Art of Problem Solving
Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus
During AMC 10A/12A testing, the AoPS Wiki is in read-only mode and no edits can be made.

Fallacious proof/2equals1: Difference between revisions

← Older editNewer edit →
Charking (talk | contribs)
editor
3,291 edits
Proposed for deletion
Line 15: Line 15:


=== Explanation ===
=== Explanation ===
The trick in this argument is when we divide by <math>a^{2}-ab</math>. Since <math>a=b</math>, <math>a^2-ab = 0</math>, and dividing by [[zero (constant) | zero]] is undefined.
The trick in this argument is when we divide by <math>a^{2}-ab</math>. Since <math>a=b</math>, <math>a^2-ab = 0</math>, and dividing by [[0|zero]] is undefined.


== Proof 2 ==
== Proof 2 ==
<center>
<math>1 + 1 - 1 + 1 - 1 \ldots = 1 + 1 - 1 + 1 - 1 \ldots</math>


<math>(1 + 1) + (-1 + 1) + (-1 + 1) \ldots = 1 + (1 - 1) + (1 - 1) \ldots</math>
<cmath>1 + 1 - 1 + 1 - 1 \ldots = 1 + 1 - 1 + 1 - 1 \ldots</cmath>
 
<cmath>(1 + 1) + (-1 + 1) + (-1 + 1) \ldots = 1 + (1 - 1) + (1 - 1) \ldots</cmath>
<math>2 + 0 + 0 \ldots = 1 + 0 + 0 \ldots</math>
<cmath>2 + 0 + 0 \ldots = 1 + 0 + 0 \ldots</cmath>
 
<cmath>2 = 1</cmath>
<math>2 = 1</math>
</center>


=== Explanation ===
=== Explanation ===
The given series does not converge.  Therefore, manipulations such as grouping terms before adding are invalid.
The given series does not converge.  Therefore, manipulations such as grouping terms before adding are invalid.


''[[Fallacy#2_.3D_1 | Back to main article]]''


''[[Fallacious_proof#2_.3D_1 | Back to main article]]''
{{delete|unnecessary page}}

Revision as of 11:50, 15 February 2025

The following proofs are examples of fallacious proofs, namely that $2 = 1$.

Proof 1

Let $a=b$.

Then we have

$a^2 = ab$ (since $a=b$)

$2a^2 - 2ab = a^2 - ab$ (adding $a^2-2ab$ to both sides)

$2(a^2 - ab) = a^2 - ab$ (factoring out a 2 on the LHS)

$2 = 1$ (dividing by $a^2-ab$)

Explanation

The trick in this argument is when we divide by $a^{2}-ab$. Since $a=b$, $a^2-ab = 0$, and dividing by zero is undefined.

Proof 2

\[1 + 1 - 1 + 1 - 1 \ldots = 1 + 1 - 1 + 1 - 1 \ldots\] \[(1 + 1) + (-1 + 1) + (-1 + 1) \ldots = 1 + (1 - 1) + (1 - 1) \ldots\] \[2 + 0 + 0 \ldots = 1 + 0 + 0 \ldots\] \[2 = 1\]

Explanation

The given series does not converge. Therefore, manipulations such as grouping terms before adding are invalid.

Back to main article


This page has been proposed for deletion. Reason: unnecessary page

Note to sysops: Before deleting, please review: • What links here • Discussion page • Edit history

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Fallacious_proof/2equals1&oldid=242740"