Having trouble understanding this question

[shredder666]

Homework Statement

In the proof that there are uncountably many real numbers between 0 and 1, one constructs a real number that turns out to be different from all the real numbers on a given (countable) list. Suppose now that the following are the first few real numbers that are on a countable list:
0.*random number*
0.*random number*
0. *random number*
0. *random number*
...
For each of the numbers below, state whether or not it could be having the beginning decimals of a number constructed according to the stipulation given in the proof. In each case, answer Yes or No.

0. *random number*
0. *random number*
0. *random number*

Homework Equations

The Attempt at a Solution

I'm really having difficulty understanding what the question meant by
"state whether or not it could be having the beginning decimals of a number constructed according to the stipulation given in the proof. In each case, answer Yes or No."
could you provide an explanation or perhaps some examples please?

 

[Quinzio]

You are obviously unaware of a famous "proof" given by a great mathematician.
Try to get yourself to the proof. If you can get it, you're really good.
I'll just give you some hints.

You've got a list of decimal numbers.
E.g.
0.546
0.625
0.792
0.796

Find a decimal number in the form 0.xxx which is differente from the 3 above.
That's stupid, ok. Eg. 0.111, 0.222, etc

Imagine the list is infinite in lenght. Imagine each number of the infinite list has infinite decimals, eg.
On enumber of the list can be:
0.9872983798749826349726397462873649260897...
Can't write don't the complete number, obvious.

Now, write down another number (with infinite decimals ) that is different from my number above.
My number and your number has infinite decimals, so we can't check them figure by figure.
So, find a general rule how we can be sure my number and your number are different.
Hint:
0.11... and 0.12.... are different
0.11... and 0.11.... are ... ?? We can't say.Now use the rule just found to write down another number which is different from a general number 0.abcdefghijk.........:

Use the rule just found to:
Write down another number (with infinite decimals ) that is different from EVERY number in the infinite list.
It is possible, it is simple, it has been already shown.

There you have the proof.

If you can't do it (it's not an easy job) look in the spoiler.

Spoiler

http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument[/SPOILER]

 

[shredder666]

thanks, but i read the proof on the link, schaum's outline and my notes... and I still don't get it...

but really, what is the "beginning decimals of a number"? so... if i have 0.63434143 the beginning decimal is 6? >.<

also if I'm understanding the question correctly is it really just asking me if the numbers above could be constructed given the list of numbers above?

 

[HallsofIvy]

The question, as you wrote it, makes no sense. You cannot talk about "For each of the numbers below" and then write all of them as "0.*randomnumber*". Are you given actual numbers rather than just "0.*randomnumber*"?

In the number 0.6343143... the "beginning decimal" ("beginning digit" would be better) but you asked about "decimals" (plural). All of "6", "3", "4", ... as far as you wish to go, are "beginning decimals".

Unfortunately, you haven't given us any numbers or the list of numbers so it is impossible to tell what you are asking.

 

[shredder666]

the question gave actual numbers, I just replaced it so that I don't feel like I'm cheating...

I think I figured it out the proof and managed to finish the assignment with minor mistakes because I still couldn't figure out what the question was REALLY asking me to do, I basically just shotgun-ed the question.