Mathematics Introduction & Formulas

Number System Number System is a writing system for expressing numbers.

Real Number:- When both rational and irrational numbers combined, the combination is defined as real number. Real numbers can be positive and negative numbers.

Classification of Real Number Natural Numbers:- includes all the counting numbers like 1,2,3,4,…….. Whole Numbers:- all natural numbers including 0. Integers:- all natural numbers and negative of natural numbers including 0. e.g., -2,-1,0,1,2,… (note:- 0 is neither positive nor negative.) Rational Numbers:- numbers that can be written in the fraction form, p/q, where q can’t be 0. Irrational Numbers:- an irrational number can’t be expressed in the form of p/q, where q can’t be 0.

Some Important Formulae Sum of first n natural numbers ………+n = 1/2n(n+1) Sum of square of first n natural numbers ……+n 2 = 1/6n(n+1)(2n+1) Sum of cubes of first n natural numbers …..+n 3 = {1/2n(n+1)} 2 Sum of first n odd numbers = n 2 Sum of first even numbers = n(n+1) Sum of odd numbers from 1 to n = (n+1/2) 2 Sum of even numbers from 1 to n = n/2(n/2+1)

Multiplication ( sign concept ) Multiplication of integers is similar to multiplication of whole numbers except the sign of the product needs to be determined. If both numbers are positive, the product will be positive. If both numbers are negative, the product will be positive. If any number is negative, the product will be negative. In other words, if the sign are same the product will be positive, if they are different the product will be negative

Division based formulae Dividend = divisor * quotient + remainder Divisor = is the number to be divided with, also said denominator. Dividend = is the number to be divided, also said numerator. Quotient = is the result found after division. Example :- a divided by b, a/b here, a is dividend or numerator b is divisor or denominator Divisor = dividend - remainder/ quotient. Quotient = dividend - remainder/ divisor

Algebraic Formulae (a + b) 2 = a 2 + 2ab + b 2 (a - b) 2 = a 2 - 2ab + b 2 a 2 - b 2 = (a + b)(a - b) (a + b) 3 = a 3 + b 3 + 3ab(a + b) (a - b) 3 = a 3 - b 3 - 3ab(a - b) a 3 + b 3 = (a + b)(a 2 - ab + b 2 ) a 3 - b 3 = (a - b)(a 2 + ab + b 2 ) (a + b + c) 2 = a 2 + b 2 + c 2 + 2(ab + bc + ca) a 3 + b 3 + c 3 = (a + b + c)(a 2 + b 2 + c 2 – ab – bc - ca)+3abc (a + b) 2 + (a - b) 2 = 2(a 2 - b 2 ) (a + b) 2 – (a - b) 2 = 4ab

If a + b + c = 0, then a 3 + b 3 + c 3 = 3abc (x + a)(x + b) = x 2 + (a + b)x + ab (x + a)(x - b) = x 2 + (a - b)x – ab Laws of Exponent (a m )(a n ) = a m+n (ab) m = a m b m (a m ) n = a mn a 0 = 1 a m /a n = a m-n √a = a 1/2 n √a = a 1/n a -n = 1/a n

HCF & LCM HCF (Highest Common Factor):- the HCF of two or more numbers is the greatest number that divides each of them exactly. The common factor of 12 and 18 is 1,2,3,6. The largest common factor is 6, so this is the HCF of 12 and 18. LCM (Lowest Common Multiple):- the LCM is the smallest number that is a common multiple of two or more numbers. Product of two number = Product of their HCF & LCM. HCF and LCM of Fractions: HCF of Fractions = HCF of numerator ÷ LCM of denominator LCM of Fractions = LCM of numerator ÷ HCF of denominator