Question & Answer

Every integer is also a/an
[a] real number
[b] rational number
[c] irrational number
[d] natural number.

ANSWER Verified Verified
Hint: Recall the definition of a real number, a natural number, a whole number, an irrational number and a real number. Check, in which of these definitions natural numbers also fit. Hence find which of these number sets also contain natural numbers

Complete step-by-step solution -

We have the following sets of numbers which play an important role in the field of mathematics.
[1] Natural Numbers: Numbers 1,2,3. … are called natural numbers These numbers are also known as counting numbers as they are used in counting, e.g. 1 egg, 20 spoons etc.
[2] Whole Numbers: Natural numbers, along with the number 0, are known as whole numbers.
[3] Integers: Natural numbers, along with their negatives and the number 0, are known as integers, e.g. 1,2,-2, -7 etc. are integers.
[4] Rational Numbers: Numbers which are either or can be expressed, in the form of $\dfrac{p}{q}$ where p and q are integers and $q\ne 0$ are called rational numbers, e.g. $1,2,\dfrac{2}{7},-1.5$ etc. are rational Numbers.
[5] Irrational Numbers: Numbers which have non-terminating and non-recurring decimal representation are called irrational numbers, e.g. $\sqrt{2},\pi $ , etc.
[5] Real Numbers: Rational numbers and irrational numbers are collectively called as real numbers, e.g. $1.3,\pi ,\gamma $ , etc.
Now since every integer n can be expressed in the form of $\dfrac{n}{1}$ and n and 1 both are integers, and 1 is non-zero, every integer is a rational number. Now, since real numbers is a collective set of rational numbers and irrational numbers, every integer is a real number.
Since every integer is a rational number, no integer is an irrational number.
Since -1 is an integer but not a natural number, every integer is not a rational number.
Hence we have options [a] and [b] are correct.

Note: [1] The set of Natural numbers is denoted by $\mathbb{N}$
[2] The set of integers is denoted by $\mathbb{Z}$
[3] The set of Rational numbers is denoted by $\mathbb{Q}$
[4] The set of Real numbers is denoted by $\mathbb{R}$
[5] We have $\mathbb{N}\subset \mathbb{Z}\subset \mathbb{Q}\subset \mathbb{R}$