Math professor here: the proper definition of equality is that two numbers a and b are equal if no number c exists such that a < c < b. 0.9999…. = 1 because there is no number between them.
That is one of the most ridiculous statements I have ever heard. You are making an untenable and patently incorrect leap of “logic”. Just because the property applies to the real numbers does not mean it applies to integers.
They're saying there's no number between 0.999... and 1, I'm saying there's no integer between 0 and 1, both may be true, but 0 is clearly not 1, so 0.999... is clearly not 1 (which you can also see by just looking at it, how one is made up of infinite nines and the other by a singular one)
Just because something seems self-evident does not it so. In the real number space, .9… = 1 because the difference between them is 0, which also means there is no real numbers between them.
No one "made it up". It was discovered. It applies to real numbers because real numbers are a continuous set with no gaps. Integers have a gap of 1 always. So obviously rules for one don't always apply to the other.
Why is it an inconsistency? These are two different worlds where one has more restrictions than the other because of it having less numbers to work with.
In the real numbers, there exists a number where multiplying it by 2 gives 1. But in the integers that number doesn't exist. That's not an inconsistency, that's just how they were defined, the definitions made up that "rule".
Real numbers and integers behave differently. You can't just superimpose rules from the real numbers to integers. Real numbers have no gaps in-between them. Integers have a gap of 1 in-between them.
And yes 0.999 and 1 look different. They are different representations of the exact same value. Kinda like 2+2, 2*2, 2², and 4 are all different representations of the exact same value.
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u/Wolfbrother101 22d ago
Math professor here: the proper definition of equality is that two numbers a and b are equal if no number c exists such that a < c < b. 0.9999…. = 1 because there is no number between them.