Question

Solve the below parts:
1. Find an irrational number between 0.1 and \[\dfrac{2}{7}\].
2. Find two irrational numbers between \[\sqrt{2}\] and \[\sqrt{7}\].

Answer 1

Hint: In this problem, we have to find the irrational numbers between given two numbers. We know that irrational numbers should have a non-terminating and non-repeating expansion. We can first write the decimal form for the given fraction or root value. We can then compare the given two numbers one by one to find the irrational number between the given numbers.

Complete step by step solution:
1. To find an irrational number between 0.1 and \[\dfrac{2}{7}\].
We know that the given two numbers are,0.1 and \[\dfrac{2}{7}\].
We can now write the decimal form of the given fraction \[\dfrac{2}{7}\].
By dividing the number 2 by the number 7, we will get
\[\Rightarrow \dfrac{2}{7}=0.285714....\]
Now we have to find the irrational number between 0.1 and 0.2.
We can see that irrational numbers between 0.1 and 0.285714… should have a non-terminating and non-repeating expansion.
Such that, 0.150150015000 ……. is an irrational number between 0.1 and \[\dfrac{2}{7}\].
Therefore, an irrational number between 0.1 and \[\dfrac{2}{7}\] is 0.150150015000…
2. To find two irrational numbers between \[\sqrt{2}\] and \[\sqrt{7}\].
We know that the given numbers are \[\sqrt{2}\] and \[\sqrt{7}\].
We know that the value of \[\sqrt{2}\] is 1.14 and \[\sqrt{7}\] is 2.64
We also know that \[\sqrt{3}\] and \[\sqrt{5}\] have non-terminating and non-repeating values.
Where,
\[\begin{align}
  & \sqrt{3}=1.7320..... \\
 & \sqrt{5}=2.2360..... \\
\end{align}\]
The above two root numbers are irrational numbers, as it has non-terminating and non-repeating values.
Therefore, two irrational numbers between \[\sqrt{2}\] and \[\sqrt{7}\] are \[\sqrt{3}\] and \[\sqrt{5}\].

Note: We should always remember that an irrational number are real numbers which have neither terminating and non-repeating numbers and it cannot be expressed as a ratio of two numbers. We should also know some root values to solve these types of problems.