Proof That dx = ln(2) Meme: The Hilarious Math Fallacy Explained

The internet is a treasure trove of content, and for those with a penchant for numbers, that includes a vibrant collection of math memes. Among the most circulated and perplexing is the 'Proof that dx=ln(2)' meme. At first glance, it appears to be a legitimate, multi-step mathematical derivation, complete with the formal 'Q.E.D.' (Quod Erat Demonstrandum – 'which was to be demonstrated') at its conclusion. However, anyone with a solid grasp of calculus will quickly realize the punchline: it concludes that the differential 'dx' is equal to a constant, ln(2), which is a fundamental misuse of mathematical notation.

Deconstructing the Fallacy: Step-by-Step

Let's dissect this viral mathematical joke, line by line, to uncover where its humor and its critical errors lie.

1. Consider ∫edx−1:

   • The Intent: To set up an integral.
   • The Error (Subtle): The expression edx is not a standard function to integrate with respect to x. While you might see eh in a limit definition, treating dx directly as an exponent within a function to be integrated with respect to x is unconventional and sets the stage for confusion. 'dx' typically denotes the variable of integration, not an exponent.

2. Multiply by dxdx​:

   • The Intent: To manipulate the expression.
   • The Error: While dxdx​=1 is technically true, this multiplication is used here to create a non-standard form for the integral in the next step, rather than for a meaningful mathematical purpose in this context.

3. ∫dxedx−1​dx:

   • The Intent: To set up a form that looks like a limit definition.
   • The Error: This notation is highly ambiguous and non-standard. The denominator 'dx' is not meant to be treated as a variable in a quotient inside an integral in this manner.

4. As dx is an infinitesimally small change in x, we may state that dx=limh→0​h:

   • The Intent: To bridge to a limit definition.
   • The Error/Misleading Statement: While dx represents an infinitesimal change, equating it directly to limh→0​h is an oversimplification that blurs the lines between a differential and a variable approaching zero. Differentials have a specific, formal definition in non-standard analysis or differential forms, but they are not just 'h' approaching zero.

5. ∫limh→0​heh−1​dx:

   • The Intent: To correctly replace the problematic term with a known limit.
   • The Correct Part: The limit limh→0​heh−1​ is indeed a fundamental limit in calculus, often used to define the derivative of ex at x=0.

6. Now evaluating limh→0​heh−1​:

   • The Intent: To show the evaluation process.

7. L'Hôpital's states that limh→0​heh−1​=limh→0​1eh​=1:

   • The Correct Part: This is a correct application of L'Hôpital's Rule to evaluate the indeterminate form 00​. The limit is indeed 1.

8. Plug in the solved limit for ∫limh→0​heh−1​dx=∫1dx:

   • The Intent: To simplify the integral.
   • The Correct Part (in isolation): Substituting the limit's value results in ∫1dx. This step, in itself, is mathematically sound given the previous (flawed) setup.

9. ∫1dx=x+C:

   • The Correct Part: This is the correct evaluation of the indefinite integral of 1.

10. dxd​∫f(x)dx=f(x), therefore dxd​(x+C)=1=edx−1:

    • The Intent: To apply the Fundamental Theorem of Calculus and link back to the original function.
    • The CRITICAL Error:
      • The statement dxd​∫f(x)dx=f(x) is correct. And dxd​(x+C)=1 is also correct.
      • However, the jump to equate 1=edx−1 is the core logical fallacy. The f(x) from the theorem should be the original function being integrated. In the initial (flawed) integral, it was implicitly something related to edx−1. But here, they are equating the derivative of the result (1) with the original (ill-defined) integrand (edx−1). This is a fundamental misapplication of the theorem and a complete breakdown of logical consistency. You cannot just equate 1 to edx−1 out of context.

11. edx−1=1 so edx=2 so dx=ln(2):

    • The Intent: To solve for 'dx'.
    • The Punchline/Absurd Conclusion: Because the equation edx−1=1 was derived from a faulty premise (Step 10), the subsequent algebraic manipulations (taking logarithms) lead to an utterly meaningless conclusion. 'dx' is a differential, not a variable that can be solved for as a constant value like ln(2). It represents an infinitesimal quantity or operator, not a fixed number.

12. Q.E.D.:

    • The Humor: This formal Latin abbreviation ("Quod Erat Demonstrandum" - "which was to be demonstrated") traditionally marks the end of a rigorous mathematical proof. Its inclusion here, after such a cascade of errors, serves as the ultimate comedic flourish, certifying a completely invalid argument as legitimate.

Why This Meme Resonates

The 'Proof that dx=ln(2)' meme is a testament to the internet's ability to blend high-brow concepts with low-brow humor. It's funny because:

• It looks convincingly like a real proof to the untrained eye, making the reveal of its absurdity even more impactful.
• It exploits common areas of confusion in calculus (like the meaning of dx or the application of fundamental theorems).
• It's relatable for anyone who has ever felt overwhelmed by complex mathematical notation or a proof that just 'doesn't click.'

In essence, this meme is a brilliant piece of mathematical satire, leveraging the precise language of proofs to derive a wonderfully incorrect outcome, giving both mathematicians and casual observers a good laugh while subtly highlighting the importance of understanding underlying mathematical principles.