Why 1+1 =2 on the 20 greatest equation list?

[viktor]

I'm doing assignment which concern the 20 greatest equation, but i can't figure out why 1+1=2 is in it?

 

[CaptainQuasar]

Maybe it's like Time's Person of the Year where they select based on influence rather than being good or bad, so they choose people like Stalin and Hitler. Maybe 1+1=2 is the Adolf Hitler of equations.

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[renigade666]

Its the first equation that everyone learns.

Hence, why it's so great.

 

[arildno]

Eeh?
Greatest equations??

Why do you think 1+1=2 is an equation?

 

[RufusDawes]

Cause it proves addition works ?

 

[twinsen]

Why do you think 1+1=2 is an equation?

Because it demonstrates equivalence which is really the most basic function of an equation isn't it?

Alex

 

[CaptainQuasar]

That is definitely an equation.

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[Carolyn]

because it's not as simple an equation as everybody thinks it is. It actually takes a lot of effort to prove it.

 

[Poop-Loops]

If you have to prove to yourself that 1 + 1 = 2, then maybe you should find a different field to work in.

 

[fasterthanjoao]

If you have to prove to yourself that 1 + 1 = 2, then maybe you should find a different field to work in.

Maybe you should try it..

 

[f95toli]

If you have to prove to yourself that 1 + 1 = 2, then maybe you should find a different field to work in.

Well, the proof can be found in Russell/Whiteheads Principia Mathematica (Yes, that really is the name of the books) and it is far from trivial.
1+1=2 is a very fundamental equation in all work on the foundations of mathematics.

 

[Poop-Loops]

Maybe you should try it..

If I have one carrot:

[PLAIN]http://www.janetscreativepillows.com/Fruit%20&%20Vegetable%20Pillows/carrot-pillow_small.gif

And then add another carrot:

[PLAIN]http://www.janetscreativepillows.com/Fruit%20&%20Vegetable%20Pillows/carrot-pillow_small.gif

I now have TWO carrots:

http://images.jupiterimages.com/common/detail/11/05/23370511.jpg

Also, for those actually debating whether 1 + 1 = 2 is an equation: wtf is wrong with you? There is an EQUALS sign right there. That is what defines an equation, no?

 

[f95toli]

If I have one carrot:
And then add another carrot:

I now have TWO carrots:

The problem is that this is only an argument (albeit a very plausible one), not a proof. It would be "good enough" in most sciences but not in mathematics.

 

[Werg22]

Adding a carrot with another to obtain two carrots just happens to be a convenient application of set theory - it doesn't prove nor show anything.

The problem is that this is only an argument (albeit a very plausible one), not a proof. It would be "good enough" in most sciences but not in mathematics.

Here is a simple argument. We define 1 + 1 as the number of objects in a set that is 1-1 with {1}U{1}. Fortunately, {1,2} is 1-1 with {1}U{1}. Now {1,2} is represented by a cardinal number, that is 2. Is there another set of the form {1, 2, ... n} that satisfies the aforementioned conditions? If there was, the set representing it would also be 1-1 with {1,2}. However, no such subset of N exists. Therefore, 2 is the unique answer to 1 + 1.

 

[Poop-Loops]

That's a lot of jargon there. I wish I understood it. :(

 

[DaveC426913]

That's a lot of jargon there. I wish I understood it. :(

That's a 'tude much more open to learning something new than 'wtf is wrong with you'.

 

[John Creighto]

Well, the proof can be found in Russell/Whiteheads Principia Mathematica (Yes, that really is the name of the books) and it is far from trivial.
1+1=2 is a very fundamental equation in all work on the foundations of mathematics.

It all depends on context. For instance, the approach would be different using:
http://en.wikipedia.org/wiki/Peano_axioms
then it would be using category theory.

 

[DeadWolfe]

Who the hell is making a list of the "20 greatest equations"?

 

[Asphodel]

I don't know, but if the Pythagorean theorem isn't on it then it's a sucky list.

1 + 1 = 2 (or for that matter a + 1 for any integer a) should be fundamental to the definition of integers. You can't construct an ordered field of integers without defining their order, which means that for any integer you should be able to trivially say what the next one is. Sometimes mathematicians try to do very simple things with complicated tools, and it's rather like using a Rube Goldberg machine instead of a screwdriver. I tend to favor more concise solutions...

 

[John Creighto]

I don't know, but if the Pythagorean theorem isn't on it then it's a sucky list.

The Pythagoras theorem is on it:
http://physicsworld.com/cws/article/print/20407/1/pwpov2_10-04

 

[LukeD]

Except that you can't prove that 1+1=2 any more than you can prove that the empty set exists. 1+1=2 is a definition of what it means to do addition. Sure, you can construct a system and then prove that in that system 1+1=2 is true, but that's not a proof of 1+1=2; it's a proof that you have working model of addition.

Yes, 1+1=2 is an equation. x=x is also an equation. So is 1=3. Equations don't even need to be true to be equations. They just need to make sense in whatever system you're interpreting them in.

 

[HallsofIvy]

Eeh?
Greatest equations??

