Rational Mathematics

Introduction

Rational numbers are among the most common types that we study in maths after integers. These numbers are in p/q, where p and q are integers and q is less than zero. Most individuals find it difficult to distinguish between fractions and rational numbers as the fundamental structure is the p/q form. Whole numbers make up fractions, whereas integers make up the numerator and denominator of rational numbers. Refer to rational study material to learn about the rational study material.

Types of Rational Numbers

We have a rational number if we can construct a fraction with both denominator and numerator as integers and a non-zero number.

It is of two types:

(i) Standard Form

(ii) Positive and Negative

Standard Form of Rational Numbers

If there are no common factors between the dividend and divisor other than one, the standard form of a rational number can be determined, and the divisor is positive.

12/36, for example, is a rational number.

However, it can be simplified to 1/3 because the divisor and dividend only have one common element. So, it wouldn’t be wrong to say that the rational number ⅓ is in its usual form.

Positive and Negative Rational Number

The rational number is written in p/q, where p and q are integers. In addition, q must be a non-zero integer. It doesn’t matter whether the rational number is positive or negative. Both p and q are positive integers if the rational number is positive. When a rational number is written as (p/q), either p or q has a negative value. It implies that

-(p/q) = (-p)/q = p/(-q).

Rational Numbers Properties

Because a real number can be a subset of an important number, it will obey all of the important numeration system’s properties. The following are some of the most essential qualities of real numbers:

If we multiply, add, or subtract two rational integers, the outcome will be a rational number.

1. If we divide or multiply the numerator and denominator with an equivalent factor, a rational number stays equivalent.

iii. If we multiply a real number by zero, we get an equivalent number.

1. Addition, subtraction, and multiplication close rational numbers.

Arithmetic Operations on Rational Numbers

Arithmetic operations are the most basic operations we perform on integers in mathematics. Let’s look at how various operations, such as a/b and p/q, can be performed on rational integers.

Addition: 

When we combine p/q and a/b, the denominator must be the same. As a result, we get (pb+qa)/qb.

Example: 1/4+5/4 = (1+5)/4 = 6/4 = 3/2

Subtraction: 

In the same way, if we subtract a/b and p/q, we must first make the denominator equal before subtracting.

Example: 5/4- 1/4 = (5-1)/4 = 4/4 = 1

Multiplication: 

When two rational numbers are multiplied, the numerator and denominator of the rational numbers are multiplied correspondingly. If a/b is multiplied by p/q, we get (p×a)/(q×b).

Example: 5/4 x 1/4 = 5/16

Division: 

When a/b is divided by p/q, the following is the result:

(a/b)÷(p/q) = aq/bp

Example: 5/4÷ 1/4 = (5×4)/(4×1) = 20/4 = 5

How do you find the rational numbers in the middle of two rational numbers?

Between two rational numbers, there exist “n” numbers of rational numbers. Two distinct strategies can find the rational numbers between two rational numbers. Let’s have a look at the two alternative approaches.

1st method:

Calculate the equivalent fractions of the given rational numbers and the rational numbers in between. Those figures should be the necessary sensible figures.

2nd method

Calculate the mean of the two rational numbers supplied. The needed rational number should be the mean value. Repeat the method with the old and newly obtained rational numbers to find more rational numbers.

Rational Numbers Tips and Tricks:

After going through the rational study material, here are some tricks to keep in mind:

• Rational numbers are all numbers that may be stated as fractions, not just fractions.
• Rational numbers include natural numbers, whole numbers, integers, fractions of integers, and terminating decimals.
• non-terminating decimals with recurring decimal patterns are rational numbers as well.
• A fraction is negative if it has a negative sign in the numerator, denominator, or in front of it. -a/b = a/-b, for example.

Conclusion

Because there are numerous quantities or measures that integers alone cannot effectively explain, rational numbers are required. The most common use of rational numbers is to measure quantities, whether they are length, mass, time, or something else. If a farmer produces and wishes to sell part of a bushel of wheat, or if a worker requires a pound of copper, rational numbers are required. In digital computers, rational numbers are used for all calculations.