Question

Is \[ - 4\] a rational number or a natural number, a whole number, irrational number or imaginary number?

Answer 1

Hint:We need to check \[ - 4\] belongs to which category. For that, we first need to know what a rational number, a natural number, a whole number, irrational number or imaginary number are.
Rational Numbers: A rational number is a number which can be written in the form \[\dfrac{p}{q}\], where \[p\] and \[q\] belong to integers and \[q \ne 0\].
Natural Numbers: Natural numbers are the positive integers starting from \[1\] till infinity.
Whole Numbers: Whole numbers are the positive integers including zero i.e. it includes all the positive integers from \[0\] till infinity.
Irrational Numbers: The numbers which cannot be expressed as a fraction (or as a ratio) of integers are known as irrational numbers.
Imaginary Number: Imaginary numbers are basically the complex numbers in which the coefficient of \[iota\]\[\left( i \right)\] is not equal to zero.

Complete step by step answer:
Now, let us consider \[ - 4\]. Since \[ - 4\] is a negative number, it cannot be a whole number or a natural number.Also, we cannot see any \[iota\] term, hence, the coefficient of \[iota\] in \[ - 4\] is zero. So, \[ - 4\] cannot be an imaginary number. Now, we know, \[a = \dfrac{a}{1}\]. So, \[ - 4\] can be written as
\[ \Rightarrow - 4 = \dfrac{{ - 4}}{1}\]
Here, if we compare \[\dfrac{{ - 4}}{1}\] with \[\dfrac{p}{q}\], we get
\[ \Rightarrow p = - 4\]
\[ \Rightarrow q = 1\]
We see both \[p\] and \[q\] belong to integers and \[q = 1 \ne 0\]. Hence, we can express \[ - 4\] in \[\dfrac{p}{q}\] form, where \[p\] and \[q\] belong to integers and \[q \ne 0\].

Therefore, by definition of rational numbers, we see that \[ - 4\] is a rational number.

Note:While categorising, we should consider each and every condition which is given in the definition. Just ignoring one single condition can change the concept and so we will obtain the wrong answer. Like, in Natural Numbers and Whole Numbers there is just a minor difference. Just ignoring one number, we will be getting different answers.