TS State VI - IX | Basics In Maths

TS VII CLASS MATHS CONCEPT FEATURE IMAGE

TS 7th Class Maths Concept

Education, TS State VI - IX / Nayini Satyanarayana Reddy

7th Class Maths Concept 7th Class Maths TS 6th Maths Concept TS 7th Maths Concept TS 8th Maths Concept TS 9th Maths Concept TS 10th Maths Concept Studying maths in VII class successfully means that children take responsibility for their learning and learn to apply the concepts to solve problems. 7th Class Maths concepts were designed by the ‘Basic in Maths’ team. These notes to do help students fall in love with mathematics and overcome fear. 1. INTEGERS Natural numbers: All the counting numbers starting from 1 are called Natural numbers. 1, 2, 3… Etc. Whole numbers: Whole numbers are the collection of natural numbers including zero. 0, 1, 2, 3 … Integers: integers are the collection of whole numbers and negative numbers. ….,-3, -2, -1, 0, 1, 2, 3,….. Integers on a number line: Operations on integers: addition of integers: 3 + 4 = 7 -2 + 4 = 2 Subtraction of integers on a number line:- 6 – 3 = 3 Multiplication of integers on a number line:- 2 × 3 ( 2 times of 3) = 6 3 × (- 4 ) ( 3 times of -4) = -12 Multiplication of two negative integers: To multiply two negative integers, first, we multiply them as whole numbers and put plus sign before the result. The multiplication of two negative integers is always negative. Ex:- -3 × -2 = 6, -10 × -2 = 20 and so on. Multiplication of more than two negative integers: • If we multiply three negative integers, then the result will be a negative integer. Ex:- -3 × -4 × -5 = -60, -1× -7 × -4 = -28 and so on. • If we multiply four negative integers, then the result will be a positive integer. Ex:- -3 × -4 × -5 × -2 = 120, -1× -7 × -4 × -2 = 56 and so on. Note:- 1. If the no. of negative integers is even, then the result will be positive. 2. If the no. of negative integers is odd, then the result will be negative. Division of integers: The division is the inverse of multiplication. When we divide a negative integer by a positive integer or a positive integer by a negative integer, we divide them as whole numbers then put negative signs for the quotient. Ex:- -3 ÷ 1 = 3, 4 ÷ -2 = -2 and so on. • When we divide a negative integer by a negative integer, we get a positive number as the quotient. Ex:- -3 ÷ -1 = 3, -4 ÷ -2 = 2 and so on. Properties of integers: 1.Closure property:- 2.commutative property:- 3.associative property:- Additive identity:- 1 + 0 = 0 + 1 = 1, 10 + 0 = 0 + 10 = 10 •For any integer ‘a’, a + 0 = 0 + a •0 is the additive identity. Additive inverse:- 2 + (-2) = (-2) + 2 = 0, 5 + (-5) = (-5) + 5 = 0 •For any integer ‘a’, a+ (-a) = (-a) + a = 0 •Additive inverse of a = -a and additive inverse of (-a) = a Multiplicative identity:- 2 × 1 = 1 × 2 = 2, 5 × 1 = 1 × 5 = 5 •For any integer ‘a’, a × 1 = 1 × a = a •1 is the multiplicative identity. multiplicative inverse:- For any integer ‘a’, 1/a × a = a × 1/a = 1 multiplicative inverse of a = 1/a Multiplicative inverse of 1/a = a. distributive property:- For any three integers a, b and c, a × (b + c) = (a × b) + (a × c). 3 × (2 + 4) = 18 (3 × 2) + (3 × 4) = 6 + 12 = 18 ∴ 3 × (2 + 4) = (3 × 2) + (3 × 4). 2. FRACTIONS, DECIMALS AND RATIONAL NUMBERS Fraction: A fraction is a number that represents a part of the whole. A group of objects is divided into equal parts, then each part is called a fraction. The proper and improper fractions: In a proper fraction, the numerator is less than the denominator. Ex: – 1/5, 2/3, and so on. In an improper fraction, the numerator is greater than the denominator. Ex: – 5/2,11/5 and so on. Comparing fractions: Like fractions: – We have to compare the like fractions with the numerator only because the like fractions have the same denominator. The fraction with the greater numerator is greater and the fraction with the smaller numerator is smaller. Ex: , and so on Unlike fractions: – With the same numerator: For comparing unlike fractions, we have to compare denominators when the numerator is the same. The fraction with a greater denominator is smaller and the fraction with a smaller denominator is smaller. Ex: – and so on. Note: – To find the equivalent fractions of both the fractions with the same denominator, we have to take the LCM of their denominators. Addition of fractions: ∗ Like Fractions: ∗ Unlike fractions: Subtraction of fractions: ∗ Like fractions: Ex: Unlike fractions: – First, we have to find the equivalent fraction of given fractions and then subtract them as like fractions Ex: Multiplication of fractions: Multiplication of fraction by a whole number: – Multiplication of numbers means adding repeatedly. Ex: – • To multiply a whole number with a proper or improper fraction, we multiply the whole number with the numerator of the fraction, keeping the denominator the same. 2.Multiplication of fraction with a fraction: – multiplication of two fractions = Division of fractions: Ex: – 2 ÷ ⇒ 6 one-thirds in two wholes Reciprocal of fraction: reciprocal of a fraction is . Note: dividing by a fraction is equal to multiplying the number by its reciprocal. For dividing a

