Question

Complete the following sentence:
Every real number is either ………... number or …………... number.

Answer 1

Hint: We need to understand the concept of real numbers to fill in the blanks. Real numbers are the Major set containing all other numbers like-Natural numbers, Rational numbers, Integers, Irrational numbers.

Complete step by step answer:
Now, Real numbers is a set of all positive and negative numbers that contains the Subsets of Integers, Rational and Irrational numbers.
Now Integers as we know are all positive or negative rational numbers, of the form $\dfrac{p}{q},q = \pm 1$
All numbers of the form $\dfrac{p}{q},q \ne 0$, are rational numbers and any number of the form $\dfrac{p}{q},q = 0$ is an irrational number because with 0 as denominator the number cannot be defined.
Thus, we can conclude that integers are a special case of real numbers, so integers belong to the set of real numbers only.
Hence a real number can be either a rational number or an irrational number.
So the sentence will become:
Every real number is either a ….…rational…... number or ……an irrational…………... number.

Therefore, the words that will be used to fill the blanks are Rational and Irrational.

Note: Definitions have to be understood and applied correctly. All integers are real numbers but all real numbers are not integers. A real number can always be expressed in the form of $\dfrac{p}{q},q \ne 0$, if it is a rational number, whether positive or negative. For example, 5 is a real number which is a rational number as it can be written as $\dfrac{5}{1}$ . Similarly $ - \dfrac{4}{5}$ is also a rational number and a real number. However the number $\dfrac{3}{0}$ is not a rational number as the denominator is 0. But, it is a real number as real numbers contain both rational and irrational numbers.