Hi, I've been reading through baby Rudin and have learnt

yes, but between any two irrational numbers, there are countably many rational numbers, but uncountably many irrational numbers. So even if there is infinite amounts of both between any two numbers, there is significantly more irrationals

>>8991492
>there is significantly more irrationals
They're infinite so there are not more of one than of the other

>>8991484
>the number of IRRATIONAL numbers is infinitely many times larger than the number of RATIONAL numbers (uncountable infinity v countable infinity).
>Yet somehow, between any two irrational numbers there are infinitely many rational numbers.
>Fuck Math!
Dont waste your time with this nonsense, study useful math instead

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>2. There are uncountably many real numbers (impossible to express by m/n)!
Since rational numbers are a subset of real numbers, so some real numbers can be expressed as m/n.
>3. Between any two real numbers there is a rational numbers.
>This is also proven, and can be even further developed, that between any two irrational numbers there are infinitely many rational numbers.
This is true as long as the two irrational numbers are distinct (nit-picking, but important)
>Here's my problem now So, ok, the number of IRRATIONAL numbers is infinitely many times larger than the number of RATIONAL numbers (uncountable infinity v countable infinity).
>Yet somehow, between any two irrational numbers there are infinitely many rational numbers.
Countability refers more to the ability to iterate through every member of an infinite set. You shouldn't think about infinite sets being larger than each other

For an example of why comparing the cardinalities of infinite sets should be taken with special care, have a look at
https://en.m.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel

>>8991582
One more thing, given any two sets that are countably infinite, you can say they have the same cardinality as there exists a bijection between them

This also means that the rationals and the integers are the same 'size'.

It's also true that an uncountable set, such as R has a cardinality greater than that of the rationals and integers.
Have a look at https://en.m.wikipedia.org/wiki/Continuum_hypothesis

>>8991484
Comparing infinitely large things is always going to be counter-intuitive.

>>8991580
>study useful math instead
Too bad this bullshit forms the foundation for most of the math out there. Analysis relies entirely on these absurd concepts.

>>8991621
>absurd concepts
have a better solution ?

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im fuckin dumb /sci/ whats wrong

>>8991621
>Too bad this bullshit forms the foundation for most of the math out there.
Newton, Leibniz, Euler, etc didnt need any of that bs to do great things. Unless you need to know it to pass an exam or something like that, it's just a waste of time imo... But to each his own

>>8991651
suppose you take anything that's literally not 3

>>8991484
>Yet somehow, between any two irrational numbers there are infinitely many rational numbers.

an easy way to see this is just to truncate the larger of the two numbers after the digit where they begin to differ.

>>8991665


you can do this a countable number of times by mapping the truncations to the integers, one integer for each point where you lop off the remaineder

>>8991651
The negation of that expression is "there exists an x such that x^2 < 0

>>8991657
>Unless you need to know it to pass an exam or something like that
>something like that
yeah, like make an actual contribution to mathematical analysis

>>8991651
Is this textbook real?

>>8991576
Set cardinality

>>8991651
the negation of for every is there exists

so the proof should be:

suppose not. then there exists a number x such that x^2<0. But x can be either negative, positive or 0. if x=0, then, x^2=0. if x>0, then x^2>0. if x<0, then let -x=y>0. then x^2=(-x)(-x)=y^2>0. So by exhaustion of cases, we have found that for all x real, x^2 cannot be negative. But we suppose such an x does exist. Contradiction