Question

Express 2.565656 .... in the form of rational number​

Answer 1

Step-by-step explanation:

t for the sake of a more complete answer, I'll assume that the 56s go on forever. That makes the decimal expansion of the number infinite, but not making it an irrational number. Why?

Because

56/990=0.056565656

So

23/10+56/990=2.356565656

(2277+56)/990=2.356565656

2333/990=2.356565656

So your number, even if it goes on forever is a rational number.

Just proving that it was a sum of two rationals was enough, but I thought that showing it as a ratio of two integers would be clearer

Thanks

Answer 2

Given:

2.565656...

To Find:

Express in the rational number form

Solution:

To express in the form of rational number we will let the given irrational number be equal to x. Please note that it can be expressed in rational form only if the decimal is repeating non terminating decimals. now let x=2.565656...

$x=2.565656...$       -(1)

Now we multiply both sides by 100

$100x=256.565656...$     -(2)

( we choose it to multiply by 10 100 1000 etc depending on the digits after which they are repeating if they are repeating after 4 digits then multiply by 10000)

Now we subtract equation 2 with equation 1, which goes as

$99x=254\\x=\frac{254}{99}$

Hence, the number 2.565656... expressed in the rational form will be 254/99.