Question

Find two irrational numbers and two rational numbers between 0.5 and 0.55.

Answer 1

Hint: Rational numbers in decimal form are number that terminate
We start by taking the number immediately to initial most value and that can be considered as rational number and its again immediate number can be considered as rational number only after verifying it lies between the given range and then checking whether its decimal is terminating or non-terminating. A rational number in its decimal form is terminating or else it is periodically recurring. Then, we find the irrational number by selecting a non-recurring and non-terminating decimal value between the given range.

Complete step by step answer:
First, we are finding the two rational numbers in the given range.
Now we first consider $0.51$
It lies in the given range and next, it is also terminating. Since it satisfies the criteria for rational numbers. We can say that $0.51$ is a rational number.
Now, we consider \[0.52\]
It lies in the given range, and next, it is definitely terminating. Since, it satisfies the criteria for rational numbers. We can say that \[0.52\] is a rational number

Now, the next half of the question. Finding irrational numbers.
Let’s consider $0.51551555155551....$
It lies in the given range and next, on observing the dotted lines, it indicates its non-terminating as well as its non-recurring. So, it satisfies the criteria. Hence, $0.51551555155551....$ is an irrational number.

Let’s consider $0.521221222122221........$
It lies in the given range and next, on observing the dotted lines, it indicates its non-terminating as well as its non-recurring. So, it satisfies the criteria. Hence, $0.521221222122221........$ is an irrational number.

Note: Irrational numbers often create a pattern after some digits. Like numbers such as 1.234234234234234…will forever continue in this way with 234 coming again and again. These types of numbers can be written as $1.\overline {234} $. And these types of numbers are irrational not rational numbers since they are never ending.