Proving 0.9999....=1 A marvellous proof #shorts

#marvellousproof
#amazingproof
#maths

[KarmaPeny]

Think about what you actually did here... you started with x=0.999... then you multiplied both sides by (10 - 1)/9, which is the same as multiplying by 1. You should get exactly what you started with. You should not end up with x=1.

Obviously if we start with x = 0.999... and then multiply both sides by (10 - 1)/9 we should end up with exactly what we started with. The reason it appears to change to x = 1 is because under the cover of the decimal system a sneaky misalignment of the terms was performed before the subtraction.

Let's do the proof again but let's replace 0.999... with the geometric series that it represents, namely 9/10 + 9/100 + 9/1000 + ... Then if we multiply it by 10 we get 90/10 + 90/100 + 90/1000 + ... then if we subtract what we started with we get 81/10 + 81/100 + 81/1000 + ... then if we divide by 9 we get back to exactly what we started with.

We can easily show that a 'shift-and-subtract' operation on geometric series can result in contradiction. We start with 1+1+1+... = 1+1+1+... and simply subtract what we started with except that on one side we 'shift the terms' before doing the subtraction. We end up with 0=1 and so this appears to be evidence of why this 'proof that 0.999...=1' is flawed.

There are other ways to demonstrate that we can't just re-arrange terms in a series. For example, we could re-write 0.999... as the series [9/10 + term1 - term1] + [9/100 + term2 - term2] + [9/1000 + term3 - term3] + ... where term1, term2 etc. are the terms of any conditionally convergent series. Then as per the Riemann Rearrangement Theorem we could make the result of the subtraction supposedly converge to any value we like.

Most people agree that the base 10 decimal and the geometric series represent exactly the same thing and so should be interchangeable. And so if we can show there is a flaw in the proof when using geometric series, then the proof is no good at all. We can't claim the proof 'works' for the base 10 decimal but not for the geometric progression because these two forms are just different ways of writing exactly the same thing.

As far as I can gather, most mathematicians think it is acceptable to define our way out of the problem. Mathematicians have simply invented a lot of rules about what we are allowed to do in terms of rearrangement/regrouping of terms. For example, we are not allowed to re-align the terms of a divergent geometric series and this allows us to claim that the example of 1+1+1+... = 1+1+1+... is not relevant. And (presumably) we are not allowed to modify an absolutely convergent series in any way where the result of a subtraction could be made to converge to more than one value.

So even though we might think that the the example of 1+1+1+... = 1+1+1+... clearly demonstrates why the misalignment of endless non-zero terms in geometric series should not be allowed, it is not enough to simply work from 1st principles. We need to be aware of every definition and every rule that has ever been accepted into the discipline called 'mathematics' just in case there are any that say "that is not allowed".

In other words, when faced with any counter argument to what mathematicians want to be true they say "that is not allowed". It is not a very elegant way to side-step problems to simply use a bunch of made-up rules that say "that is not allowed" and it raises the question of what exactly does mathematical rigour mean if mathematicians can simply define their way around any problem that they don't like.

[RSLT]

This is one of those cases where we say the emperor is naked 😂😂😂😂

[evangordon]

U assumed infinity equals infinity, which is not true, try again