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6th maths notes|| TS 6 th class Maths Concept

Education, TS State VI - IX / Nayini Satyanarayana Reddy

6th maths notes|| TS 6 th class Maths Concept 6th Maths TS 6th Maths Concept TS 7th Maths Concept TS 8th Maths Concept TS 9th Maths Concept TS 10thMaths Concept Studying maths in the 6th class successfully meaning 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. KNOWING OUR NUMBERS • Number: A number is a mathematical object used to count and measure.1,2,3…….etc. Comparing numbers: • We can compare the numbers by counting the digits in the numbers. • Now Compare 5432 and 4678… 5432 is greater as the digits at the ten thousand place in 5432 is greater than that in 4678. Order of numbers: • Ascending Order: – arrange the numbers from smallest to the greatest; this order is called Ascending order. Ex:- 23, 44, 65, 79, 100 • Descending Order: – arrange the numbers from greatest to the smallest, this order is called Ascending order. Ex:- 100,79, 65, 33, 23 Formations of numbers • Form the largest and smallest possible numbers using the digits 3, 2, 4, 1 without repetition • Largest number formed by arranging the given digits in descending order _ 4321. • Smallest number formed by arranging the given digits in ascending order _ 1234. • Greatest two-digit number is 99. • Greatest three-digit number is 999. • Greatest four-digit number is 9999. Place value • Place value is the positional notation, which defines the position of a digit. Ex:- 3458 8 is one place, 5 is tens place, 4 is hundreds place and 3 is thousands place. Expanded form • It refers to expand the numbers to see the value of each digit. Ex :- 3458 = 3000 + 400 + 50 + 8 = 3×1000 + 4×100 + 5×10 + 8×1 • Note:- 1 hundred = 10 tens 1 thousand = 10 hundreds 1 lakh = 100 thousands = 1000 hundreds 6th Maths Reading and Writing the numbers Place value table for Indian system : Example: Represents the number in 6,35,21,892 in place value table Place value table for International system : Ex:- represents the number in 635,218,924 in place value table Use of commas: • Indian system of numeration:- in the Indian system of numeration we use ones, tens, hundreds, thousands, lakhs and crores. The first comma comes after three digits from the right, the second comma comes two digits latter and the third comma comes after another two digits.E Ex:- “three crores thirty-five lakh seventeen thousand four hundred thirty” can be written as.3,35,17,430 • International system of numeration:- in the International system of numeration we use ones, tens, hundreds, thousands, millions and billions. Ex:- “ six hundred thirty-five million two hundred eighteen thousand nine hundred twenty-four” can be written as 635,218,924. Note:-10 millimetres = 1centimeter 100 centimetres = 1 meter 1000 meters = 1 kilometer 1000 milligrams = 1 gram 1000 grams = 1 kilo gram 2. WHOLE NUMBERS Natural numbers: All the counting numbers starting from 1 are called Natural numbers. 1, 2, 3… Etc. Successor and Predecessor: If we add 1 to any natural number, we get the next number, which is called the Successor. If we subtract 1 from any natural number, we get the previous number, which is called Predecessor. Ex: – successor of 23 is 24 and predecessor of 32 is 31. Note:- There is no predecessor of 1 in natural numbers. Whole numbers: Whole numbers are the collection of natural numbers. 0, 1, 2, 3 … Representation of whole number on the number line: • Draw a line mark a point on it. • Label it as ‘0’ • Mark as many points at equal distance to the right of 0. • Label the points as 1, 2, 3, 4, … respectively. • The distance between any two consecutive points is the unit distance. Addition on the number line: The distance between 2 and 4 is 2 units, like as the distance between 2 and 6 is 4 units The number on the write is always greater than the number on the left The number on the left of any number is always smaller than that number Addition of the whole number can represent on the number line Ex:- 3 + 2 = 5 Start from three, we add 3 to 2. We make two jumps to the right of the number line as shown above. We reach at 5. Subtraction on the number line: Subtraction of the whole number can be represented on the number line Ex :-5 – 3 = 2 Start from 5, we subtract 3 from 5. We make three jumps to the left of the number line shown as above. We reach at 2. Multiplication on the number line: For multiplying 2 and 3, start from 0, make 2 jumps using 3 units at a time to the right, as you reach to 6. Thus, 2 × 3 =6. Properties of whole numbers Closer property: Two whole numbers are said to be closed if their operation (+, -, ×,÷) is always closed. Addition:-Whole numbers are closed under addition. Ex: 3, 2 are whole numbers ⟹ 3 + 2 = 5 ( 5 is whole number) Subtraction:- Whole numbers are not closed under subtraction as their difference not always a whole number. Ex:- 2 – 3 = −1 ( −1 is not a whole number) Multiplication:-

