Taking issue with "well established" mathematical theories

[Kalyyn]

Alright, so google has produced no satisfactory answers, so in desperation I come to you. You see, I had an argument with my Stats teacher over 0!. He claims, and is correct, that 0! is, by definition, 1. I do not disagree with him, but the definition itself. After about half an hour on google, the only information I got was
"This has to be true for factorials to work"
and
"This definition is not arbitrary and has a reason"
Obviously they all go on to state the reason... oh wait, they don't. Not a single site I looked at offered proof that 0! is 1. So here's my line of thinking.
2! is 2x1, or 2, and 3! is 3x2x1, or 6. Therefore, 0! is 0 multiplied by... nothing. Which by logic should equal 0. However, apparently, according to the theory of factorials, logic is not a priority.
There is no possible way you can make a physical model to represent this because, unlike this mathematical "law", reality considers logic to be important. So lets look at what has happened here. We create a set of mathematical laws called factorials. They work pretty well, everything seems in order with the universe. But, wait, zero screws everything up. So now we're going to start breaking much older and more firmly established laws to make our new laws work when they rightfully shouldn't. Instead of changing the new laws to work with reality, we're going to say "fuck reality, we're right". Can you see why this frustrates me?

Anyway, if any of you out there can actually PROVE that 0!=0, through a mathematical proof or physical model, then you'll save me a world of irritation.

Edited to remove exponents

[rantus]

x^3 = x*x*x
x^2 = x*x which is x^3 divided by x
x^1 = x which is x^2 divided by x
x^0 = x^1 divided by x, which is 1.

Its maybe not exactly a proof but I always wondered why it was 1 until seeing that.

As for 0!, how can this possibly be 1?
If 2! is 2x1, and 3! is 3x2x1 would 1! just be 1? To me that sounds like 0! should be undefined, reading your post has just confused me to hell.

[Daginni]

I'm a DOC! Not a Mathematician!

[Renley]

A quick google search found me an answer for x^0, and I didn't look for 0!.

It uses the laws of indices:

(x^a)(x^b) = x^(a+b)
(x^a) / (x^b) = x^(a-b)

Now take (x^n) / (x^n), where n is any non-zero integar. This would be equivalent to saying x^(n-n), which is x^0.

[Kalyyn]

x^3 = x*x*x
x^2 = x*x which is x^3 divided by x
x^1 = x which is x^2 divided by x
x^0 = x^1 divided by x, which is 1.

Its maybe not exactly a proof but I always wondered why it was 1 until seeing that.

As for 0!, how can this possibly be 1?
If 2! is 2x1, and 3! is 3x2x1 would 1! just be 1? To me that sounds like 0! should be undefined, reading your post has just confused me to hell.

That actually does make a lot of sense.
Editing post for relevance then.

[Phlange]

3! = 4!/4
2! = 3!/3
1! = 2!/2
0! = 1!/1 = 1

Simple :P

[Renley]

Nevermind. Ignore this post.

[Magnusdragon]

3! = 4!/4
2! = 3!/3
1! = 2!/2
0! = 1!/1 = 1

Simple :P

x! = (x+1)!/(x+1)?
IT ALL MAKES SENSE NOW

[Zulandia]

x! = (x+1)!/(x+1)?

If that's actually intended to be a question maybe this will help clarify:

3! = (1X2X3X4) /4
= 1X2X3


2! =(1X2X3) / 3
= 1X2



1! =(1X2) / 2
= 1



0! = 1 /1
=1

[drinoff]

for x^0=1 Why?
Answer:If is 0 copies of the number x, all multiplied together, then should be the "empty product" with no factors multiplied together. In mathematics, the empty product is defined to be 1, because multiplying by nothing at all is the same as multiplying by 1.

for 0!=1 Why?
Answer:
Experimenting with factorials, we come up with n!=n(n-1)!. For example 17!=17x(16!):

16!=1x2x...x16
17!=(1x2x...x16)x17

That equation (n!=n(n-1)!) just dictated to us where to put the parentheses. By making n=1, we can find 0!:

1!=1(0!)
0!=1

And, it turns out that 0!=1 works very well in many situations (in probability, for example).


Addendum:

The above proof that 0!=1 is based upon n!=n(n-1)!, which is in turn based upon the definition of factorial. So, it would seem to be a valid proof. But, 0! cannot be defined directly from the definition of factorial. So, mathematicians like to define 0! as 1, without proving it. So, the proof just amounts to a demonstration that defining 0! as 1 is consistent with the definition of factorial.

