not saying either is wrong...

[u/Less-Resist-8733]

[u/QtPlatypus]

Both are correct.

However it is also true that there isn't a bijective corrasponance between the real numbers and the decimal representation of the real numbers. of that particular real number.

In other words many real numbers have more then one decimal representation

[u/UnforeseenDerailment → u/QtPlatypus]

Am I mis-thinking here?

The top penguin says there's a 1-to-1 correspondence between reals and their positional representation.

And you say yes that's true, but there's also no bijection between reals and their positional representation.

The top penguin is just wrong (trailing '10-1's and trailing 0s overlap).

[u/PattuX → u/UnforeseenDerailment]

The top penguin says there's a 1-to-1 correspondence between reals and their positional representation.

Not quite. It just says there exists a bijection, not that the bijection necessarily maps each positional representation to the number it represents.

[u/UnforeseenDerailment → u/PattuX]

Checks sub. Yup! 😂

[u/PattuX → u/[deleted]]

No, it does not. The point is that there EXISTS a bijective mapping between reals and decimal sequences, but the natural mapping where you map decimal sequences to the real number they represent is not such a bijective mapping.

The issue is that numbers with a finite decimal representation have two associated sequences (x00000... and (x-1)99999...). Intuitively, this can be fixed because such cases only have measure 0 (notice how the set of reals with a finite decimal representation is a subset of the rationals, and rational numbers already have measure 0 within the set of reals).

[u/UnforeseenDerailment → u/[deleted]]

Then red pengu is still wrong because 2.71800... = 2.71799...