Euler's "blunder" revealed in rigorous detail:

Here is the "proof" for 0.(9) = 1, as provided by Euler, in which he claims that number manipulation can magically morph a "number" assigned as a constant, into another "number"

First off, since when did

become a variable that we needed to solve for ?? It is a constant. Constants don't change their value. Try assigning a value to a constant in a computer program and see if the value ever "ends up as something else".

The huge blunder here is the combination of multiplying by 10 and subtracting a series from the resulting series, namely, 10 x 0.999... - 0.999...

There are other underlying assumptions too, but I addressed those earlier in the thread (specifically, multiplying an infinite series by anything which causes a carry or decimal shift)

Let's do 10n - n using the definition of n

Now, what Euler does, is subtract the 2 series,

...Except, what he failed to show everyone, or

*assumed *was ok, was that he shifted the second series to the right by one decimal place:

How he does this is by doing to following:

I believe Colin pointed this out at one point and said this was "ok" to do. At first glance, the fractions are all equivalent... except...

...the SERIES has changes slightly:

[

__ Notice the starting index value.__ ]

This allows the subtraction to occur as :

instead of

(The correct way)

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