# Proving 1 + 1 = 2How 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...

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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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.

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.

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.

Proving 1 + 1 = 2How 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.

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…

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.