Is there such thing as a "smallest positive number that isn't zero"?

Question

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.

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

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".

Answer 5

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.

Answer 6

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.