Mathematic Fallacy

[AHadley]

I know many of you will know this already, but for those of you who don't, this will destroy the foundation of mathematics as you know it. Yes, that's probably exaggerating

Let's say a = b. Both are non-zero integers.

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a = b


Let's multiply both sides by a.

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a^2 = ab


Take away b^2 from them.

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a^2 - b^2 = ab - b^2


Factorise it...

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(a - b)(a + b) = b (a - b)


Cancel the (a - b)s...

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(a + b) = b

a + b = b


Remember that a = b and substitute...

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b + b = b


Simplify it

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2b = b


And, finally, divide by b.

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2 = 1



Yes, you heard me right.


2 = 1.

[Clavin12]

As the title implies theres a fallacy there sumwhere. When you say

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2b = b there's only one number that can fill it, 0. Just because

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2b = b doesn't mean you can replace b with anything you want. I'm sure you've done "Find x" problems before.

[Nielk1]

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2b = b


And, finally, divide by b.

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2 = 1


INCORRECT! B is 0. Dividing both sides by B is dividing by zero. The fabric of space time breaks down when you divide by zero. Hence, 2=1

[mrtwosheds]

The fabric of space time breaks down when you divide by zero.

The fabric of space time is 0, sort of.
Mathematics can be used to model reality, but reality does not do mathematics, maths can be wrong, but reality cannot.

[AHadley]

It's called a mathematic fallacy because it is a flaw in mathematics. So MTS is exactly correct.

The flaw is that any pair of non-zero numbers never comes out as the correct answer, although it is assumed by the equation that both A and B are non-zero. I'd better edit that in.

It is at this point that algebra and actual numbers fail to match up. That is why this works as it does. I would dig up the fallacies that prove that 0 = 1 and that all numbers are equal to each other, but they are a little beyond my understanding here at AS level.

There is another one, though.

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1/3 = 0.3`

0.3` + 0.3` = 0.6`

0.6` + 0.3` = 0.9`


What is three thirds?

[ScarleTomato]

Actually (a-b) is zero. When you "Cancel the (a - b)s...", that's when you're dividing by zero.

you could find the first ignored flaw by looking at the third step.
if a = b
then a^2 - b^2 = 0
and on the other side of the equation, ab - b^2 = 0

therefore => 0 = 0

[AHadley]

That's a flaw with the mathematics though, not with the working out. If something is done to one side and the other then the values are affected in the same way and they change to the same thing - or that's the theory anyways.

[Nielk1]

There is no flaw, it is just above your understanding.

0.33333' + 0.66666' is not 0.99999'. Because both decimals extend into infinity, to claim they sum in that way suggests they do NOT continue into infinity. In reality 0.33333' + 0.66666' = 1.0

I can look for an inductive proof of such if you so require.

[AHadley]

I did point out that I'm only AS level (just starting on differentiation and integration) so most areas of mathematics are no doubt beyond me, so please do explain in terms I can understand if you can overcome the fallacy.

I do however believe that mathematics is in no way perfect. It is, as MTS said, only a model.

[Nielk1]

Math is perfect in our current physics set. The very basics of math, by that I mean Algebra, Geometry, and Calculus always hold true.

I am not even a math major. I hate proofs with a passion. But trust me, the math works perfectly.

[VSMIT]

Remember that math is the purest science.

[ScarleTomato]

That's a flaw with the mathematics though, not with the working out. If something is done to one side and the other then the values are affected in the same way and they change to the same thing - or that's the theory anyways.

The flaw has everything to do with the 'working it out' part.
By definition, a is equal to b. This fact is conveniently forgotten in the "Cancel the (a - b)s..." step.
Since a is equal to b, then a minus b is zero. period.
Therefore cancelling (a-b), also known as dividing by (a-b), is an illegal operation. A division by zero.

I agree with you on the "both sides are affected" part. In this case however, both sides equate to undefined. Therefore undefined still equals undefined.

Wikipedia may be able to explain it better that i do.

Division by zero is not technically illegal in higher math, it just means "all solutions reach infinity" or some such nonsense.

[Feared_1]

.99999...... is exactly 1. Let me prove it...

x = .999999999...

Multiply both sides by 10

10x = 9.999999999...

Because x = .99999999..., let's subtract it (x) from both sides

9x = 9

Divide both sides by 9

x = 1 = .999999...



EDIT:


ScarleTomato is exactly correct:

a = b

Let's give them number values:

5 = 5

So when you subtract a from b:

(a - b) = (5 - 5)

What's (5 - 5)?
What's (a - b)?

When the denominator of an equation =0, graphically you end up with a function where somewhere it has a vertical slope. If the slope is vertical, it has an infinite number of points "stacked up" and it does not pass the vertical line test to describe a function. In other words, when f(0) = anything divided by zero (n/0), a vertical asymptote is created where the graph doesn't actually exist at all. The canceling (a - b) from both sides step is impossible.

Here is your first post, with 5's instead of variables:

Let's say 5 = 5. Both are non-zero integers.

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5 = 5

Let's multiply both sides by 5.

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5^2 = 5^2

Take away 5^2 from them.

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(5^2) - (5^2) = ((5^2) - (25))

Factorise it...

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(5+5)(0) = 5(0)
(This should be the solution)

Cancel the (0)s...

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((5+5)(0))/0 = (5(0))/0

(dividing by 0 makes this statement incorrect, you can't actually do it, so the rest is void:)

5 + 5 = 5
Remember that 5 = 5 and substitute...
5 + 5 = 5
Simplify it
2(5) = 5
And, finally, divide by 5.
2 = 1

[mrtwosheds]

Remember that math is the purest science.

It is also pure fiction. It exists only in the minds of the beings the Universe has seen fit to create for itself. It is not real and it does not even represent reality in any meaningful way.
If you divide 1 atom by 2...you have a great deal of trouble adding it together again.

[AHadley]

.99999...... is exactly 1. Let me prove it...

x = .999999999...

Multiply both sides by 10

10x = 9.999999999...

Because x = .99999999..., let's subtract it (x) from both sides

9x = 9

Divide both sides by 9

x = 1 = .999999...

That's some clever stuff. I don't have a thing for working out equations of my own unfortunately

I shall have to point that out to my maths teacher :-d