Proof that 1 = 2

This is a silly proof that my thirteen-year old cousin showed me the other day. I'm ashamed to admit that it took me a full ten minutes to figure it out.

Let a = 1
Let b = 1

therefore:
a = b

multiply both sides by b:
ab = b^2

subtract a^2:
ab - a^2 = b^2 - a^2

factorise:
a(b - a) = (b - a)(b + a)

divide by (b - a):
[a(b - a)]/(b -a) = [(b - a)(b + a)]/(b -a)

cancel:
a = b + a

returning to our initial property:
a = 1
b = 1

therefore:
a = b + a
1 = 1 + 1
1 = 2


What is wrong with the above proof? (It's not difficult, but it's a good one to pull out to confuse people)

Division by zero .

Well done.

I can't believe it took me that long to figure it out!

Are there any 1=2 "proofs" that do not involve dividing by zero? I am trying to think of other mathematical rules that can be broken to derive such a proof.

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Originally Posted by paulrb

Are there any 1=2 "proofs" that do not involve dividing by zero?

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Here is one.

(identity)

(rearramging)

(taking square root)

(substituing

(evaluating the substitution)

(multiplying both sides by

(adding to both sides)

Quote:

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Originally Posted by TheAbstractionist

(multiplying both sides by

(adding 1 to both sides)

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The last step should lead to ...

Quote:

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Originally Posted by Chris L T521

The last step should lead to ...

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^ How?

The Abstractionist is correct!

Quote:

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Originally Posted by blueirony

^ How?

The Abstractionist is correct!

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Read what I quoted. Its correct now since he edited it.

Quote:

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Originally Posted by blueirony

^ How?

The Abstractionist is correct!

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I made a mistake in my original post, which was spotted by Chris L T521. I do this quite often, and when someone spots my mistakes, I normally thank the person who spotted them and just go back and edit my post. (Smile) (To Chris L T521: Thanks! (Happy))

By the way, did you spot the fallacy in the “proof”?

Is it that we can also write and hence i.e. -1 = -1...

Here's what could be the simplest -1 = 1 proof.


So 1 = -1.

Quote:

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Originally Posted by Deadstar

Is it that we can also write and hence i.e. -1 = -1...

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It’s along those lines, but I wouldn’t write The LHS is generally taken to mean the positive square root, so the equation is technically incorrect.

Hint:

Spoiler:

If (where and are real) then

This is logical 'proof' rather than mathematical....

Consider the following statement;
" If this statement is true, then 1= 2. " (let's call this A)

First, let's assume A to be true. on this supposition;
As statement A is true, "If statement A is true, then 1=2" is true.
and again, as statement A is true and
"if statement A is true, then 1=2" is true, we obtain 1=2.

Now we have proved that "If this statement is true, then 1=2 " is true.

Then, as 'this statement' is true, 1=2 (q.e.d.)

By using the same trick, anything can be proved.
This is called Curry's paradox..
see also: http://en.wikipedia.org/wiki/Curry%27s_paradox

A is racing against B; A wears sweater#1, B wears sweater#2.
Thet hit "finish line" at exactly same time.....WELL? (Giggle)

One empty bottle and two empty bottles contain the same amount of liquid.
If x empty bottles contain zero liquid in total, x is not immediately determineable.

Quote:

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Originally Posted by blueirony

Let a = 1
Let b = 1

therefore:
a = b

multiply both sides by b:
ab = b^2

subtract a^2:
ab - a^2 = b^2 - a^2
this is now 0=0.

factorise:
a(b - a) = (b - a)(b + a)
this is now 1(0)=(0)2, only non-zero values can be factorised.


What is wrong with the above proof? (It's not difficult, but it's a good one to pull out to confuse people)

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