Rational and Irrrational Numbers

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hitler didi

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Think of two irrational numbers X and Y such that X/Y is a rational number.
which to numbers should i select and what should i look at while thinking is there any particular method for this or we could just simply keep on dividing the numbers:confused:
 
Ya gotta show some effort here. Think of your favorite irrational number. Now finish this sentence: "anything (except 0) divided by itself equals ___"
 
In fact, given any rational number, r, we can always find irrational numbers, X and Y, such that X/Y= r. Even more, given any rational number, r, and any specific irrational number, Y, we can find an irrational number, X, such that \(\displaystyle X/Y= r\).

If you cannot find an example for your original problem you are just not trying hard enough!
 
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HallsofIvy said:
In fact, given any rational number, r, we can always find irrational numbers, X and Y, such that X/Y= r. Even more, given any irrational number, r, and any specific irrational number, Y, we can find an irrational number, X, such that \(\displaystyle X/Y= r\).

If you cannot find an example for your original problem you are just not trying hard enough!
Click to expand...
i got it sqrt of 2/sqrt of 2 .will give sqrt 1
 
hitler didi said:
Think of two irrational numbers X and Y such that X/Y is a rational number.
which to numbers should i select and what should i look at while thinking is there any particular method for this or we could just simply keep on dividing the numbers:confused:
Click to expand...

JeffM said:
Actually \(\displaystyle \dfrac{\sqrt{2}}{\sqrt{2}} = 1 = \sqrt{1}.\)
Click to expand...
The OP asks for two irrational numbers.
If \(\displaystyle X=\sqrt{2}~\&~Y=\sqrt{2}\) that is only one.

But \(\displaystyle X=\dfrac{\sqrt{2}}{2}~\&~Y=\sqrt{2}\) works nicely.
 
pka said:
The OP asks for two irrational numbers.
If \(\displaystyle X=\sqrt{2}~\&~Y=\sqrt{2}\) that is only one.

But \(\displaystyle X=\dfrac{\sqrt{2}}{2}~\&~Y=\sqrt{2}\) works nicely.
Click to expand...
Point noted but "two numbers" does not necessarily mean "two distinct numbers".
 
HallsofIvy said:
Point noted but "two numbers" does not necessarily mean "two distinct numbers".
Click to expand...
Oh yes it does!
This a favorite story about that. EH Moore began a lecture at Princeton by saying "let each of a and b be a point" Solomon Lefschetz shouted from the back of the room "But why don't you just say 'Let a and b be two points'?" Moore replied "because a may equal b". Lefschetz got up and left the room.

Now EH Moore was RL Moore advisor and I got that drilled into me by two of his students.
 
pka said:
The OP asks for two irrational numbers.
If \(\displaystyle X=\sqrt{2}~\&~Y=\sqrt{2}\) that is only one.
Click to expand...

I support HallsofIvy.

Other examples akin to the theme original question might be:


1) What/which two positive integers, whose sum is 4, give a maximum product
(when they are multiplied together, that is)?


2) The area of a rectangle is 4 square units. What are the number of units
of the rectangle's width and length for it to have a maximum area?


3) The product of two positive integers is 4. What are these two numbers if
their sum is a minimum?


I can accept the motive of offering the story in the immediately prior post to this one
if it was/is meant to be facetious.
 
lookagain said:
I support HallsofIvy.
Other examples akin to the theme original question might be:
1) What/which two positive integers, whose sum is 4, give a maximum product
(when they are multiplied together, that is)?
2) The area of a rectangle is 4 square units. What are the number of units
of the rectangle's width and length for it to have a maximum area?
3) The product of two positive integers is 4. What are these two numbers if their sum is a minimum?
I can accept the motive of offering the story in the immediately prior post to this one if it was/is meant to be facetious.
Click to expand...
@lookagain you are not a mathematician are you?
If you were to give a set ot two numbers whose product is four and you answered \(\displaystyle \{2,2\}\) you would be marked wrong.
That set has only one number in it.

So lets look at mathematical pedigree.
E H Moore

R L Moore

Now where are you?
 
pka said:
@lookagain you are not a mathematician are you? <---- rhetorical question
That's a false argument. Replace the arrow-highlighted phrase below with
"No (true) mathematician." Source: Wikipedia

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It has been suggested that Special pleading be merged into this article or section. (Discuss) Proposed since July 2012.
For the practice of wearing a kilt without undergarments, see True Scotsman.
>>> No true Scotsman < < < is an informal logical fallacy, an ad hoc attempt
to retain an unreasoned assertion.[1] When faced with a counterexample to a
universal claim, rather than denying the counterexample or rejecting the original
universal claim, this fallacy modifies the subject of the assertion to exclude rather
than denying the counterexample or rejecting the original universal claim, this fallacy
modifies the subject of the assertion to exclude the specific case or others like it by
rhetoric [as in a rhetorical question], without reference to any specific objective rule."

----------------------------------------------------

If you were to give a set ot two numbers whose product is four and you answered \(\displaystyle \{2,2\}\)
you would be marked wrong.
That set has only one number in it.

That's a false premise. I would not be presenting a "set of numbers."


So lets look at mathematical pedigree.
E H Moore

R L Moore


1) No, let's not "look at mathematical pedigree." That is immaterial
as to whether I know fact X. It is a misdirection.[/b]

> > > Now where are you? < < <
Click to expand...


2) This is a false argument, (and it is made by many novices in argument-making **).
My "pedigree" is immaterial. And asking for it is akin to asking a diner if he/she can
cook at a similar/comparable level as the cook can, as if the customers' opinions
of the food not tasting good is tied in their (the customers') ability to cook.


** The comment is parentheses, that I made of the first line of 2) above, is of the similar false
logic as you used in your sentence to me. It is akin to stating, "pka, you are not a valid argument-
maker, are you?" If you don't a particular fact X in valid argument-making, I can then falsely
claim that you are not a valid argument because you are not up to whatever standards that I have.[/b]
 
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hitler didi said:
Think of two irrational numbers X and Y such that X/Y is a rational number.
which to numbers should i select and what should i look at while thinking is there any particular method for this or we could just simply keep on dividing the numbers:confused:
Click to expand...

I believe the simple answer to your question is ...

Take any irrational number n.

Then X=an and Y=bn for any integers a and b as long as b is not zero.

It is also agreed that if a=b then it is a less interesting answer, if not incorrect.
 
SUM of Irrationals

You can also find irrationals with a rational SUM.


\(\displaystyle \phi =\frac{1 + \sqrt{5}}{2}
\text{ is an irrational number.}\)
\(\displaystyle \text{Let x = 5 + 8$\phi $ and y = 5 - 8/$\phi $}\)

Cute but not too surprising...
x+y is a rational number

What about....
\(\displaystyle \sqrt[3]{x}
\text{ + }\sqrt[3]{ y }\)

Get out your calculator and try it!
 
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