Are there other solutions to 1+1=2?

[asdfghjklqqww]

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

 

[Mentallic]

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

 

[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.

 

[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

 

[betel]

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

 

[asdfghjklqqww]

thank's for your help!
but i heard something that mathematicians could make 1+1=3 or 1+1=0

in fact i saw i video somewhere on youtube

 

[Hurkyl]

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.

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

 

[shinkyo00]

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

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

 

[Mentallic]

thank's for your help!
but i heard something that mathematicians could make 1+1=3 or 1+1=0

in fact i saw i video somewhere on youtube

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.

 

[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

 

[D H]

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

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:

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

Only if you admit 2 as a synonym for 0.

 

[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.

 

[Mentallic]

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.

Yes but judging from this post:

thank's for your help!
but i heard something that mathematicians could make 1+1=3 or 1+1=0

in fact i saw i video somewhere on youtube

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.

 

[LCKurtz]

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

 

[asdfghjklqqww]

Yay! :)

 

[Mentallic]

Yay! :)

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

 

[FizixFreak]

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

 

[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).

 

[Mark44]

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

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

 

[FizixFreak]

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

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?

 

[Gib Z]

didn't Newton wrote that book?

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

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.

You may think so.

where you think i can get info on that prove?

It's basic human understanding isn't it?

 

[FizixFreak]

It's basic human understanding isn't it?

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).

 

[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.

 

[FizixFreak]

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.

why can't 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''??

 

[rock.freak667]

why can't 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''??

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.

 

[The Chaz]

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

1 Point for calling out the flagrant bump.
+ 1 point for referencing the thread at the same time
= the new constant named in your honor - "M"

 

[Mentallic]

= the new constant named in your honor - "M"

And no shortening the name of the constant by any way shape or form, such as how the exponential e is merely called E by some students. This shall and always will be called Mentallic.

I also like your display of understanding with this M constant 1+1=M

 

[asdfghjklqqww]

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

Well it worked didn't it :P

 

[FizixFreak]

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.

so what is the approach to that prove i mean which method is used for that prove?

 

[HallsofIvy]

so what is the approach to that prove i mean which method is used for that prove?

If vector u has length 1 and makes angle $\theta$ with vector v, which also has length 1, then u+ v is a vector having length $2(1- cos(\theta))$. Since [math]cos(\theta) has maximum value 1 and minimum value -1, that length has maximum value 2 and minimum value 0. Personally, I would not refer to adding the vectors as being the same as adding their lengths I would NOT say that was a case in which "1+ 1" can be less than 2.