Sophism #4: Any Two Numbers are Equal

Consider two arbitrary numbers a and b. Let

a + b = 2c

Rearrange this in two ways to get two new valid equations:

a - 2c = -b
a = -b + 2c

Multiply the corresponding sides of the two equations:

a² - 2ac = b² - 2bc

Now add c² to each side and simplify:

a² - 2ac + c² = b² - 2bc + c²
(a-c)² = (b-c)²

Finally, take the root of both sides:

a-c = b-c
a = b

Consequently, two arbitrarily chosen numbers a and b must be equal!

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