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Proving 1 + 1 = 2How can we prove that 1+1=2? We could say that 1 pencil + 1 pencil = 2...

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jackmacc2000 | Student, Grade 11 | eNoter

Posted December 4, 2011 at 12:08 PM via web

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Proving 1 + 1 = 2

How can we prove that 1+1=2? We could say that 1 pencil + 1 pencil = 2 pencils, which is correct, but then we only know that that is true for pencils, but not for all of the other things in the world. For example, a person is trying to prove that y=x for any y and x. He tries y=2 and x=2, y=3 and x=3, and many numbers, being convinced that y=x because in all of those examples, y=x. However, we know that this is not the case since an easy example like x=3 and y=2 results in y not being equal to x. So we might think that 1 pencil + 1 pencil = 2 pencils, 1 computer + 1 computer = 2 computers, 1 fireplace + 1 fireplace = 2 fireplaces, etc., thus 1+1=2. But we might be as narrow-minded as the y=x guy; he is not thinking about any other cases, but only thought about the cases in which y is equal to x. So, is 1+1 really equal to 2? Always?

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wannam | High School Teacher | (Level 3) Educator

Posted December 5, 2011 at 12:22 AM (Answer #2)

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The problem with this question is that 1 and 2 are numbers while x and y are respresenations of numbers.  The number 1 is always 1 while x and y could be anything.  Y does not always equal x because y and x might represent different numbers at different times.  A number will never change its value.  1+1 will always equal 2 because 1 and 2 will never change their value.  X and y might be equal if the numbers they represent are equal.  Since x and y can change their values, there might be times when they are not equal as well.  Comparing 1+1 to x=y is similar to comparing a fact to an opinion.  Facts cannot change just as number values cannot change.

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dimidns14 | Student, Grade 9 | eNoter

Posted December 5, 2011 at 6:21 AM (Answer #3)

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so shes saying that 1+1=2 if it is tangelble or if it a number like 123 but not x or y because x could mean a number while y could a different.

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soccrkidz13 | Student, Grade 10 | eNoter

Posted December 6, 2011 at 8:09 AM (Answer #4)

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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 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.

and the actual site that i found out this is http://mathforum.org/library/drmath/view…

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mychmrmntic | Student, Undergraduate | eNotes Newbie

Posted December 10, 2011 at 12:53 AM (Answer #5)

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Proving 1 + 1 = 2

How can we prove that 1+1=2? We could say that 1 pencil + 1 pencil = 2 pencils, which is correct, but then we only know that that is true for pencils, but not for all of the other things in the world. For example, a person is trying to prove that y=x for any y and x. He tries y=2 and x=2, y=3 and x=3, and many numbers, being convinced that y=x because in all of those examples, y=x. However, we know that this is not the case since an easy example like x=3 and y=2 results in y not being equal to x. So we might think that 1 pencil + 1 pencil = 2 pencils, 1 computer + 1 computer = 2 computers, 1 fireplace + 1 fireplace = 2 fireplaces, etc., thus 1+1=2. But we might be as narrow-minded as the y=x guy; he is not thinking about any other cases, but only thought about the cases in which y is equal to x. So, is 1+1 really equal to 2? Always?

1 +1=3 this is simple math, which is totally true. x and y are variables which require algebra. variables are just a symbol that represents a quantity. If the equation is x=y, then the graph will show all the other senerios. Algebra usually requires a graph that shows the other cases. In a formula however, it will state either what x or y will equal in order to solve for the other.

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studybug12 | Student, Grade 11 | eNotes Newbie

Posted December 24, 2011 at 6:49 AM (Answer #6)

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1 + 1 is 2. Numbers are SET. They do not ever change. If you have 1 Sharpie Marker and 1 Crayola Marker, you will ALWAYS have 2 markers....( unless you loose one lol) 1 + 1 = 2. Simple as that. x is a variable, whose value is susseptable to change. x may not always be = y . y may not always be equal to 1. Variables are in flux, numbers are NOT.

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candy123456 | Student, Grade 9 | eNoter

Posted December 29, 2011 at 1:51 AM (Answer #7)

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u can use ur finger to prove it...

 

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enotechris | College Teacher | (Level 2) Senior Educator

Posted January 25, 2012 at 1:26 AM (Answer #8)

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As stated, x and y are variables with unknown values that can add to a sum. However, 1 is a constant, and added to itself it also produces a sum; because we know it's a constant, and we know the value of the constant, we know it always resolves to 2.

You are representing Whole Numbers when you assign one pencil, and another pencil totalling two pencils, and the properties of Whole Numbers are such that they cannot be fractions, negative, or irrational; they are simply constants of known value.

 

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