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You always need to start with something when doing math, most people use a set of axioms known as Peano axioms. The 5th one says 0 is a natural number. These axioms are the basis of math as we now know it. They are the things we assume to be true.

Answer 2: Prove existence of zero

Suppose, to the contrary that zero does not exist. Further suppose that a=b.

Then:

ab = b^2

a^2 - ab = a^2 - b^2

a( a - b ) = (a+b)(a-b). Now, since we supposed that zero does not

exist, (a-b) must be equal to some number

other than zero. Therefore,

a = (a+b) (We divide both sides by a-b, which, by

supposition, is a non-zero number).

a = (a+a) (a=b, We supposed that a=b is a given)

1a = 2a

1 = 2. We have 1=2, an obvious contradiction,

therefore, zero does exist.

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13y ago

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