1. Jul 14, 2010

### asdfghjklqqww

hi! newbie here!
i was wondering...
are there other answers to 1+1=2?
and if so what are they?
thanks

2. Jul 14, 2010

### Mentallic

Do you mean instead of the value 2, are there cases where 1+1 equals something else?

3. Jul 14, 2010

### Gib Z

In the complex numbers (and its subsets) the only answer is 2. I'm not sure if this counts as another answer, but another decimal representation is 1.9999....

However, the exist other algebraic structures such as the ring $$Z_2$$ (the ring of integers modulo 2) where 1+1 is not 2, in this structure 1+1=0.

4. Jul 14, 2010

### Gerenuk

In normal algebra 1+1=2 by definition! Basically whatever comes out for 1+1 is going to be called 2. Whatever comes out from 1+1+1 is going to be called 3. And so on.

Now you can derive that 2+1=(1+1)+1=1+1+1=3.

Of course there are other algebras. For example in
http://en.wikipedia.org/wiki/Nimber
1+1=0

5. Jul 14, 2010

### betel

If your are with respect to the binary base, then 1+1 = 10.

6. Jul 14, 2010

### asdfghjklqqww

but i heard something that mathematicians could make 1+1=3 or 1+1=0

in fact i saw i video somewhere on youtube

7. Jul 14, 2010

### Hurkyl

Staff Emeritus
Well, in that ring, 2=0, so we still have 1+1=2.

8. Jul 14, 2010

### shinkyo00

i think from all the answers posted, you can see that depending on what meanings you assign to the symbols "1", "+" and "=", you'll get the answers corresponding to those obtained by following the rules which the meanings obey

9. Jul 15, 2010

### Mentallic

This only happens when your manipulation of an equality is invalid. Somewhere you would break a rule.

Say, let x=y=1

$$x^2=xy$$
$$x^2-xy=0$$
$$x(x-y)=0$$

Dividing through by x-y gives

$$x=0$$
so $$1=0$$ ?

The problem is when we divided by x-y, since x=y this means that x-y=0 and we can't divide by 0, else we get false results like this.

10. Jul 15, 2010

### asdfghjklqqww

i think i get it now...
pretty much 1 plus 1 always equals 2
and all of the so called "alternate solutions" break one rule of mathematics or another.
is this a good summary?
thanks for all of your fantastic help

11. Jul 15, 2010

### D H

Staff Emeritus
The smallest non-trivial group has already been mentioned. It has a completely consistent set of rules for addition, subtraction, multiplication, and division by a non-zero element -- and 1+1=0.

Anticipating Hurkyl's response, ... oh wait, he already did respond:
Only if you admit 2 as a synonym for 0. :yuck:

12. Jul 15, 2010

### Landau

You admit 2 as a synonym for 1+1 in any additive group, and in this particular group it happens that 1+1=0, hence 2=0.

13. Jul 15, 2010

### Mentallic

Yes but judging from this post:
I know what he's looking for is "tricks" that use the usual mathematics where 1+1=2. This only happens when an algebraic rule is broken.

14. Jul 15, 2010

### LCKurtz

1 + 1 = 3 only for large values of 1. :uhh:

15. Aug 2, 2010

### asdfghjklqqww

Yay! :)

16. Aug 2, 2010

### Mentallic

It took you nearly 1+1=3 weeks to come up with that? :P

17. Aug 5, 2010

### FizixFreak

i heard it newton used a huge chunk of his book (pricipia mathematica) to prove 1+1=2
is that true??????????

18. Aug 5, 2010

### Char. Limit

Not sure about the author, but it sounds like something Spivak did (took him a chapter to prove 0<1 from first principles).

19. Aug 5, 2010

### Staff: Mentor

No, not Newton. It was Bertrand Russell and Alfred North Whitehead, in their multi-volume (three volumes?) book, Principia Mathematica. They didn't show the details of the proof until well into the 2nd volume

20. Aug 7, 2010

### FizixFreak

didn't newton wrote that book?????????
any ways it is quite amusing to me how some needs to spend so much time and energy just to prove that i mean why do you need to prove something like that it is just basic human understanding. where you think i can get info on that prove?

21. Aug 7, 2010

### Gib Z

No, not that one. Newton's one is called "PhilosophiĆ¦ Naturalis Principia Mathematica" which is Latin for "The Mathematical Principles of Natural Philosophy".

You may think so.

It's basic human understanding isn't it?

22. Aug 7, 2010

### FizixFreak

:rofl:that was a good one .
yes i think i do not need a degree to tell 1+1=2 i mean a person who never even heard the word MATH can tell that but what i meant to say was how do you PROVE such a thing (technically).

23. Aug 7, 2010

### HallsofIvy

Well, a person who had not even heard the word MATH will presumably think it obvious that "1+ 1= 2", giving "1", "2", "+", and "=" the very vague meanings he has assigned to them. Showing that it is true with very precisely defined meanings for "1", "2", "+", and "=" may be another matter entirely.

24. Aug 7, 2010

### FizixFreak

why cant mathematicians take things easy why do they have to prove some thing so obvious and what do you mean by precisely defined meanings of ''1'' and ''2''???????

25. Aug 8, 2010

### rock.freak667

Because '1' could be a binary number such that 1+1=10.

If '+' is defined as vector addition, then the resultant of 1+1 lies between 0 and 2.