Grade VIII Mathematics Formula Notes #GrowWithGreen
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,
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
Prime factorization of first 100 number
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) 100 + 25 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 2 + 1 forms a Pythagorean Triplet.
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. 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
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 2 + 2 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
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. 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.