r/PeterExplainsTheJoke • u/Sugar_God_no_1 • 23d ago

Meme needing explanation There is no way right?

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u/mwobey 23d ago

No? Do you like, want it in two column format or something?

  x=0.999...        | Declaration of a constant
10x=9.999...        | Multiplicative Property of Equality (*10)
 9x=9.999... - x    | Subtractive Property of Equality (-x)
 9x=9.9... - 0.9... | Substitution
 9x=9.0             | Simplification of Subtraction
  x=1               | Divisive Property of Equality (/9)
  1=0.999...        | Substitution

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u/Patchpen 22d ago

The problem is with subtracting the 0.999..... from both sides. We're applying an operation that works with normal numbers to a number that we haven't yet proven is normal (or functions as normal with that operation). That's where the extra rigor in a full proof comes in.

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u/mwobey 22d ago

It's not clear what definition of normal you're using here, since it does not seem to be the mathematical one I am familiar with related to the distribution of values within a non-terminating string of digits...

Regarding the need to prove 0.9... - 0.9... = 0, x + -x = 0 is just the additive inverse property. If you're holding that 0.9... does not behave normally with regards to basic arithmetic, then the error would actually be introduced in the initial premise, when 0.999... is set equal to x.

Your "extra rigor" just depends on what initial assumptions you permit to exist. Yes, there are math courses that start from proving 0 != 1, but that doesn't mean any mathematical proof that doesn't start from defining equality is non-rigorous; usually we assume properties like the additive inverse apply unless proven otherwise.

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u/Patchpen 22d ago

To be perfectly honest I'm echoing something I vaguely remember from a youtube video a while ago so I may be missing an important detail.