In the previous chapter, we learned about the number system, and in this chapter, we will learn about rational numbers.
Any number that can be expressed in the form of p/q, where p and q are integers and q 0, is a rational number.
Thus, rational numbers are a mathematical expression with numerator and denominator as integers, and where the denominator is not a zero. The division of any rational number will always be either a terminating or repeating decimal.
A whole number is always a rational number.
When p and q are integers, a rational number is in the form of p/q and q has to be a non-zero number. Rational numbers can be both positive and negative.
The point that differentiates both positive and negative rational numbers is that, in positive rational numbers, both numerators and denominators are of the same sign. For example, 8/9, 5/7,2/6, etc.
In negative rational numbers, the numerator and the denominator are of opposite signs. For example, -6/12, 33/-100, 45/-125, etc.
Just like whole numbers and integers, rational numbers can also be represented on a number line. The number zero is the origin of any number line, and all positive numbers are represented on the right-hand side of zero, and all negative numbers are represented in the left-hand side.
In terms of rational numbers represented on a number line, there are two types of fractions— proper fraction and improper fraction.
In a proper fraction, the value of the numerator is always less as compared to the denominator. Thus the rational number can be easily represented on a number line, as in proper fraction its value is less than 1 and more than 0.
In an improper fraction, the denominator’s value is less compared to the numerator, so the value of a given rational number is greater than 1. This calls for converting the improper fraction into a mixed fraction, and this process will help you locate the position of rational numbers on the number line.
When the numerator and denominator have no common factor other than 1 and the denominator of the rational number is a positive integer, the rational number is said to be in standard form.
We compare two negative rational numbers by ignoring their negative signs and reversing its order. Similarly, when comparing negative and positive rational numbers, a positive rational number will always be more than the negative rational numbers.
In the multiplication of any rational number, the multiplicative inverse and the reciprocal of any given fraction is the same because a rational number is denoted by p/q.
Here is an example of multiplication of a rational number
7/8 is a rational number, and the multiplicative inverse of it is 8/7, such that (7/8)x(8/7) = 1.
If a and b are two rational numbers divided, then the result of the division of these two rational numbers where b0 can be obtained by multiplication of a to reciprocal of b.
Eg. a/b is to be divided by z/q, then:
(a/b) (z/q) = aq/bz
For the addition of any two rational numbers (i.e. fraction), the denominators should be the same. For example, if a/b is to be added to z/q, then a common denominator has to be derived, such as:
(a/b) + (z/q) = (a/q+bz)/bq
For subtraction of a rational number too, the denominators have to be the same.
(a/b) – (z/q) = (a/q-bz)/bq.
Thus we now know that rational numbers are sets of fractions, integers and decimal numbers represented in the form of a/b, where a and b are integers and b 0. The set of rational numbers will always be smaller compared to irrational numbers.