Question

……….. numbers have terminating and non-terminating repeating decimals.
A) Integers
B) Whole
C) Rational
D) Irrational

Answer 1

Hint: Use the fundamental definition of the categories of numbers.
Integer: A number which is not a fraction.
Whole number: Set of non-negative integers.
Rational number: Numbers which can be written in the form of $\dfrac{p}{q}$, where p and q are integers and $q\ne 0$.
Irrational numbers: Numbers which cannot be written in the form of $\dfrac{p}{q}$ i.e. just opposite to rational numbers.
Complete step-by-step answer:
As we need to find the category of numbers which consist of terminating and non-terminating repeating decimals.
So, let us understand the definitions of the terms mentioned in options.
Integers: An integer is a number that can be written without a fractional component. i.e. a number which is not a fraction or a number complete in itself. Example: 2, 4, -3, -4, 0, 23 etc.
Whole numbers: The set of integers consisting of zero and the natural numbers, or counting numbers are called whole numbers. It is also termed as a non-negative integer. Example: 0, 1, 2, 3, 4, 5 ………..
Rational numbers: The numbers which can be represented in form of $\dfrac{p}{q}$ where p and q are integers and \[q\ne 0\]. These numbers can be terminating or repeating type. Here, terminating means the digits after the decimals (in decimal representation) are fixed, and repeating means some digits after decimal are repeating i.e. they will not terminate as well.
So, rational numbers are terminating or repeating type both. Example: $5,\dfrac{1}{3},\dfrac{4}{5},-8,-3.\overline{5}\left( -3.5555..... \right)$ etc.
Irrational numbers: It is just opposite to the definition of rational numbers. These numbers cannot be represented in the form of $\dfrac{p}{q}$. They are non-terminating and of non-repeating as well. Examples can be given as
$\sqrt{5},\sqrt{2},$ 4.413121763…………,-3.1234578032…………etc.
Hence, rational numbers have terminating and non-terminating decimals as per the definition of rational numbers given above.
So, option (C) is correct.
Note: Need to be clear with the terminologies mentioned in the options and problem as well. It is the key point of the question as well.
Students shouldn’t be confused with the numbers of type $-3.\overline{5}$ belongs to the rational number category. As one may think how to convert it into $\dfrac{p}{q}$ form, as the digits are upto the infinite after the decimal. So, it can be done as
x = -3.555555…………..---(1)
10x = -35.555555 ---(2)
Subtract equation 1 from equation 2
9x = -32
$x=\dfrac{-32}{9}$
Now, divide -32 by 9 to verify the above mentioned number in decimal form.