Understanding the Flaws in Proofs That 1 2 and the Concept of Natural Units in Physics

The supposed proof that 1 2 has been a recurring subject in mathematical discourse due to its deceptive simplicity and brilliant trickery. Let's delve into the steps of this proof and identify the critical flaw, followed by an exploration of the concept of natural units in physics, where some values are inherently constant and equal to one.

Flaws in Proofs That 1 2

The purported proof of 1 2 involves a series of algebraic manipulations that ultimately lead to a nonsensical conclusion. While the steps may seem logical at first, a critical examination reveals that the proof is fundamentally flawed due to certain invalid mathematical operations.

Step-by-Step Analysis of the Flawed Proof

Consider the following proof:

Assume: ( a b ) Multiply both sides by ( a ): ( a^2 ab ) Subtract ( b^2 ) from both sides: ( a^2 - b^2 ab - b^2 ) Factor the left side: ( (a b)(a - b) ab - b^2 ) Factor the right side incorrectly: ( (a b)(a - b) (a - b)b ) Divide both sides by ( a - b ): ( a b b ) Since ( a b ), substitute ( b ) for ( a ): ( b b b ) Simplify: ( 2b b ) Divide both sides by ( b ): ( 2 1 )

The error occurs in step 5, where the factorization is incorrectly simplified. The correct factorization of the right side should be ( ab - b^2 b(a - b) ). By dividing both sides by ( a - b ), we assume that ( a eq b ), which directly contradicts the initial assumption ( a b ). Division by zero (i.e., ( a - b 0 )) is undefined, resulting in a false conclusion.

Key Flaw: Division by Zero

The critical mistake in this proof is the division by zero. If ( a b ), then ( a - b 0 ), and dividing by zero is not permissible in mathematics. This invalidates the entire proof and demonstrates why 1 cannot be equal to 2.

Concept of Natural Units in Physics

In physics, natural units are systems of measurement in which certain constants of nature are set to 1. This simplifies equations and provides a more intuitive understanding of physical phenomena. Some common examples of natural units include Planck units, where constants like the speed of light, gravitational constant, and elementary charge are set to 1.

Elementary Charge and the Speed of Light

The elementary charge ( e ) is a fundamental constant in physics, representing the magnitude of the electric charge carried by a single proton or electron. In natural units, the charge can be set to 1, which simplifies many equations in particle physics. Similarly, the speed of light ( c ) is a key constant in relativity, and setting it to 1 (dimensionless) allows calculations to focus on other parameters.

Recent Analogous Results in High Energy Physics

A recent study by Juri Smirnov, published on March 17, 2021, on arXiv, explores the concept of 'accidentally asymmetrical dark matter.' This study uses natural units to describe the confining dark sector with heavy dark quarks. By using this hypothetical framework, Smirnov's work provides insights into the formation of dark matter, illustrating the utility of natural units in complex theoretical physics.

Accidental Asymmetry in Dark Matter

Using the concept of natural units, Smirnov's study investigates how asymmetries in the dark sector can arise naturally within certain theoretical frameworks. These asymmetries, described as 'accidental,' highlight the importance of natural units in understanding the behavior of dark matter in high energy physics.

By leveraging natural units, physicists can simplify complex equations and focus on the essential physical phenomena without the clutter of dimensional constants. This approach not only enhances the clarity of the mathematical description but also aids in uncovering new insights and phenomena in high energy physics.