I received email saying that my proof that 0!=1 is invalid because 0! is a constant, and you cannot solve for a constant. Wrong. I can find 4! (in a number of ways), and 4! is a constant. I can solve for the square root of 7, probably using my calculator, and it is a constant, too. There are numerous ways of finding pi, and it too is a constant.

n! is also the number of permutations (ways of arranging) exactly n things. It makes sense to say that there is one way to arrange zero things. So again, 0!=1.


http://www.jimloy.com/algebra/zero-f.htm

[Kalyyn]

Some excellent answers here. My mind is much more at ease knowing that there is at least some logical reason for all this madness.

[Worshaka]

Alright, so google has produced no satisfactory answers, so in desperation I come to you. You see, I had an argument with my Stats teacher over 0!. He claims, and is correct, that 0! is, by definition, 1. I do not disagree with him, but the definition itself. After about half an hour on google, the only information I got was
"This has to be true for factorials to work"
and
"This definition is not arbitrary and has a reason"
Obviously they all go on to state the reason... oh wait, they don't. Not a single site I looked at offered proof that 0! is 1. So here's my line of thinking.
2! is 2x1, or 2, and 3! is 3x2x1, or 6. Therefore, 0! is 0 multiplied by... nothing. Which by logic should equal 0. However, apparently, according to the theory of factorials, logic is not a priority.
There is no possible way you can make a physical model to represent this because, unlike this mathematical "law", reality considers logic to be important. So lets look at what has happened here. We create a set of mathematical laws called factorials. They work pretty well, everything seems in order with the universe. But, wait, zero screws everything up. So now we're going to start breaking much older and more firmly established laws to make our new laws work when they rightfully shouldn't. Instead of changing the new laws to work with reality, we're going to say "fuck reality, we're right". Can you see why this frustrates me?

Anyway, if any of you out there can actually PROVE that 0!=0, through a mathematical proof or physical model, then you'll save me a world of irritation.

Edited to remove exponents

I think you're going to be disappointed... the reason 0! = 1 is to provide consistency with our known number field otherwise the factorial calculations simply don't work. So it's been agreed that 0! = 1, not because it's proved to be so but to make it work.

In particular it has issues with C(n,k) defined as n!/(n-k)!k!. Its intuiatve that C(n,0) should equal 1 but n!/n!0! if 0! were to = 0 wouldn't be able to be calculated. If 0! = 1 then the C(n,k) operation works fine.

Now before you can say that this is bad mathematics, that's not the case... you can actually define functions to work over specific domains and do whatever you like. There is nothing wrong with the statement f(x) = x^2 for x != 3, when x = 3 f(x) = 0.

therefore we can define n! recursively as:
0! = 1
n! = n(n-1)!

So it's not so much that we prove 0! = 1, we are defining the structure of factorials in this manner because it's convienent to do so.

[Kalyyn]

I think you're going to be disappointed... the reason 0! = 1 is to provide consistency with our known number field otherwise the factorial calculations simply don't work. So it's been agreed that 0! = 1, not because it's proved to be so but to make it work.

In particular it has issues with C(n,k) defined as n!/(n-k)!k!. Its intuiatve that C(n,0) should equal 1 but n!/n!0! if 0! were to = 0 wouldn't be able to be calculated. If 0! = 1 then the C(n,k) operation works fine.

Now before you can say that this is bad mathematics, that's not the case... you can actually define functions to work over specific domains and do whatever you like. There is nothing wrong with the statement f(x) = x^2 for x != 3, when x = 3 f(x) = 0.

therefore we can define n! recursively as:
0! = 1
n! = n(n-1)!

So it's not so much that we prove 0! = 1, we are defining the structure of factorials in this manner because it's convienent to do so.

You are correct. I am disappointed.

[haxartus]

x^3 = x*x*x
x^2 = x*x which is x^3 divided by x
x^1 = x which is x^2 divided by x
x^0 = x^1 divided by x, which is 1.

Its maybe not exactly a proof but I always wondered why it was 1 until seeing that.

x^1/x^1=x^(1-1)=x^0
x^1/x^1=x/x=1
x^0=1

[Deleted]

Oh this is a nice thread
But in the end, the only answer is... there is no proof. And there will never be a proof, since there cannot even be anything proven. 0! := 0 is a definition. The factorial operator is defined (as written before) as:

0! := 0
n! := n*(n-1)!

Look at this tiny sweet colon. This is what's interesting. A definition is nothing you prove, it's something that is, otherwise it would be a derivative. And yes, there can be different definitions for same subjects, like in geometry and Non-Euclidean geometry where a line's definition is not the same. Or look at axioms... they all are definitions and so they cannot be proven.