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TS 8th Class Maths Concept

Education, TS State VI - IX / Nayini Satyanarayana Reddy

viii Class Maths || TS 8th Class Maths Concept viii Class Maths TS 6th Maths Concept TS 7th Maths Concept TS 8th Maths Concept TS 9th Maths Concept TS 10th Maths Concept viii Class Maths Studying maths in VIII class successfully meaning that children take responsibility for their own learning and learn to apply the concepts to solve problems. This notes is designed by the ‘Basics in Maths team’. These notes to do help students fall in love with mathematics and overcome fear. 1. RATIONAL NUMBERS • Natural numbers: All the counting numbers starting from 1 are called Natural numbers. 1, 2, 3… Etc. • Whole numbers: Whole numbers are the collection of natural numbers. 0, 1, 2, 3 … • Integers: integers are the collection of whole numbers and negative numbers. ….., -3, -2, -1, 0, 1, 2, 3…. • Rational numbers: The numbers which are written in the form of p/q, where p, q are integers and q ≠ 0 are called rational numbers. Rational numbers are denoted by Q. Properties of Rational numbers • Natural numbers: 1.Closure property:- 2.Commutative property:- 3. Associative property:- • Whole numbers: 1. Closure property:- 2.Commutative property:- 3. Associative property:- • Integers: 1. Closure property:- 2.Commutative property:- 3. Associative property:- • Rational numbers: 1. Closure property:- 2.Commutative property:- 3. Associative property:- Additive identity:- 1 + 0 = 0 + 1 = 1, 3/2 + 0 = 0 + 3/2 = 3/2 • For any rational number ‘a’, a + 0 = 0 + a • 0 is the additive identity. Additive inverse:- 2 + (-2) = (-2) + 2 = 0, 5 + (-5) = (-5) + 5 = 0 • For any rational number ‘a’, a+ (-a) = (-a) + a = 0 • Additive inverse of a = -a and additive inverse of (-a) = a Multiplicative identity:- 2 × 1 = 1 × 2 = 2, 6 ×1/6 = 1 × 1/6 = 1/6 • For any rational number ‘a’, a × 1 = 1 × a = a • 1 is the multiplicative identity. Multiplicative inverse:- 2 × 1/2 = 1/2 × 2 =1 For any rational number ‘a’, a × 1/a = 1/a × a = 1 • multiplicative inverse of a =1/a • Multiplicative inverse of 1/a= a. Distributive property:- For any three rational numbers a, b and c, a × (b + c) = (a × b) + (a × c) 3/2×(5/3+1/5)=(3/2×5/3)+(3/2×1/5) Representing rational numbers on a number line: Ex: represent 29/6 on a number line this lies between 4 and 5 Divide the number line between 4 and 5 into 6 equal parts. Mark 5th part counting from 4. The role of zero: • If 0 is added to any rational number, then the rational number remains the same. For any rational number ‘a’ a + 0 = a = 0 + a • 0 is the additive identity. • Natural numbers do not have an additive identity. Additive inverse:- for