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Rational, Irrational, and Integer. The correct definition.

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magic:

--- Quote from: Nominal Animal on December 21, 2022, 11:16:15 pm ---
--- Quote from: magic on December 21, 2022, 10:54:14 pm ---Like, take a unit square. Remove all points whose either coordinate is rational. Is the total area of the reminder a real number?
--- End quote ---
Again, a question with an erroneous concept at its core.  By definition, the area of a point is zero.  So, no matter how countably many points you remove, the area does not change.

--- End quote ---
"Your question is trivial and the answer is one. Therefore your question is ill-posed and I refuse to answer your other questions".

 :-DD :popcorn:

And again, the reason I ask about a lot of stupid things is not because I expect a sensible answer.
It's to show that you don't know why they are stupid and can't prove them stupid.
You could equally well say they are stupid because some heterosexual dude told you so, which is apparently how good chunk of population perceives mathematics.
I'm not even surprised ;D

ledtester:

--- Quote from: magic on December 21, 2022, 10:54:14 pm ---Like, take a unit square. Remove all points whose either coordinate is rational. Is the total area of the reminder a real number?

--- End quote ---

What you are describing is I x I where I is the set irrational numbers between 0 and 1. The unit square is [0,1] x [0,1].

According to standard measure theory, the set I is measurable and has measure 1. The cross product, therefore, also is measurable and has measure 1*1 = 1.

TimFox:
As stated above, the only numbers that can be used in digital computers are rationals, which are treated as binary integers.
All digital computation is binary arithmetic, a small subset of the broad universe of discourse in mathematics.
Progress in computation theory and practice is mainly development of algorithms to improve the efficiency of the arithmetic.

When I started grad school in physics, living in a dorm, one guy on my floor was a mathematics student who disliked the University's placement of the Department of Mathematics in the Division of Physical Sciences, instead of in the humanities where he thought it belonged.
That was a defendable statement;  I asked him what he was studying, and he stated topology and differential geometry, and he was proud that neither subject was of practical use.
I had to point out that physics used both:  topology in Feynman diagrams and differential geometry in General Relativity.

The word "stupid" has been flung about in this discussion.
Ignorance and stupidity are different concepts:  we are all ignorant, since there is far too much for one person to know; stupidity is pride in ones ignorance.

Tomorokoshi:

--- Quote from: TimFox on December 22, 2022, 04:52:21 pm ---...
All digital computation is binary arithmetic...
...

--- End quote ---

Unless one is using an IBM 650!

TimFox:

--- Quote from: Tomorokoshi on December 22, 2022, 04:55:06 pm ---
--- Quote from: TimFox on December 22, 2022, 04:52:21 pm ---...
All digital computation is binary arithmetic...
...

--- End quote ---

Unless one is using an IBM 650!

--- End quote ---

The first computer I used (ca. 1968) was an IBM 1620, which, along with the 650, used BCD (binary-coded decimal) internally.
That is another type of binary arithmetic, with slightly different rules.

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