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**Tredici****Member**- Registered: 2005-12-12
- Posts: 28

More a question.

This may have been asked before, but I've struggled finding anything vaguely similar. Anyway, to the question.

Why does 0! = 1?

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**MathsIsFun****Administrator**- Registered: 2005-01-21
- Posts: 7,626

I think I can show you why people want 0! = 1

Let's take the formula for finding permutations (the number of different ways objects can be arranged):

P(n,r) = n! / (n-r)!

(The permutation of n objects, taken r at a time)

So, for example, if we have 3 objects (a,b,c) and take them 2 at a time we can have

P(3,2) = 3! / (3-2)! = 3!/1! = (3×2×1)/1 = 6/1 = 6 (and they are: ab,ac,bc,ba,ca,cb)

If we asked for 3 objects taken 3 at a time, the formula becomes

P(3,3) = 3! / (3-3)! = 3!/0! = 6/???

So, if we say that 0! = 1 then we get the right answer P(3,3) = 6 (and they are: abc,acb,bca,bac,cab,cba)

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman

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**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

It is basically an identity created so that the rules for factorials continue to work. Like a^0 = 1

Not true. You can prove a^0=1 using group theory (an offset of set theory). But that requires quite a bit of knowledge about sets and groups. A simpler proof I did before is:

http://www.geocities.com/rshadarack/proof.pdf

The real problem is in assuming that all mathematics are precise and factual. Like any science, no matter how stringent they appear to be regarding proofs, they all rely upon arbitrary assumptions at their foundations.

The only thing math relies on are definitions. And that is simply what we call things with certain properties. Now that isn't to say there isn't ambiguity in math. But there is far more ambiguity in any science than math.

"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."

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**krassi_holmz****Real Member**- Registered: 2005-12-02
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Interesting. Why does a^0=1?

If we assept the rules (a^x)(a^y)=a^(x+y) and (a^x)/(a^y)=a^(x-y) this FACT follows from:

(a^x)/(a^x)=1=a^(x-x)=a^0=>a^0=1.

IPBLE: Increasing Performance By Lowering Expectations.

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**krassi_holmz****Real Member**- Registered: 2005-12-02
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And for 0!=1 question, we can use gamma function:

I suppose that we can use the rule n!/(n-1)!=n that is valid for all n>=1. So we can accept that it's true and for 1:

1!/0!=1 =>0!=1

But we also can define the function x! using the following definition:

(If x elem N(naturals) then x!=Product[i,{i = 1 to x}]

x!

(If x=0 then x!=1

So that definition requiers 0! to be 1 so the question Why doesn't have sence.

IPBLE: Increasing Performance By Lowering Expectations.

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**Ricky****Moderator**- Registered: 2005-12-04
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If we assept the rules (a^x)(a^y)=a^(x+y) and (a^x)/(a^y)=a^(x-y)

Power rules come from group theory, they can be proven true. No acceptance is needed.

I suppose that we can use the rule n!/(n-1)!=n that is valid for all n>=1.

We can only accept that rule because 0! = 1. You can't use it to show 0! = 1.

*Last edited by Ricky (2006-02-03 20:43:52)*

"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."

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**krassi_holmz****Real Member**- Registered: 2005-12-02
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I accept the first.

Second-my mistake: not ">=1". It's ">1"

IPBLE: Increasing Performance By Lowering Expectations.

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**mathsyperson****Moderator**- Registered: 2005-06-22
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There's a bit more discussion on this here. Pretty much everything over there has been covered here already though.

Why did the vector cross the road?

It wanted to be normal.

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**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

Try the link again, it's working fine for me.

but I just don't believe that many of its assumptions are universally true.

Such as?

"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."

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**MathsIsFun****Administrator**- Registered: 2005-01-21
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It is just a belief of mine I suppose, but Mathematics is not prone to "upheavals" as such, because the foundations are simple definitions not open to experiment etc.

There can be "revolutions" as people discover whole new territories, but the existing knowledge does not get invalidated.

When Einstein published his theory of relativity, the world of Physics changed dramatically, but Mathematics just got more important.

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**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

What we percieve as 1 is really quite different from reality.

How do you percieve 1? 1 is an idea, not a thing.

Being that we do not know our position relative to the true planes or even that they exist would continue to force us to make incorrect assumptions about what we perceive to be fact.

1 + 1 = 2. This is because of how I define 1, 2, + and =. It has nothing to do with my perception of them. If I defined them differently, 1 + 1 ≠ 2, but definitions are just want we call things. They are arbitrary, just as arbitrary as why I call my pet a dog.

In this example, something that we observe to be a singular object at a specific place would indeed be two different objects at two different places.

Since when are there objects in math?

I am not here trying to disprove any of the concepts that some place in the sector of sanctity, but rather to point out that there is always the possibility of other interpretations of the truth.

I understand where you are coming from. I share the same view as you, except that I only apply it to science. It doesn't seem to apply to math.

There can be "revolutions" as people discover whole new territories, but the existing knowledge does not get invalidated.