Why do you think 1+1=2 is an equation?

From anyone else, I would think that was a foolish post. From arildno, I think I am missing something! Arildno, I think 1+ 1= 2 is an equation because it has an "=" in it! What am I missing?

If you have to prove to yourself that 1 + 1 = 2, then maybe you should find a different field to work in.

Yes, you should be working in mathematics!

If I have one carrot:

And then add another carrot:

I now have TWO carrots:

That's a statement about carrots, not about mathematics!

Also, for those actually debating whether 1 + 1 = 2 is an equation: wtf is wrong with you? There is an EQUALS sign right there. That is what defines an equation, no?

Now this I agree with. Arildno, "wtf"?

 

[CaptainQuasar]

You can't construct an ordered field of integers without defining their order, which means that for any integer you should be able to trivially say what the next one is.

Just as a note, defining an order is not the same thing as defining a property of addition.

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[John Creighto]

Just as a note, defining an order is not the same thing as defining a property of addition.

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2 is a definition. You can only prove 1+1=2 after you define what 2 is. It is kind of like proving God.

 

[K.J.Healey]

Yes you have to define what + means too for this set.

1 + 1 = 10 should be on there too. Talk about influential...

 

[CaptainQuasar]

2 is a definition. You can only prove 1+1=2 after you define what 2 is. It is kind of like proving God.

What I meant is that you might define something like a http://mathworld.wolfram.com/PartialOrder.html" and then you can say that one object of the set is more or less than another but you don't necessarily have any operation equivalent to addition.

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[John Creighto]

What I meant is that you might define something like a http://mathworld.wolfram.com/PartialOrder.html" and then you can say that one object of the set is more or less than another but you don't necessarily have any operation equivalent to addition.

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== Arithmetic ==
The Peano axioms can be augmented with the operations of addition and multiplication and the usual total (linear) ordering on ''N''. The respective functions and relations are constructed in second-order logic , and are shown to be unique using the Peano axioms.

Addition is the function + : ''N'' × ''N'' → ''N'' (written in the usual infix notation), defined recursively as:
$$\begin{align} a + 0 &= a ,\\ a + (S (b)) &= S (a + b). \end{align}$$

http://en.wikipedia.org/wiki/Peano_axioms#Arithmetic
http://en.wikipedia.org/wiki/Peano_axioms

 

[OmCheeto]

That's a lot of jargon there. I wish I understood it. :(

I concur. Math is sometimes incomprehensible.

If 1 + 1 = 2 then 1 - 2 = -1
I disproved this at the age of 4, when my sister came home from her third grade class, all excited, and tried to teach me that when you have one pie and take away two pies, you end up with minus one pie.

"No. no. no."; I said. "You can't take away more pies than you have. There's something wrong with your math."

My sister denies remembering the conversation.

 

[CaptainQuasar]

John Creighto, the thing you quoted is talking about a set with a total order on it, not a partial order. Those are different things.

If you tried to do a square-peg-in-a-round-hole type fitting of real number arithmetic into a partially ordered set the Cartesian product ''N'' × ''N'' → ''N'' wouldn't necessarily be true. You're to be commended for your cutting-and-pasting-from-Wikipedia skills, though.

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[CaptainQuasar]

I disproved this at the age of 4, when my sister came home from her third grade class, all excited, and tried to teach me that when you have one pie and take away two pies, you end up with minus one pie.

Just think of it… you would actually lose weight eating negative pie! Get hold of your sister OmCheeto, we're all going to be rich!

Oh, wait… nevermind… that would violate Newton's Law of Conservation of Pie.

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[John Creighto]

Oh, wait… nevermind… that would violate Newton's Law of Conservation of Pie.

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Obviously a negative pie is made out of the mass energy stolen from the zero point energy field.

 

[John Creighto]

John Creighto, the thing you quoted is talking about a set with a total order on it, not a partial order. Those are different things.

If you tried to do a square-peg-in-a-round-hole type fitting of real number arithmetic into a partially ordered set the Cartesian product ''N'' × ''N'' → ''N'' wouldn't necessarily be true. You're to be commended for your cutting-and-pasting-from-Wikipedia skills, though.

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I don't believe I said that you could create addition from a partial order.

 

[CaptainQuasar]

I decided it would be fun to explain the partial order thing.

Let's say you have a set with four members, ɐ, ə, Ϟ, and ש.

You define a partial order that says ɐ < Ϟ < ə and ɐ < ש < ə.

But because this is a partial order there's no definition of a relationship between Ϟ and ש. So what would Ϟ + ש be? There's no answer, unless you define extra rules for the operation that make it unlike addition in integers or real numbers.

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[CaptainQuasar]

I don't believe I said that you could create addition from a partial order.

Uh… then why did you respond to me saying something about a partial order by quoting that?

Whatever… cut and paste from Wikipedia whatever you want and I'll say my stuff, this town is big enough for the both of us.

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[kurt.physics]

I believe that 1 +1 = 2 is quite an important equation because it would be the start of mathematics and mathematical reasoning. With out this basic principal, we would not have calculus, trigonometry, topology, cohomology!