Grade VIII. Mathematics Formula Notes. #GrowWithGreen

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Wesley Summers
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1 Grade VIII Mathematics Formula Notes #GrowWithGreen

2 Properties of rational numbers: Closure Property: - Rational numbers are closed under addition. - Rational numbers are closed under subtraction. - Rational numbers are closed under multiplication. - Rational numbers are not closed under division. Commutativity: - Rational numbers are commutative under addition. - Rational numbers are not commutative under subtraction. - Rational numbers are commutative under multiplication. - Rational numbers are not commutative under division. Associativity: - Rational numbers are associative under addition. - Rational numbers are not associative under subtraction. - Rational numbers are associative under multiplication. - Rational numbers are not associative under division. Rational numbers are distributive over addition and subtraction. i.e., for any rational numbers a, b and c, a ( b + c ) = ab + ac, a ( b c ) = ab ac 0 is the additive identity of whole numbers, integers, and rational numbers. 0 + a = a + 0 = a, where a is a rational number 1 is the multiplicative identity of whole numbers, integers, and rational numbers. a 1 = 1 a = a The additive inverse of the rational number is and vice versa. The reciprocal or multiplicative inverse of the rational number is if =1. b a b a b a c d b a c d Probability of an event = Mathematically,

3 If a natural number m can be expressed as n 2, where n is also a natural number, then m is a square number. First 100 Perfect Squares

4 Prime factorization of first 100 number

5 Perfect squares exhibit some special properties. The square of even numbers are even and square of odd numbers are odd. The unit place of a perfect square can never be 2, 3, 7 and 8. By observing the last digit of a number, we can find the last digit of the square of the number. If a number has 1 or 9 at its units place, then its square ends in 1. If a number ends with 4 or 6, then its square end with 6. If a number ends with 2 or 8, then its square ends with 4. If a number ends with 5, then its square ends with 5. If a number ends with 0, then its square also ends with 0. If a number ends with 3 or 7, then its square ends with 9. If a square number ends with 0, then the number of zeroes at the end is even. If a number ends with n number of zeroes, then its square ends with 2 n zeroes. If we add two consecutive triangular numbers, then we obtain a square number. The sum of first n odd natural numbers is n 2. Square of any odd number can be expressed as the sum of two consecutive positive integers. There are 2 n non-perfect square numbers between the squares of the numbers, n and ( n +1). We can find the squares of numbers having more than one digit by making use of the identity, ( a + b ) 2 = a ( a + b ) + b ( a + b ) The square of a number with units digit 5, say ( a 5) 2, can be written as follows. ( a 5) 2 = a ( a + 1) For any natural number m > 1, we have (2 m ) 2 + ( m 2 1) 2 = ( m 2 + 1) 2 Therefore, 2 m, m 2 1, and m forms a Pythagorean Triplet.

6 Pythagorean Triplets

Properties of cubes of numbers: Cubes of even numbers are even and the cubes of odd numbers are odd. If a number has 1 in its one s place, then its cube will also have 1 in its one s place.

7 Properties of cubes of numbers: Cubes of even numbers are even and the cubes of odd numbers are odd. If a number has 1 in its one s place, then its cube will also have 1 in its one s place. If a number has 2 in its one s place, then its cube will have 8 in its one s place. If a number has 3 in its one s place, then its cube will have 7 in its one s place. If a number has 4 in its one s place, then its cube will also have 4 in its one s place. If a number has 5 in its one s place, then its cube will also have 5 in its one s place. If a number has 6 in its one s place, then its cube will also have 6 in its one s place. If a number has 7 in its one s place, then its cube will have 3 in its one s place. If a number has 8 in its one s place, then its cube will have 2 in its one s place. If a number has 9 in its one s place, then its cube will also have 9 in its one s place. If a number has 0 in its one s place, then its cube will also have 0 in its one s place. First 20 Perfect Cubes Formula for percentage i ncrease and decrease are: C.P. = Buying price + Overhead expenses

8 Sales Tax (or VAT) = Tax % of bill amount The formulae to calculate profit and loss are: Profit % Loss % Discount = Marked Price Sale price Discount = Discount % of Marked Price If the successive discount %, d 1 %, d 2 %, d 3 % are given, then S.P. = Amount = Principal + Interest Amount (A) when interest is compounded annually is where, P = Principal, R = Rate of interest, n = Time period. Amount when interest is compounded half yearly is given by, Where, = Half-yearly rate and 2 n = Number of half years Amount when interest is compounded quarterly is given by, Where, = Quarterly rate and 4 n = Number of quarters Standard Identities: ( a b ) 2 = a 2 + b 2 2 ab ( a + b ) 2 = a 2 + b ab a 2 b 2 = (a b)(a + b) ( x + a ) ( x + b ) = x 2 + ( a + b ) x + ab For any non-zero integer, where m is a positive integer. a m is called the multiplicative inverse of a m and vice-versa. Laws of exponents (Here, a and b are non-zero integers and m and n are integers) a m a n m + n = a

9 If, where k is a positive number, then x and y vary directly. y x = k If y 1, y 2 are the values of y corresponding to the values x 1, x 2 of x respectively, then is a case of direct proportion. xy = k, where k is a positive number, then x and y vary inversely. Divisibility Rules

Numbers can be written in general form. Thus, a two digit number ab will be written as ab = 10 a + b. A three-digit number abc, a, b, and c, can be written in general form as abc = 100 a + 10 b + c.

10 Numbers can be written in general form. Thus, a two digit number ab will be written as ab = 10 a + b. A three-digit number abc, a, b, and c, can be written in general form as abc = 100 a + 10 b + c. The sum of a two-digit number and the number formed by reversing the digits is always a multiple of 11. The difference between a two-digit number and the number formed by reversing the digits is always a multiple of 9. The difference between a three-digit number and the number formed by reversing the digits is always a multiple of 99. If abc is a three-digit number, then ( abc + cab + bca ) will be a multiple of 3 and 37.

SYNA INTERNATIONAL SCHOOL LEARNING PAPERS CLASS 8 SUBJECT MATHEMATICS

CLASS 8 Page 1 SYNA INTERNATIONAL SCHOOL LEARNING PAPERS CLASS 8 SUBJECT MATHEMATICS 1 x 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 2 x 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 3