Can I add any number of zeros to 0.00 ... 1?

[nmz]

TL;DR

Please note that my word is "arbitrary" instead of "infinite". It is self-contradictory to have a non-zero number after an infinite zero, but it seems that I can add 0 to make it smaller and smaller without restriction. Does it belong to a finite number or an infinite number, and is it equal to zero?

When I add 0 in the middle of 0.001, it seems that I can add any number of 0 without restriction. If so, is it still a limited number? Of course, I can't add infinite zeros in front of it.

 

[weirdoguy]

it seems that I can add any number of 0 without restriction.

You can't. We use positional system, and 1 in 0,0...01 does not have a well defined position. Hence, the numer is not defined.

 

[PeroK]

TL;DR Summary: Please note that my word is "arbitrary" instead of "infinite". It is self-contradictory to have a non-zero number after an infinite zero, but it seems that I can add 0 to make it smaller and smaller without restriction. Does it belong to a finite number or an infinite number, and is it equal to zero?

When I add 0 in the middle of 0.001, it seems that I can add any number of 0 without restriction. If so, is it still a limited number? Of course, I can't add infinite zeros in front of it.

You have an infinite sequence of finite numbers, each of which is the previous number divided by 10. You can write these in scientific notation as $10^{-n}$, where $n$ is any whole number.

There is no last number in this sequence, as $n$ can be any whole number.

 

[nmz]

You have an infinite sequence of finite numbers, each of which is the previous number divided by 10. You can write these in scientific notation as $10^{-n}$, where $n$ is any whole number.

There is no last number in this sequence, as $n$ can be any whole number.

My friend told me that there can't be an infinite number of zeros in the decimal extension, but there is no limit to the number of zeros you can have before the first non-zero number. If I keep increasing randomly until I stop increasing somewhere, is it still equal to 0?

 

[PeroK]

My friend told me that there can't be an infinite number of zeros in the decimal extension, but there is no limit to the number of zeros you can have before the first non-zero number.

Your friend is correct. That's the difference between "arbitrarily large" and "infinite". Or, arbitrarily small and zero.

If I keep increasing randomly until I stop increasing somewhere, is it still equal to 0?

It's never equal to zero, as long as you put a 1 in some digit.

 

[javisot20]

TL;DR Summary: Please note that my word is "arbitrary" instead of "infinite". It is self-contradictory to have a non-zero number after an infinite zero, but it seems that I can add 0 to make it smaller and smaller without restriction. Does it belong to a finite number or an infinite number, and is it equal to zero?

When I add 0 in the middle of 0.001, it seems that I can add any number of 0 without restriction. If so, is it still a limited number? Of course, I can't add infinite zeros in front of it.

That doesn't happen only with 0's, you can't do the same with other numbers either, for example, you could think that between 0.999... and 1 there is the number 0,(999..)1,
but the number 0,(999..)1 is a number that cannot be constructed formally.

Same for 0,(000...)1

 

[weirdoguy]

you could think that between 0.999... and 1 there is the number 0,(999..)1,

Well, most people who struggle with 0,(9)=1 identity think that between these two is 0,(0)1

 

[FactChecker]

My friend told me that there can't be an infinite number of zeros in the decimal extension, but there is no limit to the number of zeros you can have before the first non-zero number. If I keep increasing randomly until I stop increasing somewhere, is it still equal to 0?

There is a difference between "no limit" versus undefined. The first implies that there are a definite number of zeros but it can be as many as you want. The second implies that the number of zeros is undefined, so the number itself is undefined. The notation "0.000....1" is not valid.

 

[nmz]

Your friend is correct. That's the difference between "arbitrarily large" and "infinite". Or, arbitrarily small and zero.


It's never equal to zero, as long as you put a 1 in some digit.

My friend's answer to this question is: I can add any number of zeros before 1, but even if I add trillions of zeros, when I stop. It is a finite number greater than zero (but it doesn't mean that you can't continue to add zeros). When I keep adding zeros and never stop, it is an infinite number equal to zero, because if I keep adding zeros, there is nowhere to put the last one, but I have some doubts about his point of view. How can a number be both finite and infinite?

