Need help, clever math about infinity between 0 and 1 & abou

[Nathalie]

Hi, I need help finding sites on the following 2 clever math

The first one is about infinity. Something like "How many numbers are between 0 and 1" and the answer is infinity. I would like to know if the question is phrased correctly or no. I can't seem to find website that contains that information.

The second is how many ways you can view a number. For example the number 5
You can view 5 as 5, or you can see it as
5.

5/1

5^1

then I don't know if there is other ways to view a number... please help. Thank you very much.

P.S. I hope I am posting in the right section. Thanks~!

 

[Denis]

You can view 5 as 5, or you can see it as
5.
5/1
5^1
then I don't know if there is other ways to view a number...

What's wrong with 5*1, 2+3, [sqrt(5)]^2, 10/2 .......???

 

[Denis]

Something like "How many numbers are between 0 and 1" and the answer is infinity.

Not sure what you're asking here; but infinity is correct:
1/10, 1/100, 1/1000 ...... 1/1000000000000000000000000000000000000000, .............
Get my drift?

 

[Nathalie]

You can view 5 as 5, or you can see it as
5.
5/1
5^1
then I don't know if there is other ways to view a number...

What's wrong with 5*1, 2+3, [sqrt(5)]^2, 10/2 .......???

Thank you for clarification for the first one.
This one I meant to say how many ways you can write 5 as without changing that number. Like, we usually just write 5 as 5. But 5 can be also written as 5 over 1, 5. (with the decimal) or 5^1 is still a 5. the 5 x 1 = 5 is what I meant as well. A teacher showed me the things that you don't normally see when writing one number, and I want to find out how many ways are there since I can't find the paper when he wrote it. Thank you~!

 

[daon]

There are uncountably infinite ways to "view" a number in that sense. If f is a bijective function from R to R then \(\displaystyle f^{-1}(f(x)) = x\).

Just some examples on how to "view the number n":

\(\displaystyle (n+k)-k = n\)

\(\displaystyle \frac{k}{\frac{k}{n}} = n\) (n not zero)

\(\displaystyle \sqrt[{(2k+1)}]{n^{2k+1}} = n\)

\(\displaystyle log_k(k^n) = n\) (k not negative)

 

[barson90]

There are uncountably infinite ways to "view" a number in that sense. If f is a bijective function from R to R then \(\displaystyle f^{-1}(f(x)) = x\).

Just some examples on how to "view the number n":

\(\displaystyle (n+k)-k = n\)

\(\displaystyle \frac{k}{\frac{k}{n}} = n\) (n not zero)

\(\displaystyle \sqrt[{(2k+1)}]{n^{2k+1}} = n\)

\(\displaystyle log_k(k^n) = n\) (k not negative)

Don't forget the complex form of a number, x + yi or the polar form r*cis(theta), in this case, where theta equals 0 degrees.

In other words, 5 + 0i and 5 cis(0)

 

[DrMike]

Hi, I need help finding sites on the following 2 clever math

The first one is about infinity. Something like "How many numbers are between 0 and 1" and the answer is infinity. I would like to know if the question is phrased correctly or no. I can't seem to find website that contains that information.

It would be better to say "infinitely many". It turns out there is more than one infinite number - in fact, there are infinitely many - so saying "between 0 and 1 there are infinitely many numbers" is as vague as saying "in this bag there are finitely many apples". However, "infinity" doesn't call to a mathematicians mind a specific number. It's a bit like those proverbial tribespeople who count "1, 2, many" saying "many".

 

[daon]

One of the hardest things to wrap your (well, at least my) mind around, unless you see some proofs beforehand, is why the irrationals are so much more "present" than the rationals. This quandary was one of the contributing factors which lead to my decision to pursue math. I had learned this fact after all my Calculus courses and could not believe I "knew" so much math without really knowing anything at all about its foundations. I hope to come across more and more such experiences. I actually love to learn that thinking "logically" is often counterproductive. Having not seen any arguments, why should anyone believe that there are "more" irrationals than rationals? But even after comprehending that, despite having infinitely many of these rational numbers between any two real numbers, we're expected to assimilate that the rationals make up ZERO PERCENT of all the real numbers. I've done the proofs, I've used the fact as if it were obvious in higher math classes, I understand why it is true and I've helped others understand why. Despite that it still BLOWS MY MIND.

 

[BigGlenntheHeavy]

How is this for a mind-blower.

The set of all real numbers has the same cardinality as the set of all real numbers between 0 and 1.

