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### Is 0 a natural number?

Posted: Tue Jan 13, 2015 10:30 pm UTC

Discuss.

### Re: Is 0 a natural number?

Posted: Tue Jan 13, 2015 10:38 pm UTC
Oh wow, I didn't know this was a thing. But wiki tells me it's a thing. So it must be a thing.

I agree, the answer is obviously yes.

### Re: Is 0 a natural number?

Posted: Tue Jan 13, 2015 10:42 pm UTC
Sometimes it might sneak in. When it's convenient to let it to.
Other times it is more suitable to cast it out.

But for the sake of war.
No, it is not.

### Re: Is 0 a natural number?

Posted: Wed Jan 14, 2015 2:49 am UTC
When you're counting your herd, you start with one, two, more*, ..., more sheep. If you don't have a herd, you simply don't count and say you have no sheep.

So obviously the naturals start at 1... and it's a finite set because you can't have a herd of more than many, many sheep.

0 is just a stupid idea that has no real-world application. It really is *puts on sunglasses* unnatural.

*some people tend to say "three" after two and a few may even say "four" and "five". However, eventually everyone will give up making up names and will just carve a mark on the tally stick for each sheep they count.

### Re: Is 0 a natural number?

Posted: Wed Jan 14, 2015 3:26 am UTC
The natural numbers are simply those that occur as the sizes of finite sets (including the empty set). Any other definition is unnatural.

### Re: Is 0 a natural number?

Posted: Wed Jan 14, 2015 4:59 am UTC
I say yes, because we already have a name the set {1, 2, 3, ...}, which is "positive number".

### Re: Is 0 a natural number?

Posted: Wed Jan 14, 2015 5:13 am UTC
Sizik wrote:I say yes, because we already have a name the set {1, 2, 3, ...}, which is "positive number".

Strange, I always thought 0.1 was a positive number.

### Re: Is 0 a natural number?

Posted: Wed Jan 14, 2015 5:22 am UTC
Way back when, I learned that "whole" numbers are {1, 2, ...} and natural numbers are {0, 1, ...}.

Or maybe it was the other way around. Maybe I should say "I was taught" instead.

### Re: Is 0 a natural number?

Posted: Wed Jan 14, 2015 8:03 am UTC
Thesh wrote:
Sizik wrote:I say yes, because we already have a name the set {1, 2, 3, ...}, which is "positive number".

Strange, I always thought 0.1 was a positive number.

Pedants can say "positive integers".

### Re: Is 0 a natural number?

Posted: Wed Jan 14, 2015 8:17 am UTC
We can also say "Nonnegative Integers" to describe {0, 1, 2, ...}, rendering the argument moot.

### Re: Is 0 a natural number?

Posted: Wed Jan 14, 2015 3:50 pm UTC
EvanED wrote:Way back when, I learned that "whole" numbers are {1, 2, ...} and natural numbers are {0, 1, ...}.

Or maybe it was the other way around. Maybe I should say "I was taught" instead.

I'm in the same boat, along with the slight uncertainty if it was the other way around or not. If it was the other way around, then I declare the teacher to have been wrong. I say 0 should be natural (as what is more natural than wearing 0 clothing?) and that 0 should not be whole (as clearly you can't have a whole pie if you have 0 pies).

### Re: Is 0 a natural number?

Posted: Wed Jan 14, 2015 4:17 pm UTC
You're all fools.
The "whole" numbers are clearly Z\{0}.

### Re: Is 0 a natural number?

Posted: Wed Jan 14, 2015 4:46 pm UTC
Xenomortis wrote:You're all fools.
The "whole" numbers are clearly Z\{0}.

Surely you mean Z\{0,1}.

https://en.wikipedia.org/wiki/Natural_number#History wrote:Many Greek mathematicians did not consider 1 to be "a number", so to them 2 was the smallest number.

(fun fact: all negatives are larger than 2)

### Re: Is 0 a natural number?

Posted: Wed Jan 14, 2015 5:29 pm UTC
EvanED wrote:Way back when, I learned that "whole" numbers are {1, 2, ...} and natural numbers are {0, 1, ...}.

Or maybe it was the other way around. Maybe I should say "I was taught" instead.
I'm certain that in high school I was taught that "whole numbers" include zero and naturals don't. My friend from India was taught the same thing and still insists it's the One True Way.

