Mathematics is one of the toughest subjects and has a vast syllabus in itself. For better understanding, knowing the fundamentals and solving the problems proficiently is what the students desire. Here, we will provide the necessary guidelines and help you to make you proficient in mathematics.
Chapter 1- Rational Numbers
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What is a Rational Number?
The word rational number is derived from the word “ratio.” The numbers are given as the quotient p/q of two integers such that q is not equal to zero. We can say that any fraction without a zero denominator is called a Rational number. It is considered to be a number that can be expressed as a fraction or quotient of two integers that can be either negative or positive. As per this principle, all the integers can be considered rational numbers. For example, 7 is an integer, but it also falls under the category of rational numbers because 7/1=7. Therefore, the number zero is also counted as a rational number because it can be written as 0/1, 0/2, etc. However, it can’t be considered in the denominator as it cannot be solved further.
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Types of Rational Numbers
Rational numbers are divided into two types which are standard form rational numbers and positive/negative rational numbers.
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- Standard form rational number- The standard form of a rational number is said to be when there are only two factors (Divisor and Dividend) and does not have any other common factors.
- Positive rational number- Fractions having both positive and negative numbers are called positive rational numbers.
- Negative rational number- One of the numbers in the fraction is negative, then it is said to be a negative rational number.
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Difference between Negative and Positive rational numbers
| Negative rational number | Positive rational number |
| In this type of rational number, both the denominator and numerator have opposite signs. | In this type of rational number, both denominator and numerator have the same sign. |
| These types of rational numbers are constantly less than zero. | These rational type numbers are constantly greater than zero. |
- Properties of Rational Numbers
- Closure property- The results of addition, subtraction and multiplication operations on two rational numbers (say p and q) will be a rational number. It is not applicable for the division because division with 0 is not defined.
- For eg: 4/3- 2/4 = 6/12= ½.
- Commutative property- The addition and multiplication are always commutative. It is not applicable for subtraction operations.
- The commutative law of addition is: a+b=b+a
- For eg: 2/3 + 4/3 = 6/3
- The commutative law of multiplication is: a×b=b×a
- For eg: 4/5×3/4=3/4×4/5 = 12/20 = 3/4
- Subtraction cannot be considered as commutative property as a-b ≠ b-a
- Distributive property- This property states that expressions of three rational numbers can be given as A (B+C) = (A×B) + (A×C) form.
- For eg: 1/2 (1/6 + 1/5) = -1/60
- Associative property- This property states that when the addition or multiplication of any three rational numbers, then the results remain the same. This property is applicable only during addition or multiplication calculations.
- Associative property law for addition is: (A+B) + C= A+ (B+C)
- Associative property law for multiplication is: A(BC)=(AB)C
- Multiplicative Inverse of rational numbers-
- Rational numbers are always clearly represented as a fraction, i.e., p/q.
- The reciprocal of the given fraction of rational numbers is known as the multiplicative inverse.
- Thus, p/q is converted to q/p in the multiplicative inverse.
- This concept is comparably easy to others as it does not complicate and does not contain confusing formulas.
- Additive inverse of rational numbers-
- The additive inverse of a rational number is the change in the sign of the same number, i.e., a positive sign number is converted to a negative sign number.
- The sum of the rational number and its inverse must be equal to zero for a rational number to be an additive inverse.
- Identity property- This property gives insights into the role of zero and one in the integers. Zero is considered to be the additive identity for rational numbers, while one is considered as the multiplicative identity of the rational number.
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Chapter No.1- Rational Numbers
Class 8 Chapter 1 contains Rational numbers with 8 exercises in it. The students have a misconception that rational numbers are a moderate topic, but you need to practice a lot for scoring well in this chapter. So, you have to learn the concepts and learn the formulas of this chapter.
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Study material
You can refer to RS Aggarwal Solutions Class 8 Rational Numbers for a better understanding of the concepts and problems. Now the question may arise, why RS Aggarwal Solutions? Some of the perks of using these books are:
- Even a weak student in the mathematics subject, you can use RS Aggarwal Class 8 Solutions. This will make you understand difficult problems in a simple manner.
- It provides step-by-step explanations of the problems, which helps the students to know the process of solving the problem.
- The solutions are prepared by the experts, which makes sure that it is accurate and there will be no errors.
- The basics and core of the topics are explained in a well-versed manner.
- In the end, they provide a set of questions that gives extra solving experience to the students.
- Available free on online sources and in books as well.
Summary-
Rational numbers are the integers or the numbers that are expressed in the form of a fraction, i.e., in the numerator and denominator form (p/q). In contrast, irrational numbers are numbers that can’t be expressed in the form of a fraction. Both the numerator and the denominator in the rational number are whole numbers, but the denominator cannot be equal to zero. There are two types of rational numbers, viz., positive rational numbers and negative rational numbers. Rational numbers also possess seven properties, as covered in the article.
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