1 + 1 = ?

For the discussion of math. Duh.

Moderators: gmalivuk, Moderators General, Prelates

bureau
Posts: 3
Joined: Tue Jan 20, 2009 8:52 pm UTC

1 + 1 = ?

Postby bureau » Sat Jan 31, 2009 12:01 am UTC

Sorry if this has been posted before. Is it true that it's possible for 1 + 1 to not equal 2?

auteur52
Posts: 165
Joined: Sun Mar 23, 2008 11:08 pm UTC

Re: 1 + 1 = ?

Postby auteur52 » Sat Jan 31, 2009 12:17 am UTC

That depends on your definitions of "1", "2", "+", and "=".

User avatar
Qoppa
Posts: 694
Joined: Sat Nov 24, 2007 9:32 pm UTC
Location: Yes.

Re: 1 + 1 = ?

Postby Qoppa » Sat Jan 31, 2009 12:20 am UTC

In a field of characteristic 2 (such as Z/2), 1+1=0.

Code: Select all

_=0,w=-1,(*t)(int,int);a()??<char*p="[gd\
~/d~/\\b\x7F\177l*~/~djal{x}h!\005h";(++w
<033)?(putchar((*t)(w??(p:>,w?_:0XD)),a()
):0;%>O(x,l)??<_='['/7;{return!(x%(_-11))
?x??'l:x^(1+ ++l);}??>main(){t=&O;w=a();}

User avatar
jazznaz
Posts: 18
Joined: Thu Apr 17, 2008 11:19 pm UTC

Re: 1 + 1 = ?

Postby jazznaz » Sat Jan 31, 2009 12:27 am UTC

Under modulo arithmetic (mod 2) , 1 + 1 = 0...

I'm not great on math theory, but that may be equivilent to the above statement...

User avatar
heyitsguay
Posts: 118
Joined: Thu Oct 16, 2008 6:21 am UTC

Re: 1 + 1 = ?

Postby heyitsguay » Sat Jan 31, 2009 12:27 am UTC

1+1=100010101 base i, but that's not unique.

Chet
Posts: 27
Joined: Wed Jan 28, 2009 9:07 pm UTC

Re: 1 + 1 = ?

Postby Chet » Sat Jan 31, 2009 12:55 am UTC

I think the correct answer is:
"Not in the usual sense!"

User avatar
Cleverbeans
Posts: 1378
Joined: Wed Mar 26, 2008 1:16 pm UTC

Re: 1 + 1 = ?

Postby Cleverbeans » Sat Jan 31, 2009 1:02 am UTC

1+1 can equal absolutely anything if you're willing to change the meaning of "equals" based on context. It might seem artificial at first however different flavors of equality exists for the natural, rational, real and complex numbers for "normal" math. Here is a link to the wikipedia article on the subject.

http://en.wikipedia.org/wiki/Equivalence_relation
"Labor is prior to, and independent of, capital. Capital is only the fruit of labor, and could never have existed if labor had not first existed. Labor is the superior of capital, and deserves much the higher consideration." - Abraham Lincoln

User avatar
Qoppa
Posts: 694
Joined: Sat Nov 24, 2007 9:32 pm UTC
Location: Yes.

Re: 1 + 1 = ?

Postby Qoppa » Sat Jan 31, 2009 1:17 am UTC

jazznaz wrote:Under modulo arithmetic (mod 2) , 1 + 1 = 0...

I'm not great on math theory, but that may be equivilent to the above statement...
It is. As I pointed out, Z/2 (the integers mod 2) is a field of characteristic 2.

Code: Select all

_=0,w=-1,(*t)(int,int);a()??<char*p="[gd\
~/d~/\\b\x7F\177l*~/~djal{x}h!\005h";(++w
<033)?(putchar((*t)(w??(p:>,w?_:0XD)),a()
):0;%>O(x,l)??<_='['/7;{return!(x%(_-11))
?x??'l:x^(1+ ++l);}??>main(){t=&O;w=a();}

User avatar
antonfire
Posts: 1772
Joined: Thu Apr 05, 2007 7:31 pm UTC

Re: 1 + 1 = ?

