Difference between revisions of "Proof that 2=1"

(Note:)
(Note:)
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==Note:==
 
==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.
 
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.
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==Alternate Proof==
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Consider the continued fraction <math>3-\frac{2}{3-\frac{2}{3-frac{2}{3- \cdots}}}.</math>

Revision as of 13:40, 27 January 2022

Proof

1) $a = b$. Given.

2) $a^2 = ab$. Multiply both sides by a.

3) $a^2-b^2 = ab-b^2$. Subtract $b^2$ from both sides.

4) $(a+b)(a-b) = b(a-b)$. Factor both sides.

5) $(a+b) = b$. Divide both sides by $(a-b)$

6) $a+a = a$. Substitute $a$ for $b$.

7) $2a = a$. Addition.

8) $2 = 1$. Divide both sides by $a$.

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 $a-b$ is $0$ as $a = b$, since one cannot divide by zero, the proof is incorrect from that point on.

Thus, this proof is false.

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 $3-\frac{2}{3-\frac{2}{3-frac{2}{3- \cdots}}}.$