As far as I'm concerned though, "whole number" is a synonym for "integer".

### Re: Is 0 a natural number?

Posted: Wed Jan 14, 2015 8:18 pm UTC
Sizik wrote:I say yes, because we already have a name the set {1, 2, 3, ...}, which is "positive number integers".

I agree on grounds of this being cleaner than calling {0,1,2,...} the "nonnegative integers". No point in giving both shorter names to {1,2,3,...}.

0 is extremely natural, anyway (we have 0 of most things, rather than 1 or even 2). It may have taken human civilization centuries to discover it, but that's more a condemnation of humans than a point against its naturality.

### Re: Is 0 a natural number?

Posted: Wed Jan 14, 2015 9:41 pm UTC
Another advantage to including 0 is that then the natural numbers have an identity element under addition, which I would say is a fairly natural thing to have. I think it's also a bit nicer when defining numbers in terms of set theory. 0 is the empty set, and then you define a successor function based on sets. Usually either S(n) = Power set of n, or S(n) = {n}.

### Re: Is 0 a natural number?

Posted: Tue Jan 20, 2015 9:44 pm UTC
I learned to code on C, so yes.

### Re: Is 0 a natural number?

Posted: Fri Jan 23, 2015 6:56 am UTC
Most people say yes. Maybe there is no reason and our ancestors called it a zero and wrote it like this; and they defined it means "nothing", we just follow our ancestors.

### Re: Is 0 a natural number?

Posted: Sat Jan 24, 2015 8:39 am UTC
stuffedanimalstoys wrote:Most people say yes. Maybe there is no reason and our ancestors called it a zero and wrote it like this; and they defined it means "nothing", we just follow our ancestors.

0 is not just nothing* though, it's specifically an amount. Without this concept, nothingness is only related to existence, not to amounts –either there isn't anything or you can count the number of things.

*obligatory cross-reference

### Re: Is 0 a natural number?

Posted: Mon Jan 26, 2015 9:32 am UTC
stuffedanimalstoys wrote:Most people say yes. Maybe there is no reason and our ancestors called it a zero and wrote it like this; and they defined it means "nothing", we just follow our ancestors.

Except that historically, zero was not included in the natural numbers. Including zero is a more recent innovation.

### Re: Is 0 a natural number?

Posted: Fri Feb 13, 2015 5:41 pm UTC
Flumble wrote:When you're counting your herd, you start with one, two, more*, ..., more sheep. If you don't have a herd, you simply don't count and say you have no sheep.

So obviously the naturals start at 1... and it's a finite set because you can't have a herd of more than many, many sheep.

0 is just a stupid idea that has no real-world application. It really is *puts on sunglasses* unnatural.

*some people tend to say "three" after two and a few may even say "four" and "five". However, eventually everyone will give up making up names and will just carve a mark on the tally stick for each sheep they count.

When you’re counting, you use the counting numbers. I thought that was obvious. So the counting numbers start at 1. The naturals start at 0.

### Re: Is 0 a natural number?

Posted: Fri Feb 13, 2015 7:54 pm UTC
Qaanol wrote:
Flumble wrote:When you're counting your herd, you start with one, two, more*, ..., more sheep. If you don't have a herd, you simply don't count and say you have no sheep.

So obviously the naturals start at 1... and it's a finite set because you can't have a herd of more than many, many sheep.

0 is just a stupid idea that has no real-world application. It really is *puts on sunglasses* unnatural.

*some people tend to say "three" after two and a few may even say "four" and "five". However, eventually everyone will give up making up names and will just carve a mark on the tally stick for each sheep they count.

When you’re counting, you use the counting numbers. I thought that was obvious. So the counting numbers start at 1. The naturals start at 0.

I don't understand how you even get such a ridiculous premise. Of course nature also counts from 1! There's no difference between these "counting numbers" and "natural numbers".

...except that nature probably doesn't name any of the numbers. I've never heard her count.

### Re: Is 0 a natural number?

Posted: Fri Feb 13, 2015 8:01 pm UTC
If 0 is such a natural number, then what is the natural logarithm of 0?

### Re: Is 0 a natural number?

Posted: Fri Feb 13, 2015 8:16 pm UTC
Thesh wrote:If 0 is such a natural number, then what is the natural logarithm of 0?