 

[nmz]

There is a difference between "no limit" versus undefined. The first implies that there are a definite number of zeros but it can be as many as you want. The second implies that the number of zeros is undefined, so the number itself is undefined. The notation "0.000....1" is not valid.

Yes, but I don't know what symbol to use instead.

 

[PeroK]

My friend's answer to this question is: I can add any number of zeros before 1, but even if I add trillions of zeros, when I stop. It is a finite number greater than zero (but it doesn't mean that you can't continue to add zeros). When I keep adding zeros and never stop, it is an infinite number equal to zero, because if I keep adding zeros, there is nowhere to put the last one, but I have some doubts about his point of view. How can a number be both finite and infinite?

If you never stop putting zeroes, then you simply have an infinite decimal expansion of the number zero. If you stop putting zeroes and put a 1, then you have a number greater than zero.

You either stop at some point or you don't.

 

[weirdoguy]

How can a number be both finite and infinite?

Infinite number of digits does not mean that number is infinite (there is no such thing as infinite real number). 0,(9) has inifinite number of digits but is equal to 1.

 

[PeroK]

The reason that an infinite decimal expansion can represent a finite number is that each digit represents a number ten times less than the previous digit. This similar to an infinite geometric series, which you may have studied.

 

[weirdoguy]

This similar to an infinite geometric series

Can one even rigorously define an infinite decimal expansion without using geometric series?

 

[nmz]

If you never stop putting zeroes, then you simply have an infinite decimal expansion of the number zero. If you stop putting zeroes and put a 1, then you have a number greater than zero.

You either stop at some point or you don't.

Do you think I can continue to add zeros to make it smaller after I stop adding zeros? That is to say, no matter how many zeros I add, there is no limit, but if I want to have a 1 at the end of it, I can't keep adding zeros. Even if I keep adding, I have to stop somewhere?

 

[Ibix]

Do you think I can continue to add zeros to make it smaller after I stop adding zeros? That is to say, no matter how many zeros I add, there is no limit, but if I want to have a 1 at the end of it, I can't keep adding zeros. Even if I keep adding, I have to stop somewhere?

Perhaps it would be clearer to say "if I have a decimal point followed by a large number of zeroes then a one, is there ever a number of zeroes so large that this isn't a number?" To which the answer is no, you can have any finite number of zeroes and you can always insert one more. And one more, and one more...

You couldn't have an infinite number of zeroes, though, because by definition there is no last zero to put the one after.

 

[nmz]

也许更清楚地说“如果我有一个小数点，后面跟着大量的 0，然后是 1，有没有一个 0 的数量大到这不是一个数字？答案是否定的，你可以有任意数量的零，并且你总是可以再插入一个。再来一个，再来一个......

但是，你不能有无限数量的 0，因为根据定义，没有最后一个 0 可以放在后面的 0。

Perhaps it would be clearer to say "if I have a decimal point followed by a large number of zeroes then a one, is there ever a number of zeroes so large that this isn't a number?" To which the answer is no, you can have any finite number of zeroes and you can always insert one more. And one more, and one more...

You couldn't have an infinite number of zeroes, though, because by definition there is no last zero to put the one after.

In other words, I can insert any number of zeros before 1 and after the decimal point. As long as the number of zeros is not infinite, it is a finite number greater than zero.

 

[PeroK]

Can one even rigorously define an infinite decimal expansion without using geometric series?

We need something more general than geometric series. In fact, that any bounded monotone sequence must converge to some finite number. This is effectively the Archimedean principle for real numbers.

 

[nmz]

In other words, I can insert any number of zeros before 1 and after the decimal point. As long as the number of zeros is not infinite, it is a finite number greater than zero.

Does this belong to the concept of infinitesimal?

 

[nmz]

Perhaps it would be clearer to say "if I have a decimal point followed by a large number of zeroes then a one, is there ever a number of zeroes so large that this isn't a number?" To which the answer is no, you can have any finite number of zeroes and you can always insert one more. And one more, and one more...

