Fallacious proof/2equals1: Difference between revisions

Latest revision as of 13:04, 16 May 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

@@ Line 1: / Line 1: @@
-=== 2 = 1 ===
+The following proofs are examples of [[fallacious proof]]s, namely that <math>2 = 1</math>.
+== Proof 1 ==
 Let <math>a=b</math>.
@@ Line 13: / Line 15: @@
 === Explanation ===
-The trick in this argument is when we divide by <math>a^{2}-ab</math>. Since a=b, <math>a^2-ab</math> equals 0, and dividing by zero is illegal.
+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 ==
+<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>
+<cmath>2 + 0 + 0 \ldots = 1 + 0 + 0 \ldots</cmath>
+<cmath>2 = 1</cmath>
+=== Explanation ===
+The given series does not converge.  Therefore, manipulations such as grouping terms before adding are invalid.
+''[[Fallacy#2_.3D_1 | Back to main article]]''

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Fallacious proof/2equals1: Difference between revisions

Latest revision as of 13:04, 16 May 2025

Contents

Proof 1

Explanation

Proof 2

Explanation