Postby antonfire » Sat Jan 31, 2009 2:13 am UTC

It's worth noting that in all of these cases, you still have 1+1=2. It's just that it also happens to be true that, modulo 2, 2=0.

A case where you take 1+1 to be something other than 2 is when you redefine the addition operation as a +' b=max(a,b) and multiplication as a *' b=a+b.

Also, I'm pretty sure there's been a thread about exactly this question, but I can't find it.
Jerry Bona wrote:The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma?

User avatar
Sir_Elderberry
Posts: 4206
Joined: Tue Dec 04, 2007 6:50 pm UTC
Location: Sector ZZ9 Plural Z Alpha
Contact:

Re: 1 + 1 = ?

Postby Sir_Elderberry » Sat Jan 31, 2009 4:02 am UTC

Basically, if you see someone asserting that "1+1=2", there's a few possibilities

1) They've given you a several-line proof that seems to "prove it", especially if the first line is "a=b". Chances are they divided by zero somewhere in there but hid it by dividing by "a-b", or something similar.

2) They know some high-level mathematics and are redefining the terms "one" "plus" "equals" or "two".
http://www.geekyhumanist.blogspot.com -- Science and the Concerned Voter
Belial wrote:You are the coolest guy that ever cooled.

I reiterate. Coolest. Guy.

Well. You heard him.

chapel
Posts: 121
Joined: Mon Dec 01, 2008 8:52 am UTC

Re: 1 + 1 = ?

Postby chapel » Sat Jan 31, 2009 5:19 am UTC

2 + 2 = 5 for large values of 2, as the old joke goes. :mrgreen:

keeperofdakeys
Posts: 658
Joined: Wed Oct 01, 2008 6:04 am UTC

Re: 1 + 1 = ?

Postby keeperofdakeys » Sat Jan 31, 2009 5:46 am UTC

how about 1 + 1 = 10

User avatar
skeptical scientist
closed-minded spiritualist
Posts: 6142
Joined: Tue Nov 28, 2006 6:09 am UTC
Location: San Francisco

Re: 1 + 1 = ?

Postby skeptical scientist » Sat Jan 31, 2009 6:48 am UTC

keeperofdakeys wrote:how about 1 + 1 = 10

That's just a different representation of 2.

It's all a matter of definitions, which is to say, semantics. For any reasonable definition of 2, 2=1+1, if it even makes sense to define 2 in your field ring group semigroup magma.
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.

"With math, all things are possible." —Rebecca Watson

User avatar
You, sir, name?
Posts: 6983
Joined: Sun Apr 22, 2007 10:07 am UTC
Location: Chako Paul City
Contact:

Re: 1 + 1 = ?

Postby You, sir, name? » Sat Jan 31, 2009 11:41 am UTC

jazznaz wrote:Under modulo arithmetic (mod 2) , 1 + 1 = 0...

I'm not great on math theory, but that may be equivilent to the above statement...


It doesn't change the equivalence though:
1 + 1 = 0 mod 2
2 = 0 mod 2
I edit my posts a lot and sometimes the words wrong order words appear in sentences get messed up.

Fimbulfamb
Posts: 12
Joined: Sat Sep 13, 2008 7:43 pm UTC

Re: 1 + 1 = ?

Postby Fimbulfamb » Sat Jan 31, 2009 2:06 pm UTC

I believe 1+1 was proven to be 2 around 1900 by Bertrand Russel.

Cpt. Red
Posts: 8
Joined: Sat Oct 25, 2008 4:08 pm UTC

Re: 1 + 1 = ?

Postby Cpt. Red » Sat Jan 31, 2009 3:36 pm UTC

Charles Seife wrote:Dividing by zero...allows you to prove, mathematically, anything in the universe. You can prove that 1+1=42, and from there you can prove that J. Edgar Hoover is a space alien, that William Shakespeare came from Uzbekistan, or even that the sky is polka-dotted. (See appendix A for a proof that Winston Churchill was a carrot.)

User avatar
Cass
Posts: 28
Joined: Thu Oct 04, 2007 3:04 am UTC
Location: BC, Canada

Re: 1 + 1 = ?