any rational number ‘a’ a + (-a) = 0 = (-a) + a 3 + (-3) = 0, 10 + (-10) = 0 • additive inverse of ‘a’ is ‘-a’ and additive inverse of ‘-a’ is ‘a’ The role of 1: • If 1 is multiplied to any rational number, then the rational number remains same. For any rational number ‘a’ a × 1 = a = 1 × a 1 is the multiplicative identity. Multiplicative inverse:- 3 × 1/3 = 1 = 1/3 × 3 for any natural number ‘a’ a × 1/a = 1 = 1/a ×a • Multiplicative inverse of ‘a’ is ‘1/a’ and multiplicative inverse of ‘1/a’ is ‘a’ Distributive property: For any 3 rational numbers a, b and c, a (b + c) = ab + ac Ex:- 1/3 (2/5 + 1/5) = 1/3(3/5) = 3/15 1/3× 2/5 + 1/3 × 1/5 = 2/5 + 1/5 = 3/5 Inserting rational numbers between given two numbers: • There are infinitely many rational numbers between given two numbers. • We have two methods to find rational numbers between two numbers. First method: – First we have to convert given rational numbers as the same denominator and write the rational numbers which come between given numbers. Second method: – if a and b any given rational numbers then a/bis a rational number between a and b. The decimal representation of rational numbers The decimal expansion of rational is either terminating or non-terminating repeating decimal. Note:-Decimal numbers with the finite no. of digits is called terminating Decimal numbers with the infinite no. of digits is called non-terminating decimal. In a decimal, a digit or a sequence of digits in the decimal part keeps repeating itself infinitely. Such decimals are called non-terminating repeating decimals. Terminating decimals: Consider a rational number = 0.75 Non-terminating decimals: Consider a rational number 2-3 2/3 = 0.66666… 2/3 = 2. LINEAR EQUATIONS IN ONE VARIABLE Equations: An algebraic equation is the equality of algebraic expressions involving variables and constants. It has an equality sign. The expression on the left of the equality sign is called the LHS (Left Hand Side) and right of the equality is called RHS (Right Hand Side) of the equation. In an equation, the value of RHS and LHS are equal. This happens to be true only for certain values of the variable. This value is called the solution of the equation. Linear equations: If the degree of the equation is 1, then it is called a linear equation. Ex: 2x – 3 = 5, x = 3y, 5x + 3y = 3 and so

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TS 9th Class Math Concept

Education, TS State VI - IX / Nayini Satyanarayana Reddy

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

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TS State VI – IX

TS 7th Class Maths Concept

6th maths notes|| TS 6 th class Maths Concept

TS 8th Class Maths Concept

TS 9th Class Math