# finding a rational number between a rational and irrational

Discussion in 'Undergraduate Math' started by actuarynrt, Aug 26, 2007.

1. ### actuarynrtGuest

I've been getting great help here in this discussion forum.

The question I have is:
Find a rational number between 3.14159 and Ï€.

I am thinking i could just select some number slightly larger than 3.14159 like 3.141591 and that would be it; because Ï€ is approximately equal to 3.14592654...

Am I correct in selecting 3.141591 as an answer? My text doesn't give an answer (I'm assuming because there could be many).

Thanks.

John

actuarynrt, Aug 26, 2007

2. ### actuarynrtGuest

Oops...

the ? should actually be Pi

Looking for a number between 3.14159 and Pi

actuarynrt, Aug 26, 2007

3. ### VirgilGuest

Your value is one of infinitely many rationals between them.

Virgil, Aug 26, 2007
4. ### Jeremy WattsGuest

Do you mean 'irrational' rather than rational? Because as Virgil pointed
out there will be infinitely many rationals between 3.141591 and pi, you
could take your pick...

If you meant irrational then 15001/15000 * cuberoot (31) is approximately
equal to 3.14159008, and is irrational and lies between 3.14159 and pi.

Jeremy Watts

Jeremy Watts, Aug 26, 2007
5. ### The Qurqirish DragonGuest

As others have stated, there are an infinite number of solutions for
this sort of problem. One way to get an answer is the way you did it:
1) List the digits of the decimal expansion of both the given rational
and irrational numbers, until there is a difference.
2) Extend your number by choosing digits until it is between the
givens (generally this should only take 2 more digits at most, but you
might have a 49999999 vs. 50000000 situation that needs more digits)
3) If you need a rational, stop here. If you need an irrational,
append here any irrational sequence of digits (e.g.
101001000100001....)

The Qurqirish Dragon, Aug 26, 2007
6. ### DanaGuest

Find a rational number between 3.14159 and ?.
I would say "no", as 3.14... is not a Rational Number.
The obvious answer is any number between 0 and Pi - 3.141591.
Here's a different approach at a typical computer' precision.

num = 314159/100000

Let' turn the difference into a ratio:
Rationalize[N[Pi - num]]

199421 / 75151404527

This should be the thightest ratio we can get on our computer.
This works:
num + 199420./75151404527 < Pi
True

But this is a hair too high.
num + 199421./75151404527 < Pi
False

Since there are an infinit solutions to a/b < Pi-num,
what if we picked "b", and found the range for a?
(n-Numerator, d-denominator)

Reduce[n/d < 199421/75151404527, n][]
n < (199421*d)/75151404527

Here, given a denominator 'd, we can find the range for 'a
Suppose we want to use a denominator of 1,000,000.

(199421*d)/75151404527 /. d -> 1000000

2.6535897932333796

Our numerator must be <2.6, or basically, 1 or 2.
Hence, 2/1,000,000 is ok.

num + 2./1000000 < Pi

True

but we now know this should be too high.
num + 3./1000000 < Pi

False

If we reverse the above equation, we find that the denominator must be >=
376,848

Dana, Aug 26, 2007