Question

Find a rational number and also an irrational number lying between the numbers 0.401001000100001000001…… and 0.404004000400004000004……

Answer 1

Hint:For the above question, we will have to know about the rational and irrational numbers. Rational numbers are those numbers which can be expressed as a fraction and also as positive numbers, negative numbers and zero. It can be written as \[\dfrac{p}{q}\], where q is not equal to zero. Irrational numbers could be written in decimals but not in fractions. Irrational numbers have endless non-repeating digits after decimal point.

Complete step-by-step answer:

In the above question, we have to find a rational and an irrational number between the two given numbers.
Calculating the rational number between 0.401001000100001000001…. and 0.404004000400004000004…., we get,
\[0.402=\dfrac{402}{1000}=\dfrac{201}{500}\]
Here, the number is in the form of \[\dfrac{p}{q}\] where \[q\ne 0\]. So the number 0.402 is a rational number which is lying between 0.401001000100001000001…. and 0.404004000400004000004….
Now calculating an irrational number between 0.401001000100001000001….. and 0.404004000400004000004…, we get 0.402002000200002000002…
0.402002000200002000002… is a number in decimal form which is lying between 0.401001000100001000001…. and 0.404004000400004000004…. as well as it is non-terminating and non-repeating which means that it is an irrational number.
Therefore, we have calculated a rational and an irrational number lying between the two given numbers which are 0.402 and 0.402002000200002000002….., respectively.

Note: Just remember that between any two real numbers, there are infinitely many rational and irrational numbers between them.Also, remember that the decimal expansion of a rational number is finite or recurring decimals but in irrational numbers, decimals are non-finite and non-recurring decimals.