Postby Cass » Sat Jan 31, 2009 6:03 pm UTC

keeperofdakeys wrote:how about 1 + 1 = 10


Thats correct, thought only in base 2.

Correct me if im wronge but if you defined a vector space of all real numbers and defined your axioms could you not have 1+1 = anything you want. as long as 1+(-1) = 0 ?
"I reject your reality and substitute it with my own"

User avatar
Qoppa
Posts: 694
Joined: Sat Nov 24, 2007 9:32 pm UTC
Location: Yes.

Re: 1 + 1 = ?

Postby Qoppa » Sat Jan 31, 2009 6:22 pm UTC

Well no. Vector addition has to be well defined, which means that adding two vectors should produce the same result each time.

Code: Select all

_=0,w=-1,(*t)(int,int);a()??<char*p="[gd\
~/d~/\\b\x7F\177l*~/~djal{x}h!\005h";(++w
<033)?(putchar((*t)(w??(p:>,w?_:0XD)),a()
):0;%>O(x,l)??<_='['/7;{return!(x%(_-11))
?x??'l:x^(1+ ++l);}??>main(){t=&O;w=a();}

shinjak
Posts: 6
Joined: Tue Jul 10, 2007 3:49 pm UTC

Re: 1 + 1 = ?

Postby shinjak » Sat Jan 31, 2009 6:42 pm UTC

If the additive structure of your vector space is going to have very little to do with the usual additive structure on the reals, then you might as well ditch the underlying set and call the elements of your vector space something altogether different.

User avatar
A_of_s_t
Posts: 180
Joined: Tue Aug 26, 2008 5:10 am UTC
Contact:

Re: 1 + 1 = ?

Postby A_of_s_t » Sun Feb 01, 2009 5:57 am UTC

Fimbulfamb wrote:I believe 1+1 was proven to be 2 around 1900 by Bertrand Russel.

It was. I have a picture of his original work. Its a halarious paper long proof.
Check out my web interzones powered by Web 3.0 technology running on Mozzarella Foxfire:
Spoiler:
Image
Image
Image

User avatar
Sir_Elderberry
Posts: 4206
Joined: Tue Dec 04, 2007 6:50 pm UTC
Location: Sector ZZ9 Plural Z Alpha
Contact:

Re: 1 + 1 = ?

Postby Sir_Elderberry » Sun Feb 01, 2009 2:08 pm UTC

A_of_s_t wrote:
Fimbulfamb wrote:I believe 1+1 was proven to be 2 around 1900 by Bertrand Russel.

It was. I have a picture of his original work. Its a halarious paper long proof.


He also spent a few hundred pages just laying the groundwork for it.
http://www.geekyhumanist.blogspot.com -- Science and the Concerned Voter
Belial wrote:You are the coolest guy that ever cooled.

I reiterate. Coolest. Guy.

Well. You heard him.

chapel
Posts: 121
Joined: Mon Dec 01, 2008 8:52 am UTC

Re: 1 + 1 = ?

Postby chapel » Mon Feb 02, 2009 11:05 am UTC

It can get proved assuming the Peano Postulates. The real quick and dirty of the proof is that you define addition so that a+b = (a+c)' when b = c' where x' is the successor to x (and so is an element of the naturals so long as x is), and then let 1' = 2. So,a = b = 1 implies 1 + 1 = 1' = 2. There are some key details missing, but that sketch should be good enough that someone with a list of the Peano Postulates should be able to make a real proof.

By the way, google Principia Mathematica and 1 + 1 = 2 (I went looking for the original work). Someone asked for a proof that 1 + 1 = 2 on Yahoo questions and at least a dozen people said, "If you have 1 something and someone gives you another something, then you have two somethings," before calling the guy an idiot for not being able to come up with their obviously correct proof.

User avatar
PM 2Ring
Posts: 3715
Joined: Mon Jan 26, 2009 3:19 pm UTC
Location: Sydney, Australia

Re: 1 + 1 = ?

Postby PM 2Ring » Mon Feb 02, 2009 5:57 pm UTC

Sir_Elderberry wrote:
A_of_s_t wrote:
Fimbulfamb wrote:I believe 1+1 was proven to be 2 around 1900 by Bertrand Russel.

