree the answer for real numbers should stand, but as most non-mathematicians aren't intimately familiar with formal definitions and generally work on some intuition about numbers, it might be helpful to include the surreal answer so as not to limit our question askers' imaginations by telling them "no, never!" when the asker didn't specify that they were only interested in the reals. – Shufflepants Aug 15 '16 at 15:36
  • 1
    @Shufflepants: The surreals don't have a smallest positive number either. I imagine you even have $\lim_{n \to \infty} 2^{-n} = 0$ when limit is taken over the omnific integers, although admittedly I know very little about doing calculus with them. While you can do interesting things with surreals and hyperreals and such, I feel like answers along those lines tend to change the question so that they *can* do those interesting things, but act like it was an answer to the original question. –  Aug 15 '16 at 15:57
  • @Hurkyl They don't have a smallest positive number, but they do have a number that corresponds to 0.00...01 = { 0 | 1, 1/2, 1/4, 1/8, … } = ε that is smaller than all positive real numbers. It just so happens that they also have numbers that are much smaller than 0.00...01. I also have no idea how to do calculus with them or even if many of the theorems of calculus have even been redefined for the surreals. – Shufflepants Aug 15 '16 at 15:58
  • 1
    May be worth explicitly mentioning that $ \frac{a}{n} $ is necessarily a real number as well, since sets not being closed under operations is a thing the OP should be made aware of. ;) – jpmc26 Aug 15 '16 at 22:59
  • 1
    ... and to add to @jpmc26's comment, it would be worth explicitly mentioning that $\frac an$ remains a positive number. It's pretty obvious, but it's still a key part of the proof by contradiction. – Théophile Aug 16 '16 at 04:12
  • @stepner, it was a stale joke. But here is a serious question. Assuming a is the smallest positive real number, surely that makes step 2 (divide by n) an invalid operation, and so an invalid step in the logic, because we've already defined a to be a number that is indivisible. – Bradley Thomas Aug 16 '16 at 17:56
18

You can put an infinite amount of zeroes in the decimal place before a number, (0.1, 0.01, 0.001 etc.) I am not entirely sure if our reasoning is correct though.

This claim is technically mistaken, which makes your brother's reasoning more correct than yours. You should recognize that the word "infinite" here is effectively just shorthand for "goes on forever", "doesn't have any end to it", and "always has another of the same digit coming up next". So it's a contradiction in terms to say that you can have an infinite number of zeroes and then some other digit afterward; this essential contradiction means that there's no real number like that, and thus, no smallest positive real number.

On the other hand: It would be correct to say that you can have an arbitrary number of zeroes before a 1, that is, indeed be able to find a positive decimal less than any other number someone proposed as "smallest".

Daniel R. Collins
  • 7,820
  • 1
  • 19
  • 42
  • 2
    But OP did not claim there was a smallest number. I originally read it your way too (see the revision of my answer). I think they only use "infinite" (incorrectly) to mean "not bounded" – quid Aug 15 '16 at 15:07
  • 1
    @quid I think you're correct, but this is still a useful clarification for the OP. – Kyle Strand Aug 15 '16 at 18:07
  • 1
    @quid: Fair point, I edited a bit to take that reading into account. – Daniel R. Collins Aug 15 '16 at 19:19
15

There is no smallest positive real number. The argument of your brother is correct.

Your argument is also correct. As mentioned in comments your brother divides by $2$ while your argument amounts to dividing by $10$. Note though that it is better to say that there can be arbitrarily many $0$ rather than infinitely many. (One cannot have infinitely many $0$ and then the first $1$, or non-zero digit, in a decimal expansion. But there is no bound on the number of $0$ one can have before the first non-zero digit; also in total there can be infinitely many $0$, but not before the first non-zero one.)

Of course there is a smallest positive whole number/integer, it is $1$. The halving argument does not work here, as you cannot split $1$ into two positive whole numbers.

There are various ramifications of this and you might want to look into infinitesimals or ordered sets if you are curious about such things.