Why, its log equals its reciprocal, of course! What could be more natural than that?

### Re: Is 0 a natural number?

Posted: Fri Feb 13, 2015 8:42 pm UTC
So we have a new definition: x is a natural number, if and only if ln(x) = x-1

### Re: Is 0 a natural number?

Posted: Fri Feb 13, 2015 10:01 pm UTC
Thesh wrote:So we have a new definition: x is a natural number, if and only if ln(x) = x-1

So then the only natural numbers are 0 and exp(W(1))? Or maybe there are some complex natural numbers as well.

### Re: Is 0 a natural number?

Posted: Wed Mar 04, 2015 7:27 am UTC

### Re: Is 0 a natural number?

Posted: Fri Mar 27, 2015 12:25 pm UTC
Shortly, in my opinion zero is not a number at all. It has been derived by too means, which disrupts its “natural” existence. First is - the reciprocal of infinity. Or the opposite of absolutely all things. “Absolutely all things” is not well defined, in fact it is very badly defined, so even “nothing” itself is not very clear.
Second – zero is used as middle point between symmetric quantities, such as numbers, coordinates, etc. The whole idea is totally wrong. There are no symmetric things and zero is not their center (it even doesn’t exist). Negative numbers either don’t exist or they are not a mirror of the positive ones.
1 + (-1) = 0 is nonsense.
Something cannot exist if there is such annihilation. If we put a point on a line and say this is our zero point, left and right in reference to it are not equal things. All chronology systems don’t have a zero year. In information technologies they count from zero just because the numeration system they use has the digit “0”. It is not obligatory of course, one could use a bijective system.
When discovering numbers, zero comes third, if ever comes.
First is 2 – the first real natural number. One (1) is the second and is problematic too. It is definitely not a number, but at least does exist. It is something indefinite. It is just something.
If I see one apple, in fact I don’t know that it is one apple, two apples or one hundred apples. It is just “something”. If I see another apple, then I will know what apple is and will know that I see two apples.
There is no subtracting by addition of negative numbers. Just “putting aside”. Numbers are not lost nor “annihilated” because they are limited.
The set of all natural numbers is finite. It has a final limit, which is the biggest number of all numbers. It can be called L. L is definite, finite, but unknown (yet).
Therefore it is suitable that we use a bijective ten-adic numeration system with a little variation.
If a system which uses polynomial representation looks like this:

Nk*b^k + … + N3*b^3 + N2*b^2 + N1*b^1 + N0*b^0,

where b is the base and Nk is a digit from a set of b digits.
We can present it a little different, too – adding a coefficient, say – p:

Nk*p*b^k + … + N3*p*b^3 + N2*p*b^2 + N1*p*b^1 + N0*p*b^0

and there are some conditions for the set of digits.
In our case, let b=10, p=0.5, and the digits are:

2, 4, 6, 8, A (10), C (12), E (14), G (16), I (18), K (20).

Note that 2 here means exactly 2 i.e. “two” and so on.
It will look like:

Nk*0.5*10^k + … + N3*0.5*10^3 + N2*0.5*10^2 + N1*0.5*10^1 + N0*0.5*10^0

or

(Nk*10^k )/2+ … + (N3*10^3)/2 + (N2*10^2)/2 + (N1*10^1)/2 + (N0*10^0)/2

If we count it will be:
2 1
4 2
6 3
8 4
A 5
C 6
E 7
G 8
I 9
K 10
22 11
24 12

2I 19
2K 20
42 21
44 22

II 99
IK 100
K2 101

KK 110
222 111

and so on.
Transformations . From this system to the ordinary – just apply the formula. In the opposite way, I suggest something like this:
1. If the number doesn’t contain zeroes, just replace the digits correspondingly.
For instance - 12345 is 2468A.
2. If the number contains zeros, first present it as sum of numbers ending by zero. Subtract 10 and transform them in the new system. Finally sum up.
Example – 10501
is:
10 000
+ 500
+ 1

10 000 -10 +10 = 9 990 +10
i.e. IIIK
500 -10 +10 = 490 +10,
i.e. 8IK
1 is 2.
So, the sum:
IIIK
+ 8IK =
K8IK (10500)
And:
K8IK
+ 2 =
K8K2
Eventually – 10501 is K8K2.
Someone can say – ok, if we replace 1 with 2 – what’s the difference? The difference is in thinking. Now we think of one as two halves and it is more definite, because we first call the first natural number – two.

