# Do irrational numbers contain everything infinitely times?

Watch
Announcements

I have a question about irrational numbers.

If I start writing a number of n digits does this sequence of numbers exist on every irrational number infinitely many times for all nΕΝ?

So for example if myself is nothing more than a really really large number (total information of myself) and every 10^10^10 numbers of pi is another universe (total information of the universe) do I have infinite replicas of myself ?

I guess that this happens for a completely random number, so is my question equivalent to 'Are irrational numbers completely random?'?

Is there any proof for this?

I'm a high school student ,don't expect any further knowledge from me.

If I start writing a number of n digits does this sequence of numbers exist on every irrational number infinitely many times for all nΕΝ?

So for example if myself is nothing more than a really really large number (total information of myself) and every 10^10^10 numbers of pi is another universe (total information of the universe) do I have infinite replicas of myself ?

I guess that this happens for a completely random number, so is my question equivalent to 'Are irrational numbers completely random?'?

Is there any proof for this?

I'm a high school student ,don't expect any further knowledge from me.

0

reply

Report

#2

No. A counterexample is the irrational number , which only contains the digits 1 and 0, so doesn't contain the digit sequence 2.

You could construct a number that satisfies your property. List every one digit sequence, then every two digit sequence, then three digit, then one digit, then two digit, three, four, one, two, three, four, five, one, ....

It's not yet known whether pi has this property.

You could construct a number that satisfies your property. List every one digit sequence, then every two digit sequence, then three digit, then one digit, then two digit, three, four, one, two, three, four, five, one, ....

It's not yet known whether pi has this property.

2

reply

Report

#3

Another (more general) counter example is constructing an irrational number where and are any two digits you like.

0

reply

Report

#4

(Original post by

I have a question about irrational numbers.

If I start writing a number of n digits does this sequence of numbers exist on every irrational number infinitely many times for all nΕΝ?

So for example if myself is nothing more than a really really large number (total information of myself) and every 10^10^10 numbers of pi is another universe (total information of the universe) do I have infinite replicas of myself ?

I guess that this happens for a completely random number, so is my question equivalent to 'Are irrational numbers completely random?'?

Is there any proof for this?

I'm a high school student ,don't expect any further knowledge from me.

**Quantum boy**)I have a question about irrational numbers.

If I start writing a number of n digits does this sequence of numbers exist on every irrational number infinitely many times for all nΕΝ?

So for example if myself is nothing more than a really really large number (total information of myself) and every 10^10^10 numbers of pi is another universe (total information of the universe) do I have infinite replicas of myself ?

I guess that this happens for a completely random number, so is my question equivalent to 'Are irrational numbers completely random?'?

Is there any proof for this?

I'm a high school student ,don't expect any further knowledge from me.

0

reply

(Original post by

Another (more general) counter example is constructing an irrational number where and are any two digits you like.

**Zacken**)Another (more general) counter example is constructing an irrational number where and are any two digits you like.

(Original post by

You could construct a number that satisfies your property. List every one digit sequence, then every two digit sequence, then three digit, then one digit, then two digit, three, four, one, two, three, four, five, one, ....

**morgan8002**)You could construct a number that satisfies your property. List every one digit sequence, then every two digit sequence, then three digit, then one digit, then two digit, three, four, one, two, three, four, five, one, ....

The n digit sequences will contain every sequence of 1,2,...,n-1 in them.

Doesn't this mean every sequence will be repeated infinitely?

So irrational numbers don't necessarily carry infinite information, but do pi for example satisfy this?

(Original post by

In addition to the counter-examples other folk have already posted, you might like to have a look at the notion of a normal number, the requirements for which are a bit stronger than those you advance. But you might find it amusing!

**Gregorius**)In addition to the counter-examples other folk have already posted, you might like to have a look at the notion of a normal number, the requirements for which are a bit stronger than those you advance. But you might find it amusing!

0

reply

Report

#6

(Original post by

I was talking about the 'natural' irrational numbers: pi, e, nth roots of integers. Do they all have these properties?

**Quantum boy**)I was talking about the 'natural' irrational numbers: pi, e, nth roots of integers. Do they all have these properties?

What makes those examples you've given any more "natural" than my counter-example.

Also, nth root of integers aren't always irrational, e.g: for some .

0

reply

Report

#7

**Quantum boy**)

I was talking about the 'natural' irrational numbers: pi, e, nth roots of integers. Do they all have these properties?

So normal numbers are completely random (No digit or sequence is "favored". I got the general idea ,but what is natural density anyway?

0

reply

Report

#8

(Original post by

I was talking about the

**Quantum boy**)I was talking about the

**'natural' irrational numbers**: pi, e, nth roots of integers. Do they all have these properties?No idea what you mean by natural irrationals...

0

reply

Report

#9

(Original post by

No idea what you mean by natural irrationals...