As for the smallest object in the world, this is a physics question, which has no definite answer as far as I know. But there are some theories where there is a smallest measureable length in some sense, see Planck length.

quid
  • 40,844
  • 9
  • 59
  • 101
  • 5
    AFAIK, the speculation about the Planck length is more "around this size, the notion of distance isn't a well-defined concept" and less "there is a smallest distance". –  Aug 14 '16 at 15:36
  • 4
    I do not claim any particular expertise in physics; it was intended as a side remark. Further, I did not claim that there is a smallest distance. What I wrote is "smallest measureable length in some sense" this is arguably a bit garbled but there is a 'measurable' and a 'in some sense.' – quid Aug 14 '16 at 16:06
  • Ya right its not smallest length but the smallest measurable length! We can't measure lesser length than that becoz the instruments that we are using can't able to measure it.but it MAY BE CHANGE IN FUTURE.idk. – Sathasivam K Aug 14 '16 at 16:32
  • 1
    @SathasivamK No, under that theory of quantum gravity, distances shorter than the Planck length are not measurable regardless of the instrumentation. It's a similar issue to the more familiar Heisenberg uncertainty principle. – Ian Aug 15 '16 at 00:33
  • @Ian That doesn't mean that the measurable distance cannot change. It does mean that the change necessitates a development in theory rather than instrumentation. – user66309 Aug 15 '16 at 16:33
  • @user66309 I said "under that theory of quantum gravity", which is intended to mean "under the assumption that the universe works according to that theory of quantum gravity". Under that assumption, there is no further theoretical development to be done. Of course we're not sure whether that's the case, but that doesn't mean that there is a "hole" in this theory itself which remains to be filled. – Ian Aug 15 '16 at 16:37
3

You can't have a number with an infinite number of zeros followed by a one.

Of course, in mathematics, you aren't just arbitrarily allowed to say "this is allowed" and "this isn't allowed." You have to fall back to an accepted definition to see if something makes sense.

In this case, a decimal number $0.x_1x_2x_3\ldots$ is shorthand for $\frac{x_1}{10^{1}}+\frac{x_2}{10^{2}}+\frac{x_3}{10^{3}}+\cdots$, that is $$\sum_{i =1}^\infty {\frac{x_i}{10^{i}}}.$$ Now, some discussion about what an infinite sum even means, and if it converges are needed to truly make sense of this, but even without that, we can see that the above claim is meaningless. Every digit in a number is a coefficient in the sum corresponding to a positive integer power of 10. With your suggested number, you need an integer power of 10 that you can assign 1 as a coefficient to that gives an infinite number of smaller powers of 10 to assign 0 to, in other words, a positive integer that is smaller than an infinite number of positive integers. There is no such number. In fact, every positive integer $n$ is larger than only $n-1$ smaller positive integers, which is finite.

asmeurer
  • 9,046
  • 5
  • 30
  • 50
0

Yes, there is a smallest positive number that isn’t zero… if you want there to be.

Everything in mathematics is a label for a concept. That’s why it’s popular to call mathematics a language. If there isn’t a word for something, you can make one up.

However, if you want to communicate with others, you have to speak the same language, which means agreeing on definitions and sticking to them. In mathematics, we don’t usually consider infinites (ω) or infinitesimals (ε) to be real numbers (ℝ) because they are not Archimedean. We sometimes treat them as if they were. But even then, we call them hyperreal numbers (*ℝ), and say that they are an extension of the set of real numbers.

You and your brother essentially applied the axiom of Archimedes and arrived at the generally accepted conclusion.

For any positive ε in K, there exists a natural number n, such that 1/n < ε.

You chose the natural number 10 (adding an extra zero in the decimal place before a number) and your brother chose 2. Although, asmeurer rightly points out that it is not proper to say “put an infinite amount of zeroes in the decimal place before a number”.

While it has proven useful to give infinity a name and a symbol (∞), the same can’t be said about the thing that is an infinitesimal positive distance from zero.

You should take away these two points:

  1. The thing that is an infinitesimal positive distance from zero is not a real number.
  2. There is no name or symbol for the thing that is an infinitesimal positive distance from zero.

But go ahead and call it a number and give it a name and a symbol. If you want to.