 

[daon]

How is this for a mind-blower.

The set of all real numbers has the same cardinality as the set of all real numbers between 0 and 1.

And here's a thinker, and I've asked these long ago, is the sum of all positive real numbers a real number? The product of all reals greater than 1?

Is...

\(\displaystyle \sum_{k \in \mathbb{R_+}}k \,\,= \,\,\sum_{0<k<1}k\)

\(\displaystyle \sum_{k \in \mathbb{N}}k \,\,= \,\, \sum_{k \in \mathbb{R_+}}k\)

\(\displaystyle \prod_{1<k \in \mathbb{R}}k \,\,=\,\, \sum_{k \in \mathbb{R_+}}k\)

?

 

[chrisr]

\(\displaystyle I\ asked\ a\ guy\ once,\)
\(\displaystyle What's\ the\ difference\ between\ the\ graph\ of\ \frac{x^2-9}{x-3}\ and\ the\ graph\ of\ x+3\ ?\)

\(\displaystyle Being\ familiar\ with\ what\ he\ meant,\ he\ said\ the\ first\ one\ had\ a\ hole\ in\ it.\)
\(\displaystyle If\ you\ only\ had\ the\ graph,\ how'd\ you\ ever\ detect\ a\ hole\ of\ zero\ size?\)
\(\displaystyle The\ graph\ never\ even\ starts\ to\ diverge\ around\ x=3.\)
\(\displaystyle How\ big\ is\ the\ point\ (3,k)\ ?\)

 

[daon]

\(\displaystyle I\ asked\ a\ guy\ once,\)
\(\displaystyle What's\ the\ difference\ between\ the\ graph\ of\ \frac{x^2-9}{x-3}\ and\ the\ graph\ of\ x+3\ ?\)

\(\displaystyle Being\ familiar\ with\ what\ he\ meant,\ he\ said\ the\ first\ one\ had\ a\ hole\ in\ it.\)
\(\displaystyle If\ you\ only\ had\ the\ graph,\ how'd\ you\ ever\ detect\ a\ hole\ of\ zero\ size?\)
\(\displaystyle The\ graph\ never\ even\ starts\ to\ diverge\ around\ x=3.\)
\(\displaystyle How\ big\ is\ the\ point\ (3,k)\ ?\)

Even with "infinitely powerful" eyes, no one would be able to see a "hole" in the graph. In some instances, as we might suspect, it is meaningless that it exists at all. Any function that has at most a countable (finite or infinite) collection of these holes on an interval is Reimann integrable and has the same Real value as the integral of the "simplified" function.

 

[chrisr]

\(\displaystyle We\ only\ see\ the\ graph\ of\ a\ function\ because\ we\)

\(\displaystyle grossly\ overemphasise\ the\ 'size' of\ 'all'\ the\ zero-size\ points.\)

 

[daon]

You must mean because a curve is one-dimensional. Then take the function f(x,y)=(x-1)(y-1)x/[(x-1)(y-1)]. A plane z=x is formed having no value at (1,y) nor (x,1), but "looking" at the plane, you would not see these disconnects (in actuality four separate plane-segments are formed). Then my explaination has no holes

 

[lookagain]

Re: Need help, clever math about infinity between 0 and 1 &

How is this for a mind-blower.

The set of all real numbers has the same cardinality as the set of all real numbers between 0 and 1.

And here's a thinker, and I've asked these long ago, is the sum of all positive real numbers a real number?
The product of all reals greater than 1?

Those questions are wrong, because the sum and the product here are not numbers. The question about
whether they are real or not does not apply.

 

[daon]

Re: Need help, clever math about infinity between 0 and 1 &

How is this for a mind-blower.

The set of all real numbers has the same cardinality as the set of all real numbers between 0 and 1.

And here's a thinker, and I've asked these long ago, is the sum of all positive real numbers a real number?
The product of all reals greater than 1?

Those questions are wrong, because the sum and the product here are not numbers. The question about
whether they are real or not does not apply.

Care to define what you mean by number?

 

[lookagain]

Re: Need help, clever math about infinity between 0 and 1 &

Care to define what you mean by number?

No, I don't care to define what I mean by "number."

A sum of real numbers and a product of real numbers are contained in the real numbers,
but one of your questions is analogous to asking if the product of all positive integers is
a positive integer. There is no number to pin down, so that question would not
apply either.

There is no sum of all positive real numbers.
There is no sum of all positive integers.

There is no product of all real numbers greater than 1.
There is no product of all of the positive integers.