You couldn't have an infinite number of zeroes, though, because by definition there is no last zero to put the one after.

Do you think Does this belong to the concept of infinitesimal?

 

[weirdoguy]

infinitesimal

In my opinion, infinitesimals are a very outdated concept. There may be useful in non-standard analysis, but in real one it's better to use differential forms. But my specialisation was differential geometry, so I'm not objective

 

[FactChecker]

Do you think Does this belong to the concept of infinitesimal?

That's not a bad idea, but it's too little, too late. The symbol $dx$ is already well established as an infinitesimal in the $x$ direction and it has the advantage of indicating the direction.

 

[jbriggs444]

Do you think Does this belong to the concept of infinitesimal?

There is no room for infinitesimals in the standard reals.

You are up against the problem that you have a notion. And a notation for the notion. But no valid reference for that notation to mean anything.

The standard way of thinking about decimal representation is that we have this set of positions laid out in order. The set of positions (aka the index set) is important. It has to be a countably infinite set. The ordering is also important. Not just any order will do. The standard order of the natural numbers is natural and is used. We have a decimal point somewhere in the string. This is an anchor so that we can apply place value.

Every valid decimal string of this sort has a value as a real number. That real number can be found as the sum of a geometric series, as the number converged to by a sequence of truncated decimal forms, or as the equivalence class of the set of Cauchy sequences of which this sequence of truncated decimal forms is a representative or exemplar. (If you have gone through a construction of the real numbers in terms of Cauchy sequences, this last may be particularly appealing)

Our various finite notations e.g. "3.3333..." or "0.1" are interpreted to denote a one-way infinite decimal string that can be further understood as a real number.

If you try to make a decimal string that contains infinitely many zeroes followed by a non-zero digit then you will have either have messed up the index set or the ordering of the index set. You will not have a valid decimal representation.

If you come up with a notation for this, it means that the notation does not have a recognized valid meaning.

It is possible to relax the rules on ordering and get something other than real numbers. For instance, allowing infinite digits to the left of the point you might get something like the p-adics.

 

[nmz]

In other words, I can insert any number of zeros before 1 and after the decimal point. As long as the number of zeros is not infinite, it is a finite number greater than zero.

If there are as many zeros in front of 1 as there are Grimm numbers, is this number still limited and greater than zero?
Greer's constant number is by far the largest significant mathematical number.

 

[jbriggs444]

If there are as many zeros in front of 1 as there are Grimm numbers, is this number still limited and greater than zero?
Greer's constant number is by far the largest significant mathematical number.

Graham's number is still finite. The real number whose decimal expansion contains Graham's number of zeroes to the right of the decimal point followed by a 1 is still finite and not zero.

As is the real number whose decimal expansion is one digit longer.

Or the one whose decimal expansion has $S(g_{64})$ zero digits before the 1. (This number of digits being the running time of the Busy Beaver with Graham's number of states)

For another number larger than Graham's number, see the TREE function referenced in the article on Graham's number.

 

[nmz]

In other words, I can insert any number of zeros before 1 and after the decimal point. As long as the number of zeros is not infinite, it is a finite number greater than zero.

If I can add any number of zeros before 1, then I will keep increasing until a random time stops increasing. Is it still a finite number greater than zero?

 

[PeroK]

If I can add any number of zeros before 1, then I will keep increasing until a random time stops increasing. Is it still a finite number greater than zero?

Okay, you win. $0.00001 = 0$

Happy now?

 

[phinds]

If I can add any number of zeros before 1, then I will keep increasing until a random time stops increasing. Is it still a finite number greater than zero?

As you have been told several times, yes. Asking the same question over and over isn't going to change the answer.

 

[nmz]

If I can add any number of zeros before 1, then I will keep increasing until a random time stops increasing. Is it still a finite number greater than zero?

Perhaps my clearer answer is this: If the number of zeros before 1 is so large that I don't know how many there are, is it too big to be an actual number?

 

[weirdoguy]

that I don't know how many there are

Mathematics in not about what nmz knows. If the number of zeros is finite, then it is a number. If it is not, then not. Rephrasing your questions won't change the answer.