Why Does 2 Seem to Equal 3 in This Mathematical Scenario?

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Discussion Overview

The discussion revolves around a mathematical scenario where the claim is made that 2 equals 3 through a series of algebraic manipulations. Participants explore the validity of the steps taken in the manipulation and the implications of squaring both sides of an equation.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant proposes that if 2 equals 3, then manipulating the equation leads to an apparent contradiction.
  • Another participant argues that the original claim does not prove anything and points out that squaring both sides involves multiplying by different numbers.
  • Several participants express confusion over the arithmetic presented, noting specific errors in the calculations.
  • Another participant highlights the risk of extraneous roots when squaring equations and emphasizes that assuming what one is trying to prove is a critical error.
  • A further explanation is provided regarding the logical fallacy of assuming equality between two variables without proper justification.
  • One participant clarifies that their intention was not to prove anything but to understand the mistakes in the original claim.

Areas of Agreement / Disagreement

Participants generally disagree on the validity of the original mathematical manipulation, with multiple viewpoints on the errors involved. There is no consensus on a single correct interpretation of the steps taken.

Contextual Notes

Participants note limitations in the original proof, including the assumption that 2 equals 3 and the potential for extraneous roots when squaring both sides of an equation.

PrincePhoenix
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Why is 2 = 3 over here?

Suppose 2=3

Then subtract 5/2 from both sides.
2 - 5/2 = 3 - 5/2
4-5/2 = 6-5/2
-1/2 = 1/2

Take square root
(-1/2)2 = (1/2)2
1/4 = 1/4

What's wrong? why is it proved that 2 = 3?
 
Last edited:
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You didn't prove anything. Note that when you square both sides you effectively multiply the sides by different numbers.
 
Maybe it is because I am not fully awake yet, but I do not follow your arithmetic.

2 - 5/4 = 3/4 not -1/2
3 - 5/4 = 7/4 not 1/2
 
Doc Al said:
You didn't prove anything. Note that when you square both sides you effectively multiply the sides by different numbers.
So you mean that taking square was wrong? I thought I did everything on both sides of the equation. I mean which step is incorrect?
 
pbandjay said:
Maybe it is because I am not fully awake yet, but I do not follow your arithmetic.

2 - 5/4 = 3/4 not -1/2
3 - 5/4 = 7/4 not 1/2
I corrected the mistake. Check again.
 
When you square an equation, you run the risk of creating extraneous roots. For example, [itex]-2^2 = 2^2 = 4[/itex] but this does not prove that [itex]-2 = 2[/itex]. Also, another mistake you made is assuming what you're trying to prove. To have a correct proof you would need to begin with the equaltiy [itex]1/4 = 1/4[/itex] and proceed to show that [itex]2 = 3[/itex] (which you can't do).

Edit: Just so it's clear, the biggest fallacy in your "proof" was the assumption that [itex]2 = 3[/itex]. You can't assume what you're trying to prove.
 
To absolutely clear, here's why you can't assume what you're proving . . .

Let [itex]a,b \in \mathbb{R}[/itex] and suppose that [itex]a = b[/itex]. Then clearly we have that [itex]0*a = 0*b = 0[/itex]. Hence, if [itex]a[/itex] and [itex]b[/itex] are any two real numbers, then they are equal to each other (using your same logic). A correct proof would need to start at the equality [itex]0 = 0[/itex] and work from there (which is impossible).
 
Thanks for the answers. I didn't want to PROVE anything here. This was just shown to me by a friend and I just wanted to know what was wrong.
 

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