The number  0.41856431856431856431..... is a rational or irrational number ?

Answer :

We have number  0.41856431856431856431...

That is a rational numbers as that have repeating ( 856431 ) non terminating digits , And we can express this number in form of $\frac{p}{q}$ , As  :

Let

x  = 0.41856431856431856431...

x  = 0.41 + 0.00856431856431856431...

And

0.41 $= 0.41 \times \frac{100}{100} = \frac{41}{100}$

And

y  = 0.00856431856431856431...                                   ---------- ( 1 )

Multiply by 1000000 in equation 1 , we get

1000000y  = 8564.31856431856431856431...                   ------- ( 2 )

Now Subtract equation 1 from equation 2 , we get

999999y  = 8564.31

So,

y  = $\frac{8564.31}{999999} = \frac{8564.31}{999999}\times \frac{100}{100} = \frac{856431}{99999900}$
So,

x  = $\frac{41}{100} + \frac{856431}{9999990} = \frac{40999959 + 856431}{9999990} = \frac{\mathbf{41856390}}{\mathbf{9999990}}$

So,

Our given number 0.41856431856431856431... can be expressed in form of $\frac{p}{q}$ , And get $\frac{\mathbf{41856390}}{\mathbf{9999990}}$ .

So,

Given number is a rational number