Question

Find two irrational numbers lying between 0.1 and 0.12. Enter 1 if the numbers are 0.1010010001……. and 0.1101001000100001…..otherwise, enter 0.

Answer 1

Hint: We can use the decimal method to find the two irrational numbers between 0.1 and 0.12. Remember that the decimal expansion of an irrational number is neither terminating nor recurring.

Complete step-by-step answer:
Before proceeding with the question, we should know about irrational numbers. An irrational number is a real number that cannot be expressed as a ratio of integers. Again, the decimal expansion of an irrational number is neither terminating nor recurring. Real numbers which cannot be expressed in the form of \[\dfrac{\text{p}}{\text{q}}\], where p and q are integers and \[\text{q}\ne 0\] are known as irrational numbers. For example \[\sqrt{2}\], \[\sqrt{3}\] etc. Whereas any number which can be represented in the form of \[\dfrac{\text{p}}{\text{q}}\], such that, p and q are integers and \[\text{q}\ne 0\] is known as a rational number. For example \[\dfrac{6}{13}\], \[\dfrac{3}{7}\] etc.
Let \[a=0.1\] and \[b=0.12\]. Here a and b are rational numbers.
For finding irrational numbers between any two rational number, it is our random selection but the point to be remembered is, it should be non-terminating and non-recurring and the number should be between the limits of rational numbers given.
So the two irrational numbers between 0.1 and 0.12 are 0.1010010001……. and 0.1101001000100001…..because both these numbers are non-recurring and non-terminating.
Hence we enter 1.

Note: We need to be very careful with the concept of irrational numbers and also with the meaning of non-terminating and non-recurring while solving this type of problems.