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**Anthony Lahmann****Member**- Registered: 2019-04-25
- Posts: 9

Let's prove that all numbers are equal.

What went wrong?

*Last edited by Anthony Lahmann (2020-01-05 19:21:44)*

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**ganesh****Administrator**- Registered: 2005-06-28
- Posts: 35,572

Hi,

Let us number the steps:

From (5) to (6), there is a fallacy.

The the step (5) does not necessarily equal (6).

Got it?

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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**Bob****Administrator**- Registered: 2010-06-20
- Posts: 9,357

Hi Anthony

Nice 'proof'. When applying a square root you have to remember that two roots are possible.

9 =9 but 3 ≠ -3

To check out any algebra just substitute some numbers to see what is happening.

I chose a = 5; b = 3 and so c = 8.

Here is your 'proof' with those numbers.

8 x 2 = 8 x 2

25 - 9 = 40 - 24

25 - 40 = 9 - 24

25 -40 + 32 = 9 - 24 + 32

(5-4)^2 = (3-4)^2

All so far is correct. But then we have the square root.

(5-4) ≠ (3-4)

But, back to the algebra, if we take a negative root:

a - c/2 = -b + c/2

a + b = c/2 + c/2 = c

Now it's correct but hasn't given anything new.

Bob

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

Sometimes I deliberately make mistakes, just to test you! …………….Bob

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I would like to note a different issue. You cannot start a proof by saying "a + b = c". What kind of objects are a, b, c? Do you mean that for every a, for every b, for every c, a + b = c?

For a lot of people, a proof is something that has human value. An arbitrary calculation is not a proof.

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

I'm not crazy, my mother had me tested.

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**Bob****Administrator**- Registered: 2010-06-20
- Posts: 9,357

It's ok to say let a and b be any numbers and define c to be a + b. As c 'cancels' out of the proof you're just left with a = b. That's the mystery bit. Except the proof is flawed at the step where square roots are taken.

Bob

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

Sometimes I deliberately make mistakes, just to test you! …………….Bob

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Yep, that is true. I am just saying that the incantation "Let a, b be arbitrary numbers. Set c = a + b" should be made explicit.

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

I'm not crazy, my mother had me tested.

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**jaap****Member**- Registered: 2020-09-02
- Posts: 1

The main trick occurs in step 1, multiplying both sides by a-b. While not wrong, this introduces the bogus solution a=b to the equation, and allows the oversight when taking the square roots to land on that wrong solution.

*Last edited by jaap (2020-09-02 19:35:14)*

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