Given an rational number "0,a" can you find an irrational number of the form "b,a..."?

Question

i want to know if it is possible to find such an irrational number and put it in a form that i can write as an expression of integers.

for example: for 0.41 i can write sqrt(2);

fractions are alowed and i don't care how big the integers are.

if this is possible how can this be done?

for example if 0,a = 0.41 a = 41 and if b,a... = sqrt(2) b = 1 and a..={sqrt(2)}

a can be very long but this is not an issue in my opinion.

i don't want to use the number in the expression that generates the irrational number. it would have been easy this way.

Answer 1

Saw your edit, see below...

So if I understand correctly, you start with a decimal number that has a finite number of decimals, like 0.41 or 0.87293, and you want to know if you can build an irrational number that begins with those same decimals, expressing it as a function of integers only.

In your example, this can be done with 0.41, because sqrt(2) = 1.41421... (the left side of the decimal doesn't matter to you).

Sure then. There are infinitely many ways to do this, but here's just one:

Given the number "0,a" having n decimals. So for 0.41, a=41 and n=2. Then the given number can be expressed as a/10^n.

If you take sqrt(2)=1.41421... and divide it by 10 often enough, you'll get the number 0.000...0141421... which is irrational. All you need to do then is add this to the original number. Just make sure the first 1 appears on the (n+1)th decimal so as not to perturb the first n decimals. This means dividing by 10^(n+1).

So the number a/10^n + sqrt(2)/10^(n+1) satisfies the conditions. Everything is there is integers.

In the 0.41 case, this would yield the solution 41/100 + sqrt(2)/1000 = 0.41141421356...

Edit: So you don't like it because it uses a. OK, how about this:

Define "0,x" as the square of "0,a". By adding an extra decimal to "0,x" and taking its square root, you will get "0,a" plus an irrational component. This can be expressed as

sqrt(x/10^n+1/10^(n+1))

where n is the number of decimals for "0,x". In the 0.41 example, this yields the number sqrt(1681/10000 + 1/100000) = sqrt(0.16811) = 0.410012194...

Answer 2

You cannot write 0.41 as the sqrt 2. You can write 1.4.4 Approximately = sqrt 2, but the sqrt is irrational as is pi, sqrt 3, e, and an infinite number of other examples.

You cannot do what yoou wish to do by the very definition of irrational numbers.

Answer 3

even though you have mentioned b,a.... I can provide a general method say b.a1a2a3....

1st n digits same for the irrational number

this is of the form p/q where q is power of 10

p+1/q shall have last digit different and any number between p/q and (p+1)/q shall have 1st n digits same.

take (p/q)^2 and ((p+1)/q))^2
if we take some number say (p^2+r)/q where 0
to illustrate

say starting with .1

it is between .1 and .2
(1/10) and 2/10
take square of both
1/100 and 4/100

if we take 2/100 or 3/100 and take square root it shall be irrational and start with .1
sqrt(2/100 ).14142..
sqrt(3/100) = .1732...

Answer 4

This is something strange I am coming across. And you should be specific how many digits you want after decimal points as rational numbers are sensitive to precision.