Math Is Fun Forum

Anthony Lahmann

Member

Registered: 2019-04-25

Posts: 9

Proof All Numbers are Equal

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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Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,385

Re: Proof All Numbers are Equal

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?

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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.

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Bob

Administrator

Registered: 2010-06-20

Posts: 10,619

Re: Proof All Numbers are Equal

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

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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

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 24,996

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Re: Proof All Numbers are Equal

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.

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'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.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,619

Re: Proof All Numbers are Equal

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

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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

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 24,996

Website

Re: Proof All Numbers are Equal

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.

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'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.

jaap

Member

Registered: 2020-09-02

Posts: 1

Re: Proof All Numbers are Equal

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)