It was. I have a picture of his original work. Its a halarious paper long proof.


He also spent a few hundred pages just laying the groundwork for it.


Sort of. Russell & Whitehead were attempting to formalize mathematics, from the ground up, using symbolic logic (and basic set theory). Their plan was flawed, but they didn't know about Godel's theorems*. Their proof for 1+1=2 isn't really that long, but they did have to formalize a fair amount of more fundamental material before they got to 1+1=2. So the 1+1=2 result was just an incidental result they derived for fun. The fact that it occurs after a few hundred pages illustrates that this "basic mathematical fact" isn't as basic as the lay person may think.


Disclaimer: I haven't read Russell & Whitehead's Principia Mathematica. And I don't intend to in a hurry. Reading page after page of symbolic logic makes my brain turn to jelly. :)

[*] But if it wasn't for Principia Mathematica, Godel may have not discovered his theorems. Although Godel wasn't the only one looking into these issues: there were also people like Church, Tarski & Turing looking at the limitations of formal mathematical procedures.

User avatar
Sir_Elderberry
Posts: 4206
Joined: Tue Dec 04, 2007 6:50 pm UTC
Location: Sector ZZ9 Plural Z Alpha
Contact:

Re: 1 + 1 = ?

Postby Sir_Elderberry » Tue Feb 03, 2009 12:10 am UTC

Well, yes, I know that he wasn't out to just prove 1+1=2. But nonetheless, the proof of that proposition was only possible after a volume or two of logic and formalization. Pretty hardcore stuff.

If anyone's interested in this, everything I know about this I learned from GEB.
http://www.geekyhumanist.blogspot.com -- Science and the Concerned Voter
Belial wrote:You are the coolest guy that ever cooled.

I reiterate. Coolest. Guy.

Well. You heard him.

User avatar
gmalivuk
GNU Terry Pratchett
Posts: 26824
Joined: Wed Feb 28, 2007 6:02 pm UTC
Location: Here and There
Contact:

Re: 1 + 1 = ?

Postby gmalivuk » Fri Feb 06, 2009 11:40 pm UTC

PM 2Ring wrote:So the 1+1=2 result was just an incidental result they derived for fun. The fact that it occurs after a few hundred pages illustrates that this "basic mathematical fact" isn't as basic as the lay person may think.

Well, what they really proved was the fact that logic and a few basic axioms of set theory result in a structure that contains a set corresponding to how everyone had already been treating integers for thousands of years.

The axiomatization and formalization of mathematics wasn't so much an effort to prove facts everyone already knew, as a way to provide firm logical groundwork consistent with those facts, from which to prove other things that people didn't already know, but which fit with what they did.
Unless stated otherwise, I do not care whether a statement, by itself, constitutes a persuasive political argument. I care whether it's true.
---
If this post has math that doesn't work for you, use TeX the World for Firefox or Chrome

(he/him/his)

codyhotel
Posts: 319
Joined: Mon Dec 03, 2007 3:54 pm UTC

Re: 1 + 1 = ?

Postby codyhotel » Fri Feb 13, 2009 5:16 pm UTC

I'm amazed that no one has put this, maybe out of fear, but i'll risk it.


Obviously,

1+1=1.99999...8
from a previously (over)discussed theorem.
Philwelch wrote:Would a prostitution enthusiast be a buy-sexual?

...sorry.

auteur52
Posts: 165
Joined: Sun Mar 23, 2008 11:08 pm UTC

Re: 1 + 1 = ?

Postby auteur52 » Fri Feb 13, 2009 10:40 pm UTC

codyhotel wrote:I'm amazed that no one has put this, maybe out of fear, but i'll risk it.


Obviously,

1+1=1.99999...8
from a previously (over)discussed theorem.


No. 2=1.999....

User avatar
Sir_Elderberry
Posts: 4206
Joined: Tue Dec 04, 2007 6:50 pm UTC
Location: Sector ZZ9 Plural Z Alpha
Contact:

Re: 1 + 1 = ?

Postby Sir_Elderberry » Sat Feb 14, 2009 12:15 am UTC

auteur52 wrote:
codyhotel wrote:I'm amazed that no one has put this, maybe out of fear, but i'll risk it.