**Johann von Gauss**)No idea what you mean by natural irrationals...

0

reply

Report

#10

(Original post by

It is a term used informally in analytic number theory to describe those irrational numbers that just keep popping up everywhere!

**Gregorius**)It is a term used informally in analytic number theory to describe those irrational numbers that just keep popping up everywhere!

0

reply

(Original post by

and e (transcendentals) are a different class of irrationals to irrational roots

No idea what you mean by natural irrationals...

**Johann von Gauss**)and e (transcendentals) are a different class of irrationals to irrational roots

No idea what you mean by natural irrationals...

(Original post by

Literally no clue what you're going on about...

What makes those examples you've given any more "natural" than my counter-example.

Also, nth root of integers aren't always irrational, e.g: for some .

**Zacken**)Literally no clue what you're going on about...

What makes those examples you've given any more "natural" than my counter-example.

Also, nth root of integers aren't always irrational, e.g: for some .

I meant the ones which have occured naturally : they were found in an application (ratio of length of circle to diameter, a number e so that (e^x)'=e^x,... ). The 0.deedeeeeeddddededde example is a random number generator ,it has little or no meaning.

Anyway, sorry for the misunderstanding.

0

reply

Report

#12

Actually a very interesting question.

Certainly not every irrational number does, but it would be interesting if there were a proof that a particular one has this property.

Certainly not every irrational number does, but it would be interesting if there were a proof that a particular one has this property.

0

reply

Report

#13

(Original post by

Sorry guys for this expression, I made it up.

I meant

Anyway, sorry for the misunderstanding.

**Quantum boy**)Sorry guys for this expression, I made it up.

I meant

**the ones which have occured naturally**:**they were found in an application**(ratio of length of circle to diameter, a number e so that (e^x)'=e^x,... ). The 0.deedeeeeeddddededde example is a random number generator ,it has little or no meaning.Anyway, sorry for the misunderstanding.

And what about the physical constants like the speed of light in a vacuum, etc.

You can't really prove anything about something unless it is rigorously defined, so its tough to say if what you are asking about is true or not.

0

reply

(Original post by

But you could say 0.dedeedeeedeeeed... is the simplest irrational where only 2 digits occur in the decimal expansion...

You can't really prove anything about something unless it is rigorously defined, so its tough to say if what you are asking about is true or not.

**Johann von Gauss**)But you could say 0.dedeedeeedeeeed... is the simplest irrational where only 2 digits occur in the decimal expansion...

You can't really prove anything about something unless it is rigorously defined, so its tough to say if what you are asking about is true or not.

0

reply

Report

#16

What about Sum(x=1 -> inf): 10^(-x^2)? isn't that a pretty 'simple' and 'natural-looking' irrational number that obviously doesn't contain every sequence of n digits?

[I have no idea how to insert equations!]

[I have no idea how to insert equations!]

0

reply

Report

#17

Yes

Spoiler:

Show

... in binary

More seriously your statement is not known whether you restrict to algebraic irrationals, transcendental, or what you are calling "natural" irrationals (including π, e, √2)

More seriously your statement is not known whether you restrict to algebraic irrationals, transcendental, or what you are calling "natural" irrationals (including π, e, √2)

0

reply

(Original post by

What about Sum(x=1 -> inf): 10^(-x^2)? isn't that a pretty 'simple' and 'natural-looking' irrational number that obviously doesn't contain every sequence of n digits?

[I have no idea how to insert equations!]

**Forum User**)What about Sum(x=1 -> inf): 10^(-x^2)? isn't that a pretty 'simple' and 'natural-looking' irrational number that obviously doesn't contain every sequence of n digits?

[I have no idea how to insert equations!]

(Original post by

Yes

**Lord of the Flies**)Yes

Spoiler:

Show

... in binary

More seriously your statement is not known whether you restrict to algebraic irrationals, transcendental, or what you are calling "natural" irrationals (including π, e, √2)

More seriously your statement is not known whether you restrict to algebraic irrationals, transcendental, or what you are calling "natural" irrationals (including π, e, √2)

0

reply

Report

#19

There must be an infinite amount of irrational numbers that have this property, but determining whether a number, like pi or e, does have it must be impossible.

0

reply

(Original post by

But you could say 0.dedeedeeedeeeed... is the simplest irrational where only 2 digits occur in the decimal expansion...

And what about the physical constants like the speed of light in a vacuum, etc.

You can't really prove anything about something unless it is rigorously defined, so its tough to say if what you are asking about is true or not.

**Johann von Gauss**)But you could say 0.dedeedeeedeeeed... is the simplest irrational where only 2 digits occur in the decimal expansion...

And what about the physical constants like the speed of light in a vacuum, etc.

You can't really prove anything about something unless it is rigorously defined, so its tough to say if what you are asking about is true or not.

They are in the form 'write any number from 0 to 9 randomly' .

But perhaps they are not even irrational numbers.

0

reply

X

### Quick Reply

Back

to top

to top