Question

The decimal expansion of $\pi $ is called
(a) Terminating
(b) Non terminating and non-recurring
(c) Non-terminating and recurring
(d) Does not exist

Answer 1

Hint: We will use the information about the terms that are given in the options so that we get clarity about them. Also, we will understand their differences among themselves in order to solve the question.Like what exactly we mean by terminating and non terminating likewise recurring and non recurring.

Complete step-by-step answer:
We will discuss the question by starting to understand the meaning of the terms terminating, non-terminating, recurring and non recurring. Terminating means a type of a number which stops at one point. For example if we see the number 321 then it is the terminating number. Also if we look at the number 54.8764 then, we find that it is also a terminating number. This is because both the numbers end at some point. Most of the time we consider decimal numbers to check their termination.
Now on the other hand if we consider the number 54.333333.... then, we find that this number does not end at all. So, we will consider this type of a number as a non-terminating number.
Moreover, the repeating numbers are basically recurring numbers only on the other side there is a difference between recurring and non terminating numbers. The difference between them is that the numbers which have a repetition after their decimals then those will be recurring while the numbers like 4.235522.... are called non terminating. For example the number 2.33333.... is a recurring number and we can write it as $2.\overline{3}$. Here, 3 is repeated. Now we will consider the number $\pi $. The value of $\pi $ is 3.1415926536.... So, clearly this number is non-terminating as well as non-recurring due to no repetition in it.
Hence, the correct option is (b).

Note: The decimals that we are considering here actually come from the fractions. So basically if we have the fraction like $\dfrac{4}{5}$ then the value of this fraction is 0.8. Clearly this decimal is terminating. One should not write 0.2 as 0.200000.... since, the number 0.2 is terminating but if we write it as 0.200000... then, it will be confused as non-terminating and this is considered wrong. One should also not get confused between the terms non-terminating and non-recurring. The definitions must be cleared from the solution. If we don’t understand the terms carefully then it will result in a wrong answer.