lukejanicke
  • 315
  • 1
  • 11
  • 10
    Just because you give a name for something, doesn't mean it exists. I'll try. "A circle with radius one and area 20." I'll call it a Zloik. Zloiks do not exist though. – djechlin Aug 14 '16 at 20:07
  • 2
    We’re not requiring these “things” to actually exist. Infinity doesn’t exist. But infinity is “a thing”. It’s a concept. An idea. Quite a useful one. As I said, everything in mathematics is a label for a concept, and the beauty of it is that we can give names to things that don’t intuitively (or actually) exist. A Zloik is a mathematical object with non-conventional qualities. The OPs question was really “Is there a generally accepted name for this thing?” The answer to which is “No.” I was simply pointing out that a more enlightened answer might be “Not yet” or “Not currently.” – lukejanicke Aug 14 '16 at 20:39
  • 2
    The more enlightened answer is "no, given the standard axioms for the real numbers," or "no, given Hilbert's axioms for geometry." – djechlin Aug 15 '16 at 05:48
  • This answer may be informal, but it contains the essence of the idea to [Synthetic Differential Geometry](https://en.wikipedia.org/wiki/Synthetic_differential_geometry), which has an infinitesimal $\epsilon$ and is a beautiful way to do differential geometry. I don't understand the downvotes. – Turion Aug 15 '16 at 07:41
  • 3
    @djechlin A number of which the square is -1 doesn't exist. I'll pretend it exist though, call it 'i' and get interesting results. – Florian F Aug 15 '16 at 09:13
  • 1
    @Turion I tried explaining that... "if the number you want doesn't exist - even if you have a *proof* that it doesn't exist - just make one up and maybe if you're lucky some day someone will come up with a different number system where it does exist." I don't think that's helpful to someone who is yet to understand Archimedes' principle and I don't think it's a terribly helpful way to introduce the hyperreals, which has some of the most unintuitive logical structure I've ever seen studying math. Understanding Archimedes principle comes *before* hyperreals. – djechlin Aug 15 '16 at 14:16
  • A big part of math is understanding and proving why things don't exist. A smooth vector field on the sphere that's nowhere zero. A solution in radicals to general quintics. An integer with two distinct prime factorizations -- which is a *very* important fact *even* though there are integer-like systems where such numbers do exist. Naming these things anyway because they *could* have existed is not doing math and it's worthy of a downvote when an answer does such a thing to undermine the significance of why the thing does not exist. (And seriously, don't introduce the hyperreals this way.) – djechlin Aug 15 '16 at 14:20
  • @djechlin, the hyperreals and synthetic differential geometry are two different things. – Turion Aug 15 '16 at 15:38
  • This would be a more useful/interesting answer if it mentioned internally-consistent ways to do math that *do* permit a smallest real number. The best (only?) such way that I know if is the philosophy of [ultrafinitism](http://math.stackexchange.com/q/531/52057). – Kyle Strand Aug 15 '16 at 18:11
  • 1
    Also, I think the statement that "infinity doesn't exist" is a philosophical assertion, not a mathematical one. (At this point in my life I'm inclined to agree with it, but I don't think it's necessarily a "known truth" in any real sense.) – Kyle Strand Aug 15 '16 at 18:13
  • I should have written only a down-to-earth answer from formal mathematics to answer the OP question. Instead, I spawned philosophical musings and got my answer down voted to 0. Lesson learned. Except, I would like to point out that often in mathematics we say *Let {something} = {something else}*. It’s called an assumption and we do it often in order to have a starting point that saves us from having to get into deep philosophical or existential debates. Take away that part of my answer and I think I otherwise have a pretty good reply for the OP... from formal mathematics. – lukejanicke Aug 19 '16 at 05:04
-2

Your argument and your brother argument both Are right! Let you assume N is the set of all positive number.

Now let a,b$\in Q$ where Q is a set of all rational number.

Now ,$\frac{a+b}{2}$ is again a rational number . You can prove it by taking a =$\frac{p}{q}$,b=$\frac {r}{s}$. And use properties of fraction.

Now we assume a=0,b=1. Now $\frac{0+1}{2}$ $\in Q$.which is greater than 0 and so it is positive real number.now take $\frac{0+1}{2}$=0.5 as c. Then $\frac{0+c}{2}$ is again greater than zero and $\in Q$. Therefore between two rational number there are infinite possibilities to find a new rational number .similarly for irrational number too.THERFORE YOU CAN'T FIND THE SMALLEST POSITIVE NUMBER .but 0 is the greatest lower bound of set of positive real numbers.

Sathasivam K
  • 863
  • 1
  • 9
  • 20
  • Is there any error in my answer?? Why I get negative vote? – Sathasivam K Aug 15 '16 at 12:16
  • 2
    I didn't downvote, but the logic and language are quite garbled. The proof is not complete, the issue of rationals unclear, the irrational statement unsupported, and the all-caps conclusion seem to assert that there is "the smallest positive number" but you just can't find it. – Daniel R. Collins Aug 16 '16 at 01:30
  • "all-caps conclusion seem to assert that there is "the smallest positive number" which caps conclusion you mean @Daniel R. Collins – Sathasivam K Aug 17 '16 at 14:52
  • "All-caps" means the all-capitalized-letters part. – Daniel R. Collins Aug 17 '16 at 15:05
  • You need to reconsider what your commented .I posted as "THERFORE YOU CAN'T FIND THE SMALLEST POSITIVE NUMBER".I say as "you can't" not "you can" – Sathasivam K Aug 17 '16 at 15:35