is there such thing as a “smallest positive number that isn't zero”? [duplicate]

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 etc.). 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 etc.) 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. Thank You.

marked as duplicate by T. Bongers, gamma, user223391, Zain Patel, Asaf Karagila♦Aug 16 '16 at 6:19

• 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
• 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
• @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
• @Carser: our cousins at Physics SE don't agree. – Martin Argerami Aug 14 '16 at 14:57
• @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

• 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.

• I think it is $0.\overline01$. – EKons Aug 14 '16 at 15:52
• @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
• @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
• I think I think it is actually $0.\overline{00}1$ :) – Ovi Aug 14 '16 at 22:18
• To all the folks in the comments, math.stackexchange.com/questions/979177/… – Asaf Karagila Aug 14 '16 at 23:14

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".

• 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
• @quid I think you're correct, but this is still a useful clarification for the OP. – Kyle Strand Aug 15 '16 at 18:07
• @quid: Fair point, I edited a bit to take that reading into account. – Daniel R. Collins Aug 15 '16 at 19:19

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.

• 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". – Hurkyl Aug 14 '16 at 15:36
• 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
• @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 0:33
• @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

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.

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.

• 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
• 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
• 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 5:48
• @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 9:13
• @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

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.

• Is there any error in my answer?? Why I get negative vote? – Sathasivam K Aug 15 '16 at 12:16
• 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 1:30
• "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