That's not entirely true, if read literally. But I believe what you were trying to say is this:

If math is done correctly, the results are final. No amount of time will over turn the results of a correct proof.

This is different than science in which you can do everything correctly, and still come up with an incorrect theory.

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**Tredici****Member**- Registered: 2005-12-12
- Posts: 28

Woah, a fair few responses to this thread. I've been thinking about this factorial business, and I'm wondering if a different way of thinking may make understanding why 0! is considered to equal 1 that little bit easier.

I've considered all of the above mentioned formulae during pondering this, but I fail to see how they can be used as proof. Formulae like the permutations and factorial formula where derived with knowledge that any real number less than zero cannot be applied, even fractions/decimals >0,<1 in the Gamma function. In this instance, the likes of Leonhard Euler forever disregarding the application and resultation of negative numbers as 'acceptable' seem sensible mathematicians.

Using n!/(n-1)!=n does seem sensible at first thought, but by accepting the continuation of the rule such that 1= (1)!/(0)!, you must consider the resultant of further applications, like 0= (0)!/(-1)!, and -10= (-10)!/(-11)!, and that's when it becomes slightly less reasonable.

I'm sure we're all in agreeance in respect to a factorial being the product of all positive

integers from 1 to a given number. With this definition strictly in mind, 0 can **not** be applied as it's distance from 1 in a 1,2,3.. sequence is restricted by the realms of negative infinity, using the term 'negative infinity' very loosely. This is all obviously ignoring a vector like existance, where 0 is -1 from 1, which cannot be used here.

If we think more application as opposed to the confinements of laws, rules and constraining equalities maybe it begins to make sense. For example, what exactly is factorial? Well, factorial is just a simpler version of the permutations forumla, where n = r. Therefore, we're given all the different arrangements of n objects which are taken n at a time.

With this in mind, Let's imagine I've just spent a beautiful day by the lake side and have caught myself two wonderfully sized fish, they're so large in fact I'm going to name them Lance and Roy. Now, how should they sit on my plate when I have them for dinner? Well, I can have Roy, Lance or Lance, Roy? (2!=2). Hmmm, decisions, decisions.

Now, the day after, the clouds swell and stain black, and a powerful rainstorm hits the villiage. Being such a poor man, I rely entirely on fish, and must go out, once again, and try to catch some. But nope, not today. I get home, sling my rod to the floor, reach for the cupboard, grab a plate and stare, green with envy, nostalgic with the thought of yesterday's plentiful supper in mind. With the painful sound of my stomach rumbling and the food depriving rain attacking my straw roof, I think, how many times can 0 fish sit on my plate, and that's only 1 way, with no fish whatsoever. .:. 0! = 1

I realise I've just wasted about 5 minutes of your lives, but oh well, atleast I'm now entirely sure you've all gotten my point .

*Last edited by Tredici (2006-02-04 15:15:39)*

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**Tigeree****Member**- Registered: 2005-11-19
- Posts: 13,883

~ Anton Chekhov

Cheer up, emo kid.

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**krassi_holmz****Real Member**- Registered: 2005-12-02
- Posts: 1,905

We can create another "math" which have different axiomes. But the result still will be group- and set- theory result, because all known math may be expressed in terms of group theory and set theory.

IPBLE: Increasing Performance By Lowering Expectations.

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**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

It is a dark day in any science when its followers vehemenently deny the possibility that other foundations would perhaps advance their understanding of truth.

Yes, it is. Good thing no one else but you is even talking about science.

It is an even darker day when its followers would prove the validity of their theories by simply stating that that is the way they have defined truth.

What? We aren't defining truth. We are defining ideas. What is a prime number? Prime is a definition. If you wish to call it something else but prime, or if you wish to call a wigglysnug something that is divisible by itself and one, you may. But there is no point. Names don't mean anything, they just make it easier for us to express ideas.

The Pythagoreans elevated their mathematic science to the level of a religion.

Is anyone else here agreeing to that philosophy? Then why bring it up?

Irspow, we aren't talking about truth. There is no truth value of "any number that is divisble by only 1 and itself." You can't say true or false to it. It's just a type of number. Will that type of number ever change? No. Even if we call it something different in the future, or if we call something else prime, that type of number will still exist.

The same is true for all definitions. Whether it be a function, operation, group, set, field, irrationality, even, odd, and so on. It's just what we call things that have certain properties.

*Last edited by Ricky (2006-02-05 04:45:31)*

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**Ricky****Moderator**- Registered: 2005-12-04
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I guess I underestimated the level of indignation that could be aroused at the mere mention of a differing idea.

It's not that you are mentioning a different idea, it's the idea that you mention and how you mention it. And calling me a zealot just because I disagree with you only makes you the zealot.

All you said in that last post is that I disagree with you. I, on the other hand, have been saying why I think you are wrong. Please, do the same to me.

The definition of science does include the interpretation of it as skill set for solving problems.