### Re: Is 0 a natural number?

Posted: Fri Mar 27, 2015 4:11 pm UTC
keed wrote:Shortly, in my opinion zero is not a number at all. It has been derived by too means, which disrupts its “natural” existence. First is - the reciprocal of infinity. Or the opposite of absolutely all things. “Absolutely all things” is not well defined, in fact it is very badly defined, so even “nothing” itself is not very clear.

Second – zero is used as middle point between symmetric quantities, such as numbers, coordinates, etc. The whole idea is totally wrong. There are no symmetric things and zero is not their center (it even doesn’t exist). Negative numbers either don’t exist or they are not a mirror of the positive ones.

1 + (-1) = 0 is nonsense.
Something cannot exist if there is such annihilation. If we put a point on a line and say this is our zero point, left and right in reference to it are not equal things. All chronology systems don’t have a zero year. In information technologies they count from zero just because the numeration system they use has the digit “0”. It is not obligatory of course, one could use a bijective system.

When discovering numbers, zero comes third, if ever comes.
First is 2 – the first real natural number. One (1) is the second and is problematic too. It is definitely not a number, but at least does exist. It is something indefinite. It is just something.
If I see one apple, in fact I don’t know that it is one apple, two apples or one hundred apples. It is just “something”. If I see another apple, then I will know what apple is and will know that I see two apples.

There is no subtracting by addition of negative numbers. Just “putting aside”. Numbers are not lost nor “annihilated” because they are limited.
The set of all natural numbers is finite. It has a final limit, which is the biggest number of all numbers. It can be called L. L is definite, finite, but unknown (yet).

Someone can say – ok, if we replace 1 with 2 – what’s the difference? The difference is in thinking. Now we think of one as two halves and it is more definite, because we first call the first natural number – two.

### Re: Is 0 a natural number?

Posted: Sun Mar 29, 2015 9:17 am UTC
I got lost at the bit about four simultaneous 24-hour days.

### Re: Is 0 a natural number?

Posted: Fri Apr 17, 2015 8:47 pm UTC
Copper Bezel wrote:I got lost at the bit about four simultaneous 24-hour days.

That's because 4 is the first natural number. Anything less than 4 is just bad. You can't have a cube without four sides. All numbers can be derived from 4, including 4 AND 24. So, you see, this proves timecube fact.

### Re: Is 0 a natural number?

Posted: Fri Apr 17, 2015 9:05 pm UTC
And don't forget about the Fourier Transform.

### Re: Is 0 a natural number?

Posted: Thu Mar 16, 2017 1:35 pm UTC
We have Z^+ for {1,2,3,.....}. So N may as well include 0, lest we have to write N\{0} all the time.

But if you're using English, "nonnegative integers" and "positive integers" are far more unambiguous and thus superior.

### Re: Is 0 a natural number?

Posted: Tue Mar 21, 2017 4:15 am UTC
Xenomortis wrote:And don't forget about the Fourier Transform.

Now I'm wondering if there is an algorithm to find the fouriest base for a given number that is better than a linear search.

### Re: Is 0 a natural number?

Posted: Tue Mar 21, 2017 9:25 pm UTC
Derek wrote:
Xenomortis wrote:And don't forget about the Fourier Transform.

Now I'm wondering if there is an algorithm to find the fouriest base for a given number that is better than a linear search.

How about base (1 - 4/n)?

### Re: Is 0 a natural number?

Posted: Wed Mar 22, 2017 2:09 am UTC
Qaanol wrote:How about base (1 - 4/n)?

For a base less than 1 wouldn't it be simpler to use 1/b? Using 1/(1-4/n), then 0.444... = n. At least for n > 4, for small n is gets weird. n=3 and n=1 are negative bases, and it doesn't work at all for n=4 (undefined base) and n=2 (base -1, 0.444... does not converge). For all n it's non-standard notation, since you wouldn't normally use the digit 4 for bases with absolute value less than or equal to 4.

If we stick to positive integer bases with standard notation, I would expect base 5 to be the fouriest base for the majority (almost all?) integers, but that's the best conclusion I can reach.