+413 THIS IS FOR CLASS 8 USERS FOR CHAPTER 1 RATIONAL NUMBERS NOTES I WILL KEEP UPLOADING NOTES AS PER THE SYLLABUS
THANK YOU
RATIONAL NUMBERS
Natural Numbers:
Natural Numbers are counting numbers. We can represent Natural Numbers indefinitely to the right of 1 on the number line.
Whole Numbers:
Whole Numbers are Natural Numbers including zero. We can represent Whole Numbers indefinitely to the right of Zero on the number line.
Integers:
Integers are a collection of numbers consisting of all Natural Numbers, their negatives, and zero. We can represent Integers in definitely on both sides of Zero on the number line.
Rational Numbers:
Rational number is a number that is expressed in the form pq , where p and q are Integers and q ≠ 0.
In case of a Rational number, the denominator tells us the number of equal parts into which the first unit has been divided, while the numerator tells us ‘how many’ of these parts have been considered.
Lowest form of a Rational number
A rational number pq is said to be lowest form or simplest form if p and q have no common factor other than 1.
Equality of Rational numbers
If two rational numbers pq and rs are said to be equal then p x s = q x r (or) pq = rs.
We can also represent Rational Numbers in definitely on both sides of Zero on the number line.
Addition of Rational Numbers
Numbers with same denominators
Sum of two rational numbers have the same denominator, follow the following steps.
1) Obtain the numerators of two given rational numbers and their common denominator.
2) Add the numerator of two given rational numbers obtained in step 1.
3) Write a rational number whose numerator is the sum of two given rational numbers obtained in step 2 and whose denominator is the common denominator of the given rational numbers.
Numbers with distinct denominators
Sum of two rational numbers which do not have the same denominator, follow the following steps.
1) Obtain the rational nmbers and see whether their denominators are positive or not. If the denominator of one or both of the numbers is negative rewrite the denominators becomes positive.
2) Obtain the denominators of the rational numbers in step 1.
3) Find the LCM of denominators in step 2.
4) Express each one of the rational numbers in step 1 so that the LCM obtained in step 3 becomes their common denominator.
5) Write a rational number whose numerator is equal to the sum of the numerators of rational numbers obtained in step 4 and denominator as the LCM obtained in step 3.
6) The rational number obtained in step 5 is the required sum.
Subtraction of Rational Numbers
If aband cd are two rational numbers, then subtracting cd from ab means adding additive inverse (negative) of cd to ab.
The subtracting of cd from ab is written as ab - cd
Thus, we have ab- cd = ab + (- cd), [∴ Additive inverse of cd is - cd]
Multiplication of Rational Numbers
The product of two given fractions is a fraction whose numerator is the product of the numerators of the given fractions and whose denominator is the product of the denominators of the given fractions.
Product of two given fractions = Product of their numeratorsProduct of their denominators
Division of Rational Numbers
Division of fractions is the inverse of multiplication.
If m and n two rational numbers such that n ≠ 0, then the result of dividing m by n is the rational number obtained on multiplying m by the reciprocal of n.
When m is divided by n, we write m ÷ n. Thus m ÷ n = m × 1n.
Rational Numbers between two Rational Numbers
If m and n be two rational numbers such that m < n then 12 (m + n) is a rational number between m and n.
There are a finite number of Natural Numbers between any two Natural Numbers. Similarly there are a finite number of Whole numbers between any two Whole Numbers. But there are infinitely many Rational Numbers between any two Rational Numbers. The idea of mean helps us to find Rational Numbers between two Rational Numbers.
+33615 kes, you have written "Rational number is a number that is expressed in the form pq , where p and q are Integers and q ≠ 0."
This is incorrect. It should read: "Rational number is a number that is expressed in the form p/q , where p and q are Integers and q ≠ 0." Note the "p/q" not "pq".
This comment affects some of your other statements as well. I recommend that you edit your note to correct these.
.
+33615 kes, you have written "Rational number is a number that is expressed in the form pq , where p and q are Integers and q ≠ 0."
This is incorrect. It should read: "Rational number is a number that is expressed in the form p/q , where p and q are Integers and q ≠ 0." Note the "p/q" not "pq".
This comment affects some of your other statements as well. I recommend that you edit your note to correct these.
.
