Pondering a Proof

[zhul4nder]

today, i overheard my nerd friends (alright, my friends =.=') that the proof for 1*1=1 does not exist. So I'm just curious, how do you prove that 1*1=1? Is this just an intuitive fact that us mathematicians as true? Is it something like questioning the fundamentals of time, a human invention just as math is?

discuss

[rzr]

Unless 1 is just a variable, I don't really see how it can't be 1.

[devonin]

1 group of 1 unit = 1 unit...makes sense to me.

[zhul4nder]

I did some googling and found the proof for 1 + 1 = 2

It looks quite complicated with fancy postulates that i don't understand:

http://mathforum.org/library/drmath/view/51551.html

i was expecting for something like that...but with 1*1 = 1

-edit-

lolz i found a video of it
http://video.google.com/videosearch?...um=5&ct=title#

oh wait, it's 1 + 1 = 0...and it's flawed. *sigh* googling attempts foiled yet again by misleading titles!


-edit-

Originally Posted by devonin

1 group of 1 unit = 1 unit...makes sense to me.

For every simple understanding, there's a very complicated reasoning behind it.

It's like sex.

[Hypnotic Malfunction]

Wouldn't it just be:

1*1=1
1=1

[zhul4nder]

well for 1 + 1 = 2...

the same logic is as you said

1 = 1

YET THERE HAS BEEN A PROOF DERIVED

[MooMoo_Cowfreak]

There are postulates out there. I'm guessing if there isn't a proof for 1 X 1 it's just a postulate.

[QED Stepfiles]

So, a lot of mathematics really is just built on a set of axioms that we deem is true in some system. In most cases in some structure, the existence of some "unit" 1 such that 1*g = g for any other element g in this structure is one of the given axioms. There really isn't much to say about this - it's just something that's assumed to be true and which allows us to study structures without having to really deal with what this unit "1" actually means.

In terms of a proof, it really goes down to breaking down mathematics in terms of whether or not certain axioms are correct or not, and whether or not the mathematical system we've developed is consistent in all cases. This gets VERY, VERY convoluted, which is why the proof to 1+1=2 is quite long, since all of mathematics essentially has to be dissected and put back together. If you're interested in stuff like this, definitely look out for a class in Mathematical Logic when you go to college, or some equivalent. There you'll formally break down what mathematical systems mean and prove a number of important results that will help you get a firmer grasp on what all these systems actually represent.

[lord_carbo]

In the group of multiplication with real numbers, 1 is considered the identity element. Therefore for all n, n * 1 = n. Granted even that proof doesn't dig into why 1 is the identity element, but we're getting closer.

The word "axiomatic" is sometimes meant to mean, "taken for granted." Basically, we take axioms in math for granted. As far as 1 * 1 = 1 or anything like that, you don't need a proof. It's obvious. We can take it for granted.

[QED Stepfiles]

Originally Posted by lord_carbo

In the group of multiplication with real numbers, 1 is considered the identity element. Therefore for all n, n * 1 = n. Granted even that proof doesn't dig into why 1 is the identity element, but we're getting closer.

The word "axiomatic" is sometimes meant to mean, "taken for granted." Basically, we take axioms in math for granted. As far as 1 * 1 = 1 or anything like that, you don't need a proof. It's obvious. We can take it for granted.

Well I think it generalizes even further - in ANY group with any law of composition, there must exist some unique identity element, denoted "1", such that 1*g = g for all elements g in the group. (Note that * does not denote multiplication - it just denotes whatever law of composition exists on the group structure - this could be addition, multiplication, function composition, etc) Yes, we do take this for granted, and it's very convenient to do so. However, it is still a fair question as to WHY this is a convenient choice and HOW this affects the structure of mathematics and how we think about these things. There are plenty of ways I could define a set of axioms for a new structure that would be completely wrong and fail to be consistent with the rest of mathematics. So, the question becomes, why is the existence of such an identity element consistent? Is mathematics a consistent system anyways? Is it possible to "break" mathematics?

So yes, on the surface this seems like a question that requires a one line answer - "well, of COURSE 1 * 1 = 1, that's how we defined it!" But on a deeper level this is a very important issue that has plagued mathematicians, and more specifically logicians, since the establishment of mathematics as a rigorous system.

I think it boils down to - is there really any reason to have complete faith in mathematics as an absolute system?

[Cavernio]

"I think it boils down to - is there really any reason to have complete faith in mathematics as an absolute system?"

Yes, if and until if becomes broken. It becomes broken when using it wrongly describes some aspect of the universe. If it doesn't describe it properly, it simply hasn't been 'invented' yet.

[tha Guardians]

Let's try it with x*1.

x*1 = x
Why? Because it's one of x.

1x*1 = 1x

Anything that exists tangibly or theoretically exists in the form of 1, possibly manipulated by an action or equation.

So if x = x then 1x = 1x and 1 = 1.

Any one of something is that same something. It's one of it. This reasoning implies that any negative number has a positive counterpart though, as something must exist before it can unexist. Physically speaking though, it always existed, just in a different form.

