NCERT Solutions for Class 8 Maths Chapter 1
At its heart, mathematics is a language for quantifying the world, and rational numbers are a fundamental part of that vocabulary. They are the numbers we often encounter in daily life—when dividing a pizza, measuring ingredients, or calculating a discount. Formally, a rational number is defined as any number that can be written in the form p/q, where p and q are integers, and q is not zero. The condition that q cannot be zero is crucial, as division by zero is undefined. This simple definition of a fraction opens the door to a rich and intricate numerical system.
The Many Faces of a Rational Number
One of the most important ideas to grasp is that a single rational value can wear many different disguises. Consider the value one-half. It can be represented not only as 1/2 but also as 2/4, 3/6, 50/100, or -1/-2. These are called equivalent rational numbers. They look different on paper but hold the same value on the number line. This concept is the very foundation of simplifying and manipulating fractions.
This leads us to the standard form of a rational number. A rational number is in its standard form when two conditions are met: first, the numerator and denominator are co-prime, meaning they share no common factors other than 1 (their greatest common divisor is 1); and second, the denominator is a positive integer. For example, to express -4/6 in standard form, we first simplify it by dividing both numerator and denominator by 2, giving us -2/3. Since the denominator is already positive, this is the standard form. The standard form provides a unique, canonical representation for every rational number, eliminating ambiguity and making comparison easier.
Comparing and Ordering Rationals
How do we determine if 3/5 is greater than 4/7? The most reliable method is to convert them into equivalent fractions with a common denominator. The denominators here are 5 and 7, so the least common denominator is 35. We convert 3/5 to 21/35 and 4/7 to 20/35. It now becomes clear that 21/35 (or 3/5) is larger than 20/35 (or 4/7). This technique allows us to arrange any set of rational numbers in ascending or descending order, just as we would with whole numbers.
The Arithmetic of Fractions
The operations of addition, subtraction, multiplication, and division follow specific, consistent rules for rational numbers.
- Addition and Subtraction: These require a common denominator. Once the fractions are expressed with the same denominator, we simply add or subtract the numerators while keeping the denominator unchanged. For instance, 1/4 + 1/6 requires a common denominator of 12, becoming 3/12 + 2/12 = 5/12.
- Division: To divide by a rational number, we multiply by its reciprocal (the fraction flipped). So, (3/4) ÷ (2/5) is the same as (3/4) * (5/2) = 15/8.
Fundamental Properties
Rational numbers adhere to the fundamental properties that govern the integers, making them a well-behaved number system.
- Commutative Property: The order of numbers does not change the result in addition and multiplication (e.g., a + b = b + a).
- Associative Property: The grouping of numbers does not change the result in addition and multiplication (e.g., (a + b) + c = a + (b + c)).
- Distributive Property: Multiplication distributes over addition (e.g., a × (b + c) = a×b + a×c).
These properties ensure that the algebraic techniques we learn for integers can be confidently applied to rational numbers.
Visualizing on a Number Line and the Concept of Density
Every rational number has a unique, precise location on the number line. To plot a fraction like 3/2, we find the point 1.5, exactly midway between 1 and 2. For a negative number like -5/4, we find -1.25, which is one-quarter of the way from -1 to -2.
Perhaps the most mind-bending property of rational numbers is their density. This means that between any two distinct rational numbers, no matter how close they are, you can always find another rational number. For example, between 0.1 and 0.2, we have 0.15. But between 0.1 and 0.15, we have 0.125, and so on, ad infinitum. This process never ends. This incredible property implies that rational numbers are packed infinitely closely together, forming a dense set on the number line.
In essence, rational numbers represent a monumental leap from the world of whole integers. They allow us to precisely express parts of a whole, enabling accurate measurement, fair division, and sophisticated calculation. They form a robust and dense system that underpins much of arithmetic and algebra, providing an indispensable tool for problem-solving and a gateway to understanding even more complex number systems, like the irrational and real numbers.
class 8 maths chapter 1
Exercise 11.1
Name the property under multiplication used in each of the following:
Ans :
i) -4/5 * 1 = 1 * -4/5 = -4/5
Commutative property of multiplication.
ii) -13/17 * -2/7 = -2/7 * -13/17
Commutative property of multiplication.
iii) -19/29 x 29/-19 = 1
Multiplicative inverse
2. Tell what property allows you to compute
Ans : The associative property of multiplication is being used.
3. The product of two rational numbers is always a …………
Ans : Rational number.
