Difference between revisions of "Fallacious proof/2equals1"
m |
Flamefoxx99 (talk | contribs) m (→Explanation) |
||
(2 intermediate revisions by 2 users not shown) | |||
Line 1: | Line 1: | ||
− | === | + | The following proofs are examples of [[fallacious proof]]s, namely that <math>2 = 1</math>. |
+ | |||
+ | == Proof 1 == | ||
Let <math>a=b</math>. | Let <math>a=b</math>. | ||
Line 13: | 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 | + | 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. |
+ | |||
+ | == 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> | ||
+ | |||
+ | <math>2 + 0 + 0 \ldots = 1 + 0 + 0 \ldots</math> | ||
+ | |||
+ | <math>2 = 1</math> | ||
+ | </center> | ||
+ | |||
+ | === Explanation === | ||
+ | The given series does not converge. Therefore, manipulations such as grouping terms before adding are invalid. | ||
− | [[ | + | ''[[Fallacious_proof#2_.3D_1 | Back to main article]]'' |
Latest revision as of 07:29, 5 June 2013
The following proofs are examples of fallacious proofs, namely that .
Contents
[hide]Proof 1
Let .
Then we have
(since )
(adding to both sides)
(factoring out a 2 on the LHS)
(dividing by )
Explanation
The trick in this argument is when we divide by . Since , , and dividing by zero is undefined.
Proof 2
Explanation
The given series does not converge. Therefore, manipulations such as grouping terms before adding are invalid.