Question

Find two irrational numbers between \[0.16\] and \[0.17\].

Answer 1

Hint: Here we will find two irrational numbers between \[0.16\] and \[0.17\] by using the definition of an irrational number. Between any two rational numbers, there are infinite irrational numbers, also irrational numbers have a non-terminating and non-recurring decimal expansion. We will use these points to select a number between the two numbers given which is irrational.

Complete step-by-step answer:
We have to find two irrational numbers between \[0.16\] and \[0.17\].
Irrational numbers are numbers that cannot be expressed in the form of a fraction or ratio of two integers. In other words, we can’t express them as \[\dfrac{p}{q}\] the form where \[p\] and \[q\] are integers and \[q \ne 0\].
Some examples of Irrational numbers are values of \[\pi \], Square root of any number except perfect square numbers.
We know by the definition of irrational numbers that they are non-terminating and non-recurring in nature. So we can take any decimal value between the two numbers given having non-terminating value.
Let us take the values as \[0.1655727752.......\] and \[0.1617840374892.......\]. These numbers are greater than \[0.16\] and smaller than \[0.17\]. Therefore, they both lie in between the two given numbers and they both are irrational.

Therefore the required answer is \[0.1655727752.......\] and \[0.1617840374892.......\].

Note: This question can have many different answers as the irrational numbers between two rational numbers are infinite such as \[0.163513626236.....\], \[0.1685234236.......\] Irrational numbers are inverse of rational numbers. An irrational number has associative properties of addition. These numbers are not closed under multiplication, division, addition, and subtraction. As real numbers are uncountable sets and rational are countable sets, so the complementary set of irrational i.e. irrational numbers are uncountable.