How to find rational number betn two irrational numbers ?

Question

Hi,

Is there a standard way (any algorithm) to find a rational number between two irrational numbers say square_root(6) and square_root(7) ?

Answer 1

Square the numbers, in this case obtaining 6 and 7
Since the difference is only 1, multiply by 100, obtaining 600 and 700. Find a square in between.
600 < 625 < 700
6.00 < 6.25 < 7.00
√6 < 2.5 < √7
An alternate method is to extract the square roots, average the results, and truncate the average:
. 2.6457513110645905905016157536393
+2.4494897427831780981972840747059
. 5.0952410538477686886988998283452/2 =
. 2.5476205269238843443494499141726
Now you have a whole plethora of rational numbers between √6 and √7

Answer 2

Piece of cake.

1. Represent the numbers e.g. as decimal numbers, up to one digit past the one they start to differ. E.g.

sqrt(6) = 2.4494897427831780981972840747059...
sqrt(7) = 2.6457513110645905905016157536393...

They differ already in the first digit after the point, so represent them as

sqrt(6) = 2.44...
sqrt(7) = 2.64...

2. Truncate the remaining digits in both numbers and Increase the last digit of the lower number in this representation by one:

sqrt(6) < 2.45
sqrt(7) > 2.64

3. Take any rational number between the two, e.g.

2.5

Answer 3

I think that between any 2 irrational numbers there is at least one rational and between any 2 rational numbers there is at least one irrational, so between sqrt(6) and sqrt(7) and given that sqrt(6)= 2.449... and sqrt(7)=2.645... you can find many rational numbers (ie 2.46, 2.47, 2.5, 2.6, 2.64, 2.641...). One approximate method would be to take an approximation of those two numbers (ie 3 decimal digits or as many needed so that the approximations are different) and find the mean value. That would be rational number definitely between those two.

Answer 4

one simple way would be rounding the irrational numbers to a certain place, say, the millionths place, and then find the average of the "trimmed-up" numbers.

Note: Of course the two irrational numbers must be sufficiently distant, thats to say, not all-same-digits up to the millionths place. Well, you would have to really work hard to find so close numbers, anyway.

Answer 5

Any truncation to a finite number of decimals of the larger irrational number will give you a rational number less then this larger one and larger than the other

Answer 6

it depends what are the numbers
now if you approximate the numbers, say
x = 1.231478432319054................irrational
y= 1.231478432319055................irrational
then a = 1.2314784323190540000000(only 0-s)
is between x and y.