# If 1+1=2, what's the proof?

See, we always know that 1+1=2, but what is the proof that the equation is true?

Relevance

The proof starts from the Peano Postulates, which define the natural

numbers N. N is the smallest set satisfying these postulates:

P1. 1 is in N.

P2. If x is in N, then its "successor" x' is in N.

P3. There is no x such that x' = 1.

P4. If x isn't 1, then there is a y in N such that y' = x.

P5. If S is a subset of N, 1 is in S, and the implication

(x in S => x' in S) holds, then S = N.

Then you have to define addition recursively:

Def: Let a and b be in N. If b = 1, then define a + b = a'

(using P1 and P2). If b isn't 1, then let c' = b, with c in N

(using P4), and define a + b = (a + c)'.

Then you have to define 2:

Def: 2 = 1'

2 is in N by P1, P2, and the definition of 2.

Theorem: 1 + 1 = 2

Proof: Use the first part of the definition of + with a = b = 1.

Then 1 + 1 = 1' = 2 Q.E.D.

Note: There is an alternate formulation of the Peano Postulates which

replaces 1 with 0 in P1, P3, P4, and P5. Then you have to change the

Def: Let a and b be in N. If b = 0, then define a + b = a.

If b isn't 0, then let c' = b, with c in N, and define

a + b = (a + c)'.

You also have to define 1 = 0', and 2 = 1'. Then the proof of the

Theorem above is a little different:

Proof: Use the second part of the definition of + first:

1 + 1 = (1 + 0)'

Now use the first part of the definition of + on the sum in

parentheses: 1 + 1 = (1)' = 1' = 2 Q.E.D.

Source(s): and the actual site that i found out this is http://mathforum.org/library/drmath/view/51551.htm...
• K
Lv 4

1+1=2

That's an interesting question if you really think about it. Who ever came up with that idea? It's just something we're told when we're 3 and remember forever.

1 + 1 = 2 isn't always true. If you use the 2 digit system 1 + 1 actually = 10

• Anonymous

the proof is 1 + 1 = 2

• Anonymous

Bertrand Russell and Alfred North Whitehead wrote the three volume 'Pincipia Mathematica' in the early twentienth century, and after 200 or 300 pages came up with "Therefore, 1 + 1 = 2." Russell later said it was the most difficult thing he ever did and it took something out of him that he never recovered from! Overthinking can kill!

1 apple + 1 apple

= 2 apple

∴ 1 + 1 = 2

I want to be a jerk here, but I'm not sure how old this person is, so I'll just make a joke I guess.

1+1= my 2 points

The proof is in the pudding.

• Anonymous

This can be proved with some common sense and just a tiny bit of knowledge, if u have one apple, and ur friend gives u another apple, u have 2 apples. The number one, represents a single item, person or anything for that matter, wen another single object is joined with this item, there becomes 2 items. Simple mate

u ought to get an education....

Source(s): COMMON SENSE!!!
• Anonymous