Obviously,

1+1=1.99999...8
from a previously (over)discussed theorem.


No. 2=1.999....


Considering that 1.9999....8 doesn't make any sense in any notation I'm aware of, yes.
http://www.geekyhumanist.blogspot.com -- Science and the Concerned Voter
Belial wrote:You are the coolest guy that ever cooled.

I reiterate. Coolest. Guy.

Well. You heard him.

User avatar
Deep Fried Pickles
Posts: 14
Joined: Fri Feb 13, 2009 11:42 pm UTC

Re: 1 + 1 = ?

Postby Deep Fried Pickles » Sat Feb 14, 2009 2:57 am UTC

I have a friend who can use circular logic to prove that 1 + 1 is 0

User avatar
Freiberg
Posts: 171
Joined: Thu Oct 30, 2008 4:09 am UTC

Re: 1 + 1 = ?

Postby Freiberg » Sat Feb 14, 2009 4:04 am UTC

Deep Fried Pickles wrote:I have a friend who can use circular logic to prove that 1 + 1 is 0


Off topic, but he actually proved that 1 + 1 = 0/0

xkcdaddict
Posts: 5
Joined: Sat Feb 14, 2009 3:05 pm UTC

Re: 1 + 1 = ?

Postby xkcdaddict » Sat Feb 14, 2009 3:20 pm UTC

(x-x)(x+x) = x²-x²
(x-x)(x+x) = x*(x-x)
x+x = x
2x = x
2 = 1
1+1 = 1, 2, 3, or 4.

1+1 is clearly ambiguous.

:D

spot the error haha

another one:

(-1) * (-1)= 1

log (-1 * -1) = log 1
log -1 + log -1 = log 1 = 0
log -1 = - (log -1)
(log -1)/(log -1) = -1
1 = -1

1+1 = 0, +-2

:D

User avatar
Sir_Elderberry
Posts: 4206
Joined: Tue Dec 04, 2007 6:50 pm UTC
Location: Sector ZZ9 Plural Z Alpha
Contact:

Re: 1 + 1 = ?

Postby Sir_Elderberry » Sat Feb 14, 2009 7:18 pm UTC

Divisions by zero.
http://www.geekyhumanist.blogspot.com -- Science and the Concerned Voter
Belial wrote:You are the coolest guy that ever cooled.

I reiterate. Coolest. Guy.

Well. You heard him.

Tac-Tics
Posts: 536
Joined: Thu Sep 13, 2007 7:58 pm UTC

Re: 1 + 1 = ?

Postby Tac-Tics » Mon Feb 16, 2009 8:57 pm UTC

xkcdaddict wrote:(-1) * (-1)= 1

log (-1 * -1) = log 1
log -1 + log -1 = log 1 = 0
log -1 = - (log -1)
(log -1)/(log -1) = -1
1 = -1

1+1 = 0, +-2

:D


I like this one. I haven't seen it before.

The trick is it is NOT true that log xy = log x + log y for all real x and y. Namely, log -1 = pi * i, so log (-1)(-1) = 0 /= pi * i + pi * i. The identity does seem to hold when working in mod 2-pi arithmetic.

Buttons
Posts: 858
Joined: Wed May 02, 2007 3:27 pm UTC
Location: Somerville

Re: 1 + 1 = ?

Postby Buttons » Mon Feb 16, 2009 9:57 pm UTC

Tac-Tics wrote:The trick is it is NOT true that log xy = log x + log y for all real x and y.

Sure it is. It's well known that log 1 = 2pi*i, among other things. There's nothing wrong with the rule that log xy = log x + log y, so long as you don't assume that the result is the only value log xy equals.

User avatar
Macbi
Posts: 941
Joined: Mon Apr 09, 2007 8:32 am UTC
Location: UKvia

Re: 1 + 1 = ?

Postby Macbi » Mon Feb 16, 2009 10:00 pm UTC

Buttons wrote:
Tac-Tics wrote:The trick is it is NOT true that log xy = log x + log y for all real x and y.

