Why does 1 + 1 = 2

I've seen a proof or two that I can't even remember on this topic; and I was wondering if any readers out there can show the proof of 1 + 1 = 2.


I actually give Sklansky a 50% chance of knowing the right answer.

The rest of you, well not much chance. But surprise me if you can.

Honestly You are a freak buddy, your lost !!!


regards,

-MJ

In college, I actually took a course in this stuff. It has something to do with the fact that 1 has a successor. I'll let others elaborate.

rest of the proof is fairly straight forward. The hardest numbers to prove are 0 and 1. Of course, I took that course in 1988, and I sure don't remember it off the top of my head. Good luck!

that's what happened with me and the little lady...a family man now...used to be a line in a who song...behind blue eyes...gl

Yes, 1 + x is defined as the successor to x, and 2 is the successor to 1.

I suppose you can just turn it around.

2 is defined as the addition of 1+1. How do you prove it? You can do something simple and take an object and say that this represents the number 1. A second object, together with the first, represents the number two. If your definition of the number 1 is different from ours, then so be it. But your number 1 will have no basis for what is in the real world.


In the end, though, you don't need to prove anything. You merely need to accept that definition at 2 = 1 + 1. If you don't accept it, fine. Then YOU must prove that it is false. We do not need to prove that it is true.

spikey, on many boards, this question usually yields the response, "do your own homework."


the club

A good bit of a question like this gets folded into standard ontological questions about the existence of God, "justice," and other matters pertaining to such debatable topics of subjective reality as ethics and morality.


Mathematics is immutable; that is to say that it simply IS, and will not find itself subjected to changes brought about by ANY factors, so to speak.


So, then: 1+1=2. Proof: because 1+1=2. In semantics, it's called begging the question. In mathematics, I believe it's just called a fact.


Or I'm just blowin' smoke.

(This is from mathforum.org)


The proof starts from the Peano Postulates, which define the natural

numbers N. N is the smallest set satisfying these postulates:


P1. 1 is in N.

P2. If x is in N, then its "successor" x' is in N.

P3. There is no x such that x' = 1.

P4. If x isn't 1, then there is a y in N such that y' = x.

P5. If S is a subset of N, 1 is in S, and the implication


(x in S => x' in S) holds, then S = N.


Then you have to define addition recursively:

Def: Let a and b be in N. If b = 1, then define a + b = a'


(using P1 and P2). If b isn't 1, then let c' = b, with c in N


(using P4), and define a + b = (a + c)'.


Then you have to define 2:

Def: 2 = 1'


2 is in N by P1, P2, and the definition of 2.


Theorem: 1 + 1 = 2


Proof: Use the first part of the definition of + with a = b = 1.


Then 1 + 1 = 1' = 2 Q.E.D.


Note: There is an alternate formulation of the Peano Postulates which

replaces 1 with 0 in P1, P3, P4, and P5. Then you have to change the

definition of addition to this:

Def: Let a and b be in N. If b = 0, then define a + b = a.


If b isn't 0, then let c' = b, with c in N, and define


a + b = (a + c)'.


You also have to define 1 = 0', and 2 = 1'. Then the proof of the

Theorem above is a little different:


Proof: Use the second part of the definition of + first:


1 + 1 = (1 + 0)'


Now use the first part of the definition of + on the sum in


parentheses: 1 + 1 = (1)' = 1' = 2 Q.E.D.

No, that's about my answer.


If you look up "two" in the dictionary, it says "one more than one".


1+1=2 by definition.

Tell a bank that you are conducting a scientific experiment. Open a savings account with 1 dollar. Go back on the next day and deposit 1 dollar in the same account. Read the bank receipt - what is the total amount in the account? I will bet 2 dollars that you will have 2 dollars in the account. Of couse, I could be wrong. Experiments can blow up in your face sometimes.


In the proof supplied by tewall (via mathform.org) P5 depends on set theory - and may be a questionable assumption. But that is a guess on my part. I for one am happy with the proof and will sleep well tonight knowing that the bank is doing its proper job in accounting for my money. Or is it?


-Zeno

because someone already proved that it was, in fact, $2. Not the other way around.

The proof completely depends on the axiomatic structure of the numer theory you chose to use, e.g.:


Let ' be the successor function, and 1 and 2 shorthands for 0' and 0''


With the axioms (1) X+Y'=X'+Y and (2) X+0=X the proof is simple:


0'+0' = 0''+0 (axiom 1) = 0'' (axiom 2)


cu


Ignatius

There is a mathmatical proof that proves that 1 does not equal 1 (1 = 1), that 1 = 2. So go figure.

such as in modular number systems. what about a binary system of 0 and 1? does 1+1 = 2? i don't think so. so when you ask why does 1+1 = 2, you have to specify what set of numbers and rules we are using here. and once you have done that, well, then it is almost arbitrary...

yeap, another one comes from (FZ) set theory where you define 0 = {} (null set) and n + 1 = n U {n}. Cardinality defines the numbers. From that the proof is easy.