I'm confusing myself. xD
Good luck.

[Sullyman2007]

1+1=2
1(1)=1

I know this is CT, but why are you debating over something so clear and obvious? I am not really what you would call 'mathematically oriented', but this is definitely some logic I have never pondered..

[Patashu]

Originally Posted by zhul4nder

today, i overheard my nerd friends (alright, my friends =.=') that the proof for 1*1=1 does not exist. So I'm just curious, how do you prove that 1*1=1? Is this just an intuitive fact that us mathematicians as true? Is it something like questioning the fundamentals of time, a human invention just as math is?

discuss

by definition

peace

[NFD]

1 times 1 times 1 times 1 times 1 = 1.

It's like an infinite loop!!11!!1!one!!11!

[QED Stepfiles]

Originally Posted by Cavernio

"I think it boils down to - is there really any reason to have complete faith in mathematics as an absolute system?"

Yes, if and until if becomes broken. It becomes broken when using it wrongly describes some aspect of the universe. If it doesn't describe it properly, it simply hasn't been 'invented' yet.

Again, that's a bit of an oversimplification. How do we know when math has become "broken"? As far as we know, it may be broken already, even if we have not found a way to prove it is so. What you're suggesting is really a proof by lack of counterexample... which is not a very good way to rigorously define things in mathematics. The fact of the matter is that rigorous proofs of whether or not mathematics as a system is consistent are few and far between... at least as far as I can tell (I'm not a logician though, so don't quote me on that).

Math doesn't necessarily become broken just when using it incorrectly. Sure, that's a possibility, but we must ask ourselves if axiomatic mathematics is justified in the first place. With a set of bad axioms, even when using it correctly, you can prove a lot of really stupid things that are obviously false. If you want, you can google the proof that Winston Churchill is a carrot, just based on the stupid initial assumption that 1=2.

Perhaps I'm just misinterpreting your statement... I'm not quite sure what you mean by "describing some aspect of the universe." Of course, this brings up a whole other issue of: is math actually physical? Sure, we can apply math to physical systems, but aren't these but shoddy impure tangible copies of what we think of as the ideal mathematical objects? You can look at a basketball and think of a sphere, but really a sphere is an idealized mathematical object that does not really exist in nature. This brings up the issue of - what about a basketball allows us to invariably make the connection between it and a sphere? At this point, I may be getting off track... so I think I'll stop while I'm ahead.

The truth, and most of the posts in this topic confirm this, is that almost everybody takes math for granted. Oh sure, it's obvious that 1*1=1. But is it really? Not when you really think about it. I'm pretty sure if you were to grow up being taught that 1*1=2, and building mathematical models off of that fact, then when faced with the question "why is 1*1=2?" you would most likely scoff and go "well, it's obvious, isn't it?" Unfortunately, it's not that simple.

[Patashu]

it's obvious that 1*1=1 because it's part of the definition of the integers

math is axiomatic, it works the way it does because we make the rules

if 1*1 equaled 2 then it would be a different system

god

[QED Stepfiles]

Originally Posted by Patashu

it's obvious that 1*1=1 because it's part of the definition of the integers

math is axiomatic, it works the way it does because we make the rules

if 1*1 equaled 2 then it would be a different system

god

Actually, you'd be pretty hard pressed to define a coherent mathematical system based on the axiom 1*1=2. There's a reason why 1*1=1... because given the most natural set of axioms, it is important to define an identity element.

Well, then, what exactly are the integers? In terms of naive set theory, there's a very rigorous process by which the integers are constructed. So, really, we can say that 1*1=1 if 1 is considered to be an integer because of the axioms of set theory. Of course, then we must wonder whether or not the axioms of set theory hold in any case. Do we really have any intuition as to what a "set" actually is? Can we just assume the existence of sets?

So no, unfortunately you're making a gross oversimplification. In fact, you're pretty much arguing in a circle -

"1*1=1 works because we defined it that way. We defined it that way because it works."

But no, how really do we know that it works? This is really a more serious issue than most people who have posted in this thread believe...

[devonin]

The problem is that if you're only concerned with the actual statement 1*1=1, the simple expression of that as "one group of objects wherein each group contains one object is a set that contains one total object across all groups" is obvious not because of the nature of integers, not because of the way set theory works, but because of the lingusitic definitions of the words "one" and "group"

The only way this is complicated is if you somehow want to question the linguistic definitions involved.

[MrRubix]

I hate questions like this because no matter what, you're going off SOME sort of assumption about how things relate. 1*1=1 because you have one 1. Easy. I mean, why do you need a "proof" of this? It's like saying I am holding one apple. Prove I have one apple. By definition 1*1=1! I never saw the point in these kind of questions.

As for those "proofs" showing that 1=something other than 1, those have never impressed me because every single one basically plays off mathematical ignorance of the majority by violating some fundamental step along the way or making some assumption that is clearly, in itself, not a valid assumption.