Sure it is. It's well known that log 1 = 2pi*i, among other things. There's nothing wrong with the rule that log xy = log x + log y, so long as you don't assume that the result is the only value log xy equals.
If you don't treat your functions like functions the Mathematical Fairy stabs you to death with set squares.
    Indigo is a lie.
    Which idiot decided that websites can't go within 4cm of the edge of the screen?
    There should be a null word, for the question "Is anybody there?" and to see if microphones are on.

Tac-Tics
Posts: 536
Joined: Thu Sep 13, 2007 7:58 pm UTC

Re: 1 + 1 = ?

Postby Tac-Tics » Mon Feb 16, 2009 10:39 pm UTC

Macbi wrote:
Buttons wrote:Sure it is. It's well known that log 1 = 2pi*i, among other things. There's nothing wrong with the rule that log xy = log x + log y, so long as you don't assume that the result is the only value log xy equals.
If you don't treat your functions like functions the Mathematical Fairy stabs you to death with set squares.


Thank you Macbi!

This is the math forum, not the physics forum. Functions are single-valued. Always. If you need to choose bewteen two alternative "branches", then pick on and be done with it!

Buttons
Posts: 858
Joined: Wed May 02, 2007 3:27 pm UTC
Location: Somerville

Re: 1 + 1 = ?

Postby Buttons » Mon Feb 16, 2009 11:06 pm UTC

You'd rather throw out the rule log xy = log x + log y than accept that ex is not injective? Inverses of noninjective functions are multivalued. There's nothing physics-y about that.

I'm all for narrowing the range in a given situation, of course. But I would hardly say that log x = 2pi*i "has no solutions" in general.

Tac-Tics
Posts: 536
Joined: Thu Sep 13, 2007 7:58 pm UTC

Re: 1 + 1 = ?

Postby Tac-Tics » Mon Feb 16, 2009 11:26 pm UTC

Buttons wrote:You'd rather throw out the rule log xy = log x + log y than accept that ex is not injective? Inverses of noninjective functions are multivalued. There's nothing physics-y about that.


I'm just saying that a "multivalued" function isn't a function. It's a relation or something else, but I'd rather give up the log xy = log x + log y rule than screw around with multivalued functions. Both result in the same answer, but one has a strong formal foundation and the other is just lazy mathematicians playing loose-and-fast with notation.

User avatar
t0rajir0u
Posts: 1178
Joined: Wed Apr 16, 2008 12:52 am UTC
Location: Cambridge, MA
Contact:

Re: 1 + 1 = ?

Postby t0rajir0u » Mon Feb 16, 2009 11:27 pm UTC

[imath]e^z[/imath] is a continuous homomorphism from the additive group [imath]\mathbb{C}[/imath] to the multiplicative group [imath]\mathbb{C}^{*}[/imath] with kernel the integer multiples of [imath]2 \pi i[/imath], hence restricts to an isomorphism
[math]\mathbb{C}/2 \pi i \mathbb{Z} \to \mathbb{C}^{*}.[/math]
There's nothing mysterious about this. The preimage of any element of [imath]\mathbb{C}^{*}[/imath] is a coset of the kernel like it always is, and this is perfectly rigorous; mathematicians play around with preimages all the time.

FatPhil
Posts: 3
Joined: Tue Feb 17, 2009 3:23 pm UTC

Re: 1 + 1 = ?

Postby FatPhil » Tue Feb 17, 2009 9:48 pm UTC

Tac-Tics wrote:
Macbi wrote:
Buttons wrote:Sure it is. It's well known that log 1 = 2pi*i, among other things. There's nothing wrong with the rule that log xy = log x + log y, so long as you don't assume that the result is the only value log xy equals.
If you don't treat your functions like functions the Mathematical Fairy stabs you to death with set squares.


Thank you Macbi!

This is the math forum, not the physics forum. Functions are single-valued. Always. If you need to choose bewteen two alternative "branches", then pick on and be done with it!


Functions may be single-valued, but it's perfectly valid for the codomain to be a set of equivalence classes, rather than a single value. Of course, you've then got to use the members of those sets appropriately, but if you're prepared to do the pedantry to start the process, you've probably got the skills to pull it off.


Return to “Mathematics”

Who is online

Users browsing this forum: No registered users and 10 guests