class 9 maths chapter 1

ix Class Maths Concept ix Class Maths  6th Maths Concept  7th Maths Concept  8th Maths Concept 9th Maths Concept 10th Maths Concept TS IX CLASS MATHS CONCEPT  Studying maths in IX class successfully means that children take responsibility for their own learning and learn to apply the concepts to solve problems. This note is designed by the ‘Basics in Maths’ team. These notes to do help students fall in love with mathematics and overcome fear. 1.REAL NUMBERS Rational numbers:- The numbers which are written in the form of, where p, q are integers and q≠ 0 are called rational numbers. Rational numbers are denoted by Q. ex:-  3/2, 3/5, 2, 1 and so on Natural numbers, Whole numbers, and Integers are rational numbers. The rational numbers do not have a unique representation.   Representation of rational number:       Represent    To find a rational number between given numbers:  Mean method:- A rational number between two numbers a and b is   Ex:- insert two rational number between 1 and 2 To find a rational number in a single step:-  Ex:- insert two rational number between 1 and 2  To find two rational numbers, we 1 and 2 as rational numbers with same denominator 3     (∵ 1 + 2 = 3)   The decimal form of rational numbers: Note:- Every rational number can be expressed as a terminating decimal or non-terminating repeating decimal. Converting decimal form into a fraction: Terminating decimals:-  (i) 1.2 = 12/10 = 6/5                   (ii) 1.35 =135/100 = 135/100 = 27/20 Non-Terminating repeating decimals:- Irrational numbers: The numbers which are not written in the form of, where p, q are integers and q ≠ 0 are called rational numbers. Rational numbers are denoted by QI or S. Every irrational number can be expressed as a non-terminating decimal or non-repeating decimal. Ex:-    Calculation of square roots: There is a reference of irrationals in the calculation of square roots in Sulbha Sutra. Procedure to find   value: Representing irrational numbers on a number line:           Ex:- Locate    on a number line At ‘O’ draw a unit square OABC on a number line with each side 1 unit in length. By Pythagoras theorem                          OB2 = OA2 + AB2 =  12 + 12 OB2 = 2 OB = Using a compass with centre O and radius OB, draw an arc on the right side to O intersecting the number line at the point The location of is now at k. Note:-  If a and b are two positive rational numbers such that ab is not a perfect square, this an irrational number lies between a and b.    Real numbers The collection of all rational and irrational numbers is called real numbers. Real numbers cover all the points on the number line. Every real number is represented by a unique point on the number line. Ex:-   are some examples of real numbers. Representing real numbers on the number line through successive magnifications:- locating 2. 775 on a number line Operation on real numbers The sum, difference, product and quotient of irrational numbers need not be an irrational number. Irrational numbers are not closed under addition, subtraction, multiplication, and division. For any two real numbers a and b Rationalizing the denominator: Rationalizing factor(R.F):-If the product of two irrational numbers is rational, then each of the two is the rationalizing factor to others. The rationalizing factor of a given irrational number is not unique. It is convenient to use the simplest of all R.F.s of given irrational number. Note:- Law of exponents for real numbers: 2. POLYNOMIALS AND FACTORIZATION Polynomial: An algebraic expression in which the variables involved have only whole number powers is called a polynomial. Ex: x2 , x3 + 1, x2 + xy + y2  and so on. Polynomials in one variable: The polynomials which are in the form of (a constant) × (some power of variable) are called polynomials in one variable.     Ex: 2x, 4x, 3×2 + 1 and so on. Degree of the polynomial: The degree of a term is the sum of the exponent of its variable factors. The degree of the polynomial is the highest power of its variable term. Ex:  degree of 3×2 + 2×3 + 1 is 3         degree of 5x2y3 + 2xy + 3×3 is 5 a polynomial in one variable x of degree n is anxn + an-1xn-1 + …+a1x + a0. Where a0, a1…an are constants and an ≠ 0. Types of polynomials:   According to no. of terms:  No. of non-zero terms Name of the polynomial Examples Terms 1 Monomial 3x 3x 2 Binomial -2 x + 7 -2x, 7 3 Trinomial 5×2 + 4x + 2 5×2, 4x, 2 More than 3 Multinomial 6×3 – 5×2 + 7x – 3 6×3, -5×2, x, -3  According to a degree:    Degree of the polynomial Name of the polynomial Example Not defined Zero polynomial 0 0 Constant polynomial -12, 4, 7 etc. 1 Linear polynomial 2x+3, x – 3 etc. 2 Quadratic polynomial 2×2 + 3x + 1, x2 – 4 etc. 3 Cubic polynomial 3×3 – 4×2 + 2x + 6 4 Bi quadratic polynomial 4×4 + 2×3 + 45×2 +9x + 7 Zero of the polynomial: Let p(x) be a polynomial, if p(x) = 0 then, x is the zero of the polynomial p(x). Ex: p(x) = 2x – 2 P(1) = 2(1) – 2 = 2 – 2 = 0 ∴ 1 is the zero of the polynomial. Zero of the linear polynomial in one variable: Linear polynomial Zero of the polynomial x+ a       – a x – a a ax + b -b/a ax – b b/a  Dividing polynomials: If p(x) is divided by g(x), then there exists quotient polynomial q(x) and remainder r(x) such that p(x) = q(x) × g(x) + r(x) this