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# Is there a rational number between any two irrationals? Watch

1. Question in the title.

I feel intuitively the answer should be yes, but can't really seem to prove this... Any help?
2. Yes, all you have to do is look at the digits of both irrational numbers until there is a difference between two digits, truncate the irrationals after this digit (so you have a rational number) add the two rational numbers together, divide by two and you've got a rational number between the two irrationals.

So for example, suppose you've got the two following irrational numbers and wlog write them as

Where the are the digits.

The reason we can say they're between 0 and 1 is because it doesn't matter what the digits are to the left of the decimal point, it's just extra notation if we add them in.

Let's say that for , so what we're saying is there is a difference between the two irrationals at the digit in place

So denote the rational numbers

Then the number is also rational, but note that

This number will have the first digits identical to the digits of but the last two digits will be such that
3. Yes, there is.

Hint for the proof: consider the decimal expansions of the numbers.
4. (Original post by Maths Failure)
Question in the title.

I feel intuitively the answer should be yes, but can't really seem to prove this... Any help?
Yes

Edit : as Noble said

Edit : even more what Noble said
5. Oh ok thanks guys
6. (Original post by Maths Failure)
Oh ok thanks guys
I've just updated my reply with a more mathematical explanation which you might find more helpful in trying to understand.
7. (Original post by TenOfThem)
Yes

Edit : as Noble said

Edit : even more what Noble said
I think you accidentally negged me, with that god awful -8 reputation power.

Either that or you're getting your own back for me finding your facebook picture the other day
8. (Original post by Noble.)
Yes, all you have to do is look at the digits of both irrational numbers until there is a difference between two digits, truncate the irrationals after this digit (so you have a rational number) add the two rational numbers together, divide by two and you've got a rational number between the two irrationals.

So for example, suppose you've got the two following irrational numbers and wlog write them as

Where the are the digits.

The reason we can say they're between 0 and 1 is because it doesn't matter what the digits are to the left of the decimal point, it's just extra notation if we add them in.

Let's say that for , so what we're saying is there is a difference between the two irrationals at the digit in place

So denote the rational numbers

Then the number is also rational, but note that

This number will have the first digits identical to the digits of but the last two digits will be such that
Why does have to be bounded? If was infinite could an argument be made that and are still possibly different numbers?
9. (Original post by Maths Failure)
Why does have to be bounded? If was infinite could an argument be made that and are still possibly different numbers?
No, if the numbers do not differ for some finite then they are the same number. You can argue it in a similar way to how using infinite series and sequences because if the digits do not differ for any finite then the infinite sum of both are exactly the same.
10. (Original post by Noble.)
I think you accidentally negged me, with that god awful -8 reputation power.

Either that or you're getting your own back for me finding your facebook picture the other day
Balls

Let me see if I can redress

xx

I will come back and personal rep tomorrow but I am out today

Sowwy
11. (Original post by TenOfThem)
Balls

Let me see if I can redress

xx
Haha don't worry about it, I've got plenty of rep.
12. (Original post by Noble.)
Haha don't worry about it, I've got plenty of rep.
You call a +6 "plenty"

Hah

[obviously I have more cos i am soooooooo much older]
13. (Original post by TenOfThem)
You call a +6 "plenty"

Hah

[obviously I have more cos i am soooooooo much older]
Haha, I've just seen you joined this site in Sep 2011 and have 15.5k posts - I think we can safely conclude you like helping people too much, what with you being a maths teacher. You and Mr M should start some Avengers like team of maths teachers.
14. (Original post by Noble.)
Haha don't worry about it, I've got plenty of rep.
Attempted to redress balance

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