A strange fact as well as big doubt

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Discussion Overview

The discussion revolves around the differentiation of the expression x^2 represented as x added to itself x times. Participants explore the implications of this differentiation and the validity of the steps taken, questioning whether the conclusion that 1=2 is a mistake or a misunderstanding of the differentiation process.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions the validity of differentiating x^2 as x added to itself x times, leading to the conclusion that 1=2.
  • Another participant clarifies that the expression should be interpreted as x+x+x... (x times) and highlights issues with differentiating when the number of terms varies with x.
  • Concerns are raised about the continuity of the function f(x) = x+x+x... (x times), particularly when x is not a nonnegative integer, which affects the ability to differentiate.
  • A participant points out that for differentiation to be valid, all instances of x must vary, and holding "x times" fixed leads to confusion in the differentiation process.
  • Some participants express interest in the problem, noting it as an intriguing thought experiment despite the confusion surrounding the definitions used.

Areas of Agreement / Disagreement

Participants generally do not agree on the validity of the differentiation process and the interpretation of "x times," indicating multiple competing views and unresolved questions regarding the mathematical handling of the expressions.

Contextual Notes

Limitations include the dependence on the definition of "x times" and the continuity of the function, which are not clearly established in the discussion. The mathematical steps involved in the differentiation process remain unresolved.

iitjee10
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We know x^2 = x*x*x*x... x times
If we differentiate both sides with respect to x, we have
2x=1+1+1+... x times
i.e. 2x=x
i.e 1=2
How is it possible or am i making a mistake?

Please helpp!
 
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iitjee10 said:
We know x^2 = x*x*x*x... x times
If we differentiate both sides with respect to x, we have
2x=1+1+1+... x times
i.e. 2x=x
i.e 1=2
How is it possible or am i making a mistake?

Please helpp!
Erm, well to start with x2 = x*x
 
I'm assuming you meant x+x+x+x... x times.

The problem with trying to take the derivative of that with respect to x is first, the number of terms changes as well as each term itself, and second, you can't really write it like that if x is not a nonnegative integer.
 
And even if x is a nonnegative integer, your function, f(x) = x+x+x... (x times), is only defined at the nonnegative integers, so your function f(x) isn't continuous anywhere, so it can't be differentiated.
 
Your way of handling addition and multiplication is confused as you have not prperly defined what x times means. That aside the problem with your differentiation is that for a derivative to be correct all x in the function must be allowed to vary, you hold your x times fixed.
x^2=x+x+x+... xtimes
differentiate
2x=x'+x'+x'+... xtimes
+x+x+x+... x' times
=1+1+1+... x times
+x+x+x+... 1 times
=x+x=2x

So the problems were holding the x in "x times" fixed and not properly defining what "x times means.
 
lurflurf said:
Your way of handling addition and multiplication is confused as you have not prperly defined what x times means. That aside the problem with your differentiation is that for a derivative to be correct all x in the function must be allowed to vary, you hold your x times fixed.
x^2=x+x+x+... xtimes
differentiate
2x=x'+x'+x'+... xtimes
+x+x+x+... x' times
=1+1+1+... x times
+x+x+x+... 1 times
=x+x=2x

So the problems were holding the x in "x times" fixed and not properly defining what "x times means.

Very interesting actually. I had never thought of that.
 

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