Logically wrong

1 = 2

Let a = b = 1
Multiply both sides by a : a² = ab
substract b² : a²-b² = ab-b² = b(a-b)
Hence (a+b)(a-b) = b(a-b) and a+b = b, that is 2 = 1 !
Why wrong ?

Other proof

Sum x terms equal to x :    x+x+x+x+...+x = x times x = x²
If we differentiate both sides, we get :
1+1+1+1+...+1 = x times 1 = x, and differentiating x² = 2x
Hence x = 2x for all x, that is 1 = 2 !
Why wrong ?

Did you like it ? One more time !

Let x solution of x²+x+1=0. x is then different from 0 and we can divide by x : x+1+1/x = 0
But x+1 = -x², hence -x²+1/x = 0 that is x² = 1/x, or x3 = 1 hence x = 1.
If we substitute into the first equation, we get 3 = 0 !
Why wrong ?

Even better, all the numbers are equal !

Let a et b any two numbers and m=(a+b)/2 their mean value.
a+b = 2m. Multiplying by a-b : (a+b)(a-b) = 2m(a-b) that is a²-b² = 2ma-2mb
Separate the a and b terms : a²-2ma = b²-2mb. Adding m² :
a²-2ma+m² = b²-2mb+m² that is (a-m)² = (b-m)² or a-m = b-m
hence a = b, and finally any two numbers a and b are equal !
Why wrong ?

Really imaginary !

Let i the imaginary number with i² = -1, that is i=√(-1) = √(1/-1) = 1/√(-1) = 1/i
Hence i² = 1, that is -1 = 1 !
Why wrong ?

 

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