Science, the most basic definition, is the study of our natural world. Math is not part of our natural world. It can be used as a tool to probe into the universe, but it is not a part of it. Ironic that it was the Pythagoreans who believed math *was* a part of our natural world...

If you don't wish to reply, I'm not pulling your leg to make you do so. If you want this conversation to end, that is fine. I, on the other hand, love debates, especially when I'm right (just to be clear, that was a joke). But I implore you, if there is one thing you get from this, it is to argue appropriately.

They laughed at Newton, they laughed at Galileo, but they also laughed at Bozo the clown.

I have dealt with pseudoscience for the past 3-4 years of my life, being a skeptic. Homeopaths, psychics, ghost hunters, you name it. One of the few things that unite all of these groups is what you just did in your previous post. Claiming that you were silenced ("executed") because you proposed a new idea. When this happens, a huge red flag appears in my mind.

It is this argument that comes from those with no other arguments to go off of. Please, don't make it. You know I'm not trying to silence you. I'm trying to have open discourse, I thought that's what this forum was for.

If you think I'm wrong, then state why you think I'm wrong. Point out what you think are errors in my posts. But, if you're not sure why you think I'm wrong, but you still do, then just say it. I have been in that situation many times. There's nothing wrong with it.

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**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

My argument tends more to a philosophical level. Philosophical arguments, er debates, can only be won when one of the contributors is willing to change core beliefs. In my case that will not happen on this subject.

That's fine, but we can't just talk about it?

I think that all types of definitions just represent the limit of understanding at the time in which they were created. So I feel to defend definitions simply to protect the status quo is silly.

And just to reiterate, I think that definitions are the same as us calling a sock, a sock.

Can't we agree to disagree? I am not here to debate anything, so I probably should not have offered my viewpoint in the first place.

Sure we can. But even if we do so, we can still discuss why it is we disagree. And like I said, if you don't want to debate about it, that's fine. But your viewpoint is always welcomed.

Actually this type of discussion is not even appropriate in a math forum anyway.

Sure it is. It's about math, isn't it? So what if it's in a philosophical context.

Oh, I did not attach any negative connotation by using the word zealot. There is no such one implied in the definition provided either. I used it in its strictest sense only.

Zealot sure does have a negative connotation. And besides, look how you used it:

of the zealots who serve to protect [pseudoscience]

You're talking about zealots who protect pseudoscience in reference to me. If that doesn't have a negative connotation, I'm not sure what does.

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**MathsIsFun****Administrator**- Registered: 2005-01-21
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Ummm ... great discussion irspow and Ricky ... but I am worried that you may be getting a little heated over the definition of "definition" ... or something.

But I am sure that if there were a breakthrough on some fundamental aspect of mathematics you would both embrace it with open arms.

On an earlier matter:

Tredici wrote:

With the painful sound of my stomach rumbling ... I think, how many times can 0 fish sit on my plate, and that's only 1 way, with no fish whatsoever. .:. 0! = 1

However, I don't think the "none" arrangement is counted

So, P(0,0) = 0!/(0-0)! = 1/1 ... what???

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**Tredici****Member**- Registered: 2005-12-12
- Posts: 28

MathsIsFun wrote:

However, I don't think the "none" arrangement is counted

So, P(0,0) = 0!/(0-0)! = 1/1 ... what???

I've had many a discussion with Mathematicians, and they've all agreed that's the logical reasoning behind it, albeit not 'proof'. Admittedly, from my point of view, it seems most reasonable. I'll continue to disregard plugging 1 into the permutations formula and stating it wouldn't work unless 0! = 1 as 'proof', but hey, I'll happilly admit I'm the least knowledgable in Math amongst the members of this community, and hence, I'd understand if you'll disagree.

Just for a bit of fun, my Mechanics teacher noticed somewhat of a sequence within the factorials:

4! = 24

÷4

3! = 6

÷3

2! = 2

÷2

1! = 1

÷1

0! = 1

The following division would be by 0, and hence, 0! is the lowest of the factorials. Also, MathsIsFun, you've put 0 into the permutations formula and ended with 1/1, and then questioned it. Isn't that what I was saying, that 0 could be arranged only 1 way?

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**MathsIsFun****Administrator**- Registered: 2005-01-21
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That "divide by" method makes good sense.

0 could be arranged only 1 way

Only if "0" were an object, possibly a donut.

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**Ricky****Moderator**- Registered: 2005-12-04
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0 could be arranged only 1 way

A similar statement, there is only 1 mapping from one empty set to another empty set, also leads to the conclusion that 0! = 1.

There is only one way to map the null set to the null set. That is, 0^0 = 1. But 0^0, just like 0!, is the product of no numbers at all. So the product of no numbers at all must be 1.

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**Ricky****Moderator**- Registered: 2005-12-04
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Unfortunately, irspow, that would be 0*0, not 0^0. Otherwise, if I deposit 2 dollars 3 times, I now have 8 dollars in my account.

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**Tredici****Member**- Registered: 2005-12-12
- Posts: 28

Oops, sorry for spam.

*Last edited by Tredici (2006-02-10 06:17:01)*

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