Expert: Paul Klarreich - 1/6/2010

Question
Dear Sir,
I am houswife and I am going through with the follwoing problem in calculus subject.

For A = { r belongs to  Q /r^2 is lesser than 2} and B = {r belongs to  Q / r^2 greater than 2}
Does there exists a largest element of A in Q? and Does there exist a smallest element of B in Q? If not, how we can fill the gaps by irrationals at real number line?

Please clarify this with clear details. pleaseeeeeeee.

Thanks in Advance,
Pavani

Answer
Questioner: mahima
Country: India
Category: Calculus
Private: No
Subject: calculus
Question: Dear Sir,
I am houswife and I am going through with the follwoing problem in calculus subject.

For A = { r belongs to  Q /r^2 is lesser than 2} and B = {r belongs to  Q / r^2 greater than 2}
Does there exists a largest element of A in Q? and Does there exist a smallest element of B in Q? If not, how we can fill the gaps by irrationals at real number line?

Please clarify this with clear details. pleaseeeeeeee.

Thanks in Advance,
Pavani
.....................................

NOTATION:  Where you wrote:

"A = { r belongs to  Q /r^2 is lesser than 2} and B = {r belongs to  Q / r^2 greater than 2}"

I shall write:

A = { r in Q | r^2 < 2} and B = {r in Q | r^2 > 2}

which makes the notation more concise.

This is your classic Dedekind cut example (which you will look up)

Now your question:

Does there exist a largest element of A in Q? and Does there exist a smallest element of B in Q?

No, to both questions.

But, when you write:

"If not, how we can fill the gaps by irrationals at real number line?"

your question does not make sense, but you may be referring to the fact that we can specify every real number, rational or not, wih this definition:

A Dedekind Cut (or just a 'cut') is a pair of sets A,B having the properties that if a in A and B in B:

1. a,b are both rationals
2. a < b  [so no number is in both]
3. Every rational is either in A or B.

You can call this cut x, and then:

A. if A has a largest element or B has a smallest, (it is not possible to have both) then x is rational.
B. otherwise x is irrational.

So x = sqrt(2) is defined by the pair of sets you have written, provided you take care of the negative numbers in some way.  See:

en.wikipedia.org/wiki/Dedekind_cut

for more detail and a careful specification of sqrt(2).

Fun?  In fact, all arithmetic can be defined in terms of cuts.