Are All Integers Rational Numbers?

# Are All Integers Rational Numbers?

Are All Integers Rational Numbers?

You have learnt about counting numbers or natural numbers, whole numbers, integer, fractions, rational numbers, irrational numbers and real numbers.

Natural numbers are the counting numbers such as 1, 2, 3, 4, … . By including zero to natural numbers, the system was extended to whole numbers, i.e., 0, 1, 2, 3, 4, … . We further extended the system by including negatives of natural numbers to the whole numbers to get a set of integers. Integers are …, –3, –2, –1, 0, 1, 2, 3, …

We can add, subtract or multiply integers to get an integer, but what about division of integers?

It is observed that division of integers may not always yield an integer. In order to get a solution for such problems, we extend the system to system of fractions. As fraction is a part of a whole, it is always positive. We do come across situations where fractions may be negative. A distance of 250 m above sea level is represented by ¼ km, which is a fraction. What about representation of 250 m below sea level in kilometres? We denote the distance of ¼ km below sea level by –¼. It is observed that –¼ is not a fractional number. Thus, there is a need to extend our number system to include such numbers.

This new system is the system of rational numbers.

Rational numbers can be written in the form of p/q, where p and q are integers and q ≠ 0. If a number cannot be written in the form of p/q, then it is called an irrational number.

The collection of rational and irrational numbers is called a set of real numbers.

To remember all these numbers is very confusing. You can understand and remember these numbers using a simple Venn Diagram given below.

By looking at the above diagram, you can say that all natural numbers, whole numbers and integers are rational numbers. Because they can be written in the form of p/q using denominator as 1.

Hence, all the integers are rational numbers.

Are Rational Numbers Fractions?

No. All rational numbers are not fractions. Because rational numbers can have negative fractional value but fractions cannot be negative. For example, -3/4 is a rational number but it is not a fraction.

Are Fractions Rational Numbers?

Yes. All fractions are rational numbers. Because all the fractions are of the form p/q. For example, 2/5, 4/7, 5/9, 4/11, etc. are all fractions and they are rational numbers as well.

Are All Whole Numbers Rational Numbers?

Yes. All whole numbers are rational numbers. Because whole numbers are 0, 1, 2, 3, 4, 5, 6, …… . They all can be written in the form p/q by taking 1 as a denominator.

Are Decimals Rational Numbers?

Yes. All terminating decimals and non-terminating repeating decimals are rational numbers. For example, all decimals such as 2.35, 5.125, 7.8463, 9.3755, 16.365, etc. can be written in the form of p/q. Therefore, they are rational numbers. But the non-terminating non-repeating decimals such as 2.37819476841…., 5.385629419462…., etc. are not rational numbers because they cannot be written in the form of p/q.

Are Rational Numbers Integers?

No. Rational numbers are not integers. Because rational numbers are of the form p/q such as 2/3, -5/8, -3/7, ¼, etc. and they are not integers.

Are Rational Numbers Real Numbers?

Real numbers are the collection of rational numbers and irrational numbers. So, all the rational numbers and irrational numbers are real numbers.

In the Venn diagram given above, you can see that real numbers are the super set of all types of numbers, so, all the numbers are real numbers.

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