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/sci/ - Science & Math

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12480927 No.12480927[DELETED]  [Reply] [Original]

Prove there exists an irrational number between a, b such that a and b are irrational and b>a

>> No.12480931

no u

>> No.12480952

>>12480927
a=√2 < n=√3 < b=√5
op btfo

>> No.12480956
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12480956

GO AWAY REALS

>> No.12480962

>>12480927
(a+b)/2 and (a+2b)/3 can't both be rational, so one of them is irrational and satisfies your condition.

>> No.12480973

>>12480962
Why can't both be rational?

>> No.12480976

Is a+b/2 is irrational, it lies between them. If not, a+b is rational, and you can map the interval (a,b) onto (3,4) which contains pi which is irrational.

>> No.12480981

>>12480973
If x=(a+b)/2 and y=(a+2b)/3 are both rational, then
3y-2x = a + 2b - a - b = b is rational.

>> No.12480983

>>12480973
if a+b=k for rational k, a+2b=k+b which is irrational if b is irrational

>> No.12480987

>>12480976
>If not, a+b is rational, and you can map the interval (a,b) onto (3,4) which contains pi which is irrational
Elaborate. How do you map the interval?

>> No.12480994

>>12480981
>>12480976
>>12480962
Also note here that the proof works with the weaker condition that only one of a,b is irrational. This is because both a and b are rational linear combinations of x,y.
Also, note that none of this actually makes any sense because "real numbers" is a fake system.

>> No.12480995

Just apply a rational coefficient to sqrt(2) so that it ends up inside (a,b)

>> No.12480999

>>12480987
you map a to 3 and b to 4, and then all the linear combinations of a and b to all the rational numbers between 3 and 4.

>> No.12481010

>>12480999
> and then all the linear combinations of a and b to all the rational numbers between 3 and 4.
How do you do this? What is the actual map that you're using?

>> No.12481014

>>12480995
Prove that this can be done.

>> No.12481030

>>12481010
You don't need to know the actual map, you just need to know that the continuum is bigger than Q (by Cantor's diagonalization argument, if I must), and then the pigeonhole principle guarantees that you can't get em all with any map.
I am not going to spoonfeed you, /sci/.

>> No.12481049

>>12481030
Honestly I've got no fucking idea what you're talking about. You're mapping all linear combinations of a and b which is the same as saying the whole interval [a,b] to all the rational numbers between 3 and 4. Pigeonhole gives you that some rational numbers get mapped to infinitely often. And?? What do you conclude from this? How does pi come in?
Maybe you're drunk.

>> No.12481058

>>12481030
>>12480976
Either a troll or an idiot.

>> No.12481095

Just take a+1/n where n is sufficiently large. This is possible because of the Archimedean property.

>> No.12481102

>>12481095
>This is possible because of the Archimedean property.
What is the Archimedean property? Prove it.