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### Why does 2+2 = 4?

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Date: 6/4/96 at 22:6:35
From: Garikai Campbell
Subject: 2+2 =? 4

I have a question. I was nosing around your site and saw the question
why does 2+2 = 4?
(in http://mathforum.org/dr.math/problems/gora10.27.html   )
In your discussion you say that the hard thing to show is that 1+1 = 2,
but you say that 4 is just another name for 1+1+1+1.

Isn't this a little incongruent? Also, I am not sure I know what takes
so long to prove. Doesn't this all (if you are saying that 2 is a name
for 1+1 and 4 is a name for 1+1+1+1) boil down to the construction of
integers as a rank 1 free abelian group?

Garikai "KAI" Campbell

P.S. Who is Dr. Math really?
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```
Date: 6/5/96 at 1:25:38
From: Doctor Pete
Subject: 2+2 =? 4

First, I'd like to address your question of "Who is Dr. Math"....

"Dr. Math," in short, is not a single individual, but a collective
group of people (many of them Swarthmore undergraduates) who provide
the service of answering math questions via E-mail and the World Wide
Web.  So perhaps one should ask, "Who *are* Dr. Math?"  (In
particular, I am an undergrad at Caltech with an avid interest in
geometry, Galois theory, and in getting a job that wouldn't result in
my becoming a Fed or being locked up in an ivory tower.)

As for the question about the construction of the integers, it is more
one of set-theoretical result than an algebraic one.  Remember that
"hard" is a relative term; for a college student, "hard" might entail
finding the Galois group of a polynomial; for a child in the 3rd
grade, it would probably mean something more on the level of
multiplying two ten-digit numbers.  So when kids are told by their
math teachers that proving why 1+1 = 2 is "hard," it usually means two
things:  (1) The familiarity with concepts required to construct the
integers is not attainable by most elementary schoolchildren, and
(2) aside from this, the teacher him/herself either doesn't know
remember/know how to show it, or doesn't want to talk about it with
the person who asked the question.

In any case, there are a few things to be seen from this - primarily,
that the notion of "1+1 = 2" is not necessarily a trivial one, nor is
it an axiom which children are told to assume; rather, it is a
(constructible/"provable") result.

-Doctor Pete,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```

```
Date: 6/5/96 at 9:15:23
From: Garikai Campbell
Subject: 2+2 =? 4

The reason I asked is: I am a graduate of Swarthmore and am now
finishing up (or at least trying to) at Rutgers. I thought Dr. Math
might be Swat faculty/students.

> As for the question about the construction of the integers, it is
> more of set-theoretical result than an algebraic one.

I don't know if I buy that, especially given what you were willing to
can you get something, you may want to start set-theoretically by
saying that 1 (namely something) is the set consisting of nothing and
from there the construction is really algebraic in nature. All you
really do is construct words (as in the construction of free groups).
I hope this makes sense, it is not exactly the clearest exposition.

reader to agree that 4 was just a name for 1+1+1+1. Isn't that what
the symbol 2 is also? And so I must not understand the following
comment that

> the notion of "1+1=2" is not necessarily a trivial one, nor is it an
> axiom which children are told to assume; rather, it is a
> (constructible/"provable") result.

Let me just expand on exactly what I mean, but with a few notes:
1. I will not consider inverses in this discussion and
2. I will assume that 1 has been defined.
Consider that the only "number" you had was 1 and you had an operation
called +. Other numbers are defined to be just strings of 1's
seperated by the + symbol.And that's it! Isn't this more accurate and
just as easy to convey (using a few more words) to a third grader.

Garikai "KAI" Campbell
```

```
Date: 6/5/96 at 17:17:53
From: Doctor Ce
Subject: Re: 2+2 =? 4

Hello,

Construction of the natural numbers does not boil down to the
construction of the integers as a rank 1 free abelian group.

The reason is that a rank 1 free abelian group is a stronger concept
than the natural numbers, which (with 0) are a monoid with identity
(there are no inverses).

In any case, before you can say 4 = 1+1+1+1, you have to show that
addition is associative.  This is a "difficult" theorem starting with
the Peano axioms. It is necessary to show this so that you can even
write 1+1+1+1 and make sense (no parentheses).

-- Doctor Ce
```

```
Date: 6/6/96 at 8:30:1
From: Garikai Campbell
Subject: Re: 2+2 =? 4

I understand that. My point was not to construct the natural numbers
but rather to use enough of a piece of the construction of the
integers to illustrate that showing 1+1+1+1=1+1 + 1+1 may be easier
using this construction.

> In any case, before you can say 4 = 1+1+1+1, you have to show that

I sincerely do not understand why - what I am saying is that 4 is a
name given to the string of symbols 1+1+1+1. The confusion may be in
the fact that I am using this well known symbol +. I think what I am
saying is better said using no plus sign. In other words, what if I
say you have the symbol A and the group operation is concatenation.
Throw in the symbols B and 0 (where B = A^(-1)). The group is the set
of all reduced words (i.e. a free group with one generator). We then
call 4 the word AAAA and 2 the word AA. Then it is more easy to see
that 2 "+" 2 = 4 since what this really means is does AA AA = AAAA. In
this explanation I never need to talk about many of the hard issues
that come up in showing in general that the object I am talking about
is in fact a group.

believe that showing 1+1 = 2 is hard but to assume that 1+1+1+1 is
just a name for 4. This seemed  terribly incongruent, and hence all
the hooplah.

Thanks for your quick response Doctor Ce. I am very interested to know

Garikai Campbell
```

```
Date: 6/14/96 at 2:14:30
From: Doctor Ethan
Subject: Re: 2+2 =? 4

Hey,

I am Ethan, a very recent graduate of Swat and very good friend of the
answerer of the question under discussion.

I hate to jump into a conversation where I am not exactly
qualified but I do want to say that you are right. In that answer
there is some confusion about whether we are just calling 4 a label
for 1+1+1+1 or if there is something else going on.

Particularly since it is 2:15 and I have been doing Doctor Math for 5
hours I can't tell what is going on in this problem

I can tell you though that after this conversation is over I will take
what I can from here and try to update that answer to make it a little
more right.

Perhaps you would like to help by replying to this with how you would

Why does 2+2 = 4?

-Doctor Ethan,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
College Number Theory

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