Spot the mistake
Here's my attempt at proving that 2 = 1 . In which step did I first commit a flaw in my logic?
1 2 2 2 3 2 = 1 = 2 + 2 = 3 + 3 + 3 (2 times) (3 times)
Step 1: For any positive integer:
x 2 = x + x + ⋯ + x (x times)
Step 2: Now, differentiating with respect to x :
2 x = 1 + 1 + ⋯ + 1 (x times) .
Step 3: Summing this up, we get
2 x = x .
Step 4: Dividing by x , we get
2 = 1 .
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2 solutions
Even the first step is wrong. You have written 'for any non-zero number'; but equation (1) is true for positive numbers only.
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1.5squared does not equal 1.5+1.5 hence step 1 is wrong. The result works only for positive integers. Since the function's domain is not over a continuous set, differentiation is not defined.
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It is written x times x. It makes some sense that ( 1 . 5 ) 2 = 1 . 5 times 1 . 5 . So, it looks correct to me.
But you can't write it as a sequence of x. So, this step should be written true for positive integers only.
Fun fact is that:
( 1 . 5 ) 2 = one times and a half times of 1 . 5 . Resulting 1 . 5 ∗ 1 + 1 . 5 ∗ 0 . 5 . So , it looks true.
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@Sachin Vishwakarma – I stated 1.5 squared is not, equal to the sum of 1.5 and 1.5. You misread my post
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@Thomas Hesbach – Why did you represented the ( 1 . 5 ) 2 , as sum of 1.5 two times? Isn't that should be 1.5 times?
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@Sachin Vishwakarma – Yes. 1.5 + 1.5/2(BDMAS) = 2.25 = 1.5^2
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@Jase Jason – unless you interpret it that way,the step is wrong,but my main argument is that it doesn't make any sense for negative numbers
also x times doesn't make sense unless x is a natural number,which makes the statement wrong
@Rajdeep Bharati is correct,the answer is (1) if he's asking for the first mistake
I agree with pankaj
X²= x+x+.....+x
Here the right side of the equation is dependent on variable x, not only for the value but also for the No. Of terms in the equation. ( i.e X*X)
So if we differentiate only it's individual terms then it ll be like considering the equation to be x*1 ...
So we have to differentiate it again and add (same like a differentiation of multiplication)
2x = (1+1+...+1) (1) + (1) (1+1+...+1)
2x = x + x
2x = 2x
(2)
The second step is violating the linearity principle of differentiation operator.The right side of the equation is dependent on variable x, so the diff cannot be applied to each individual x.