[ryoukthebloodelf]

Stats the class that makes me loose the will to live

[Serenais]

Oh this is a nice thread
But in the end, the only answer is... there is no proof. And there will never be a proof, since there cannot even be anything proven. 0! := 0 is a definition. The factorial operator is defined (as written before) as:

0! := 0
n! := n*(n-1)!

Look at this tiny sweet colon. This is what's interesting. A definition is nothing you prove, it's something that is, otherwise it would be a derivative. And yes, there can be different definitions for same subjects, like in geometry and Non-Euclidean geometry where a line's definition is not the same. Or look at axioms... they all are definitions and so they cannot be proven.

0! = 1. What you wrote would essentially make any n! = 0.
Also, you do have mathematical proofs, and quite a few types. You can define all you want, but before you can prove that your theory is correct, it won't be anything more than a curious thought. It's all nice and stuff saying that 1+1=2, but you have to show that it can't also happen that 1+1=3. With the definition of "+", it is quite easy, provided that you do it on real numbers.
If you want to read a bit more on the subject, there's always wikipedia, that has quite exhaustive material on mathematical proof.

[endersblade]

I'm sorry, but after reading this thread, I figured out exactly how friggin' stupid and ignorant I am. Gonna go take some aspirin or something, my head is about to explode from all the math in here >.> I /bow to your superior intellects.

[ownager]

@Serenais : Since your so eager with this topic, u should google Kurt Gödel and his " Incompleteness Theorem". He has shown , that you cannot decide the correctness of a theorie if it is sufficient large enough. And if you do, you should also google what a theory in science means, because if you talk about a theory it is by far not this structure , if a scientist talks about it.


Or i simply tell you :

In science , a Theory is a set of fundamental assumptions (Axioms) united with every conclusion that can be logically derived by these Axioms.

So my theory of fractionals would lead to :

Axiom 1 : 0! = 1
Axiom 2 : n! = n*(n - 1)!

Conclusions :

1! = 1
2! = 2
...

Theory : {0! = 1, n! = n*(n - 1)! , 1! = 1, ... }

I could define another theory by :

Axiom 1 : 0! = 5
Axiom 2 : n! = (n - 1) + 2

Conclusions :

1! = 7
2! = 9

...

No mathematic will ever oppose you, as long as you logically derive the conclusions (by the laws of logic). The question here is simply, does this second definition of the axioms make sense? Mathematical definitions are mostly chosen because they make sense.

It's all nice and stuff saying that 1+1=2, but you have to show that it can't also happen that 1+1=3

You can show this by using the field axioms. But you cannot show that the field axioms (basic definitions) are correct. They are simply making sense.

[Serenais]

@Serenais : Since your so eager with this topic, u should google Kurt Gödel and his " Incompleteness Theorem". He has shown , that you cannot decide the correctness of a theorie if it is sufficient large enough. And if you do, you should also google what a theory in science means, because if you talk about a theory it is by far not this structure , if a scientist talks about it.


Or i simply tell you :

In science , a Theory is a set of fundamental assumptions (Axioms) united with every conclusion that can be logically derived by these Axioms.

So my theory of fractionals would lead to :

Axiom 1 : 0! = 1
Axiom 2 : n! = n*(n - 1)!

Conclusions :

1! = 1
2! = 2
...

Theory : {0! = 1, n! = n*(n - 1)! , 1! = 1, ... }

I don't dispute any of those, and I believe I didn't say what exactly a theory is Neither do I dispute the fact that the more complex theory and the field it describes gets, the harder the proof is, sometimes making it impossible to have a proof for a theory that would be universaly valid. Even my example of 1+1=2 and not 3 on real numbers fits this, because both "+" and real numbers are well defined subsets of functions and numbers. I just said that a theory requires some proof to be considered valid on at least some area of definition, otherwise it is just that, a set of assumptions that happens to fit conclusions derived from those assumptions. You need to show that at least sometimes, those assumptions are correct. To fit the topic of this forum, I might assume that a paladin can block up to 100%, and therefore state a theory of a paladin being able to solo everything with the right setup of gear. However, my assumption is proven wrong by looking at the battle.net armory - my initial assumption was wrong, and therefore my theory of immortal paladin is invalid. Similarly, I can't say that 1+1 always equals 2 - the mathematic science is large enough to find examples where that is not true, and, for example, 1+1=1. A proof therefore needs a well defined set of conditions under which it becomes valid, and with it, the theory.