Irrational Numbers

1, 2, 3, 4, ... are natural numbers. But 1, 4, 9, 16, ... are perfect squares to boot. Because all have square-root viz 1, 2, 3, 4, .... But there are some numbers which act nasty. For example 2, 3, 5, 6, 7, 8, 10, etc. Whose square roots can't be found exactly. Remember 'exactly'. In other words these number are known as irrational numbers.If you take 10 for example's sake, it comes after 9 and before 16. Which means its square root is greater than 3 and smaller than 4.
Now if we go into more details, we can know that it square root up to 2 digits is 3.16. Up to 3 digits 3.162 and up to 4 digits 3.1622. And the sequence of a never ending calculation starts. But the value 3.1622 is a nearer value than 3.162. And 3.162 is more accurate than only 3.16. But it is also worth noticing than, no value of any of the above is exact.
As you keep increasing the decimal places, you will reach a more accurate and nearer value. In that ascending order the square root of 10 will be 3.1, 3.16, 3.162, 3.1622, 3.16227, 3.162277, 3.1622776, 3.16227766, .... And the process never ends. Manually it make bulk of time to calculate the values of such irrational numbers. But now a days, with the help of computers values in the region and probably above 5 lac decimal places can be found within a matter of fraction of a second.
But again one thing needs to be clarified that even 5 lac decimal places' value found with the help of computers doesn't mean exact values. They still are irrational numbers and so 5 lac decimal places' value still is an approximate one and not perfect. Only the plus point is that, it is far more nearer to the unknown value.