Difference between revisions of "Proof that 2=1"

Latest revision as of 14:11, 19 February 2025

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-==Proof==
+#REDIRECT [[Fallacy#2=1]]
-) <math>a = b</math>. Given.
-) <math>a^2 = ab</math>. Multiply both sides by a.
-) <math>a^2-b^2 = ab-b^2</math>.  Subtract <math>b^2</math> from both sides.
-) <math>(a+b)(a-b) = b(a-b)</math>.  Factor both sides.
-) <math>(a+b) = b</math>. Divide both sides by <math>(a-b)</math>
-) <math>a+a = a</math>.  Substitute <math>a</math> for <math>b</math>.
-) <math>2a = a</math>.  Addition.
-) <math>2 = 1</math>.  Divide both sides by <math>a</math>.
-==Error==
-Usually, if a proof proves a statement that is clearly false, the proof has probably divided by zero in some way.
-In this case, the quantity of <math>a-b</math> is <math>0</math> as <math>a = b</math>, since one cannot divide by zero, the proof is incorrect from that point on.
-<b>Thus, this proof is false.</b>
-==Note:==
-If this proof were somehow true all of mathematics would collapse. Simple arithmetic would yield infinite answers. This is why one cannot divide by zero.
-==Alternate Proof==
-Consider the continued fraction <math>3-\frac{2}{3-\frac{2}{3-\frac{2}{3- \cdots}}}.</math> If you set this equal to a number <math>x</math>, note that
-<math>3-\frac{2}{x}=x</math> due to the fact that the fraction is infinitely continued. But this equation for <math>x</math> has two solutions, <math>x=1</math> and <math>x=2.</math> Since both <math>2</math> and <math>1</math> are equal to the same continued fraction, we have proved that <math>2=1.</math> QED.