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?

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  • 1 decade ago
    Favorite Answer

    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

    definition of addition to this:

    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 decade ago

    1+1=2

  • 1 decade ago

    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
    1 decade ago

    the proof is 1 + 1 = 2

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  • Anonymous
    1 decade ago

    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 decade ago

    1 apple + 1 apple

    = 2 apple

    ∴ 1 + 1 = 2

  • 1 decade ago

    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
    1 decade ago

    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
    1 decade ago

    Honestly i se that as relative to what objects you are adding, and relative to what you call "adding." If you are really going in quantitative direction yes i think that WOULD be 2. But if you are adding two things together like salt and water to create a homogenous substance, wouldn't that still equal one?

    Source(s): My mind which is wrong on many occasions
  • okdan
    Lv 4
    1 decade ago

    You might say that 1+1 is the DEFINITION OF 2, so there is nothing to prove.

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