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Prove that C+C is not equal to 2C

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I don't think that's possible. Replace the C with a random number like 5. That's like saying "Prove that 5 + 5 is not equal to 2(5)". On the other hand, they are both expressions and expressions have no solutions. So it could just be because one is addition and the other multiplication.

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Just google 1=0. There is a proof there that can be modified. It's wrong of corse but it's hard to find the flaw.

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2C is 2xC

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It is.
1) Make C = 3 (or any other number)
2) C+C = 3 + 3 = 6
3) 2C means 2x3 =6
Thus C+C = 2C
You could make the argument that the "2" represents the number of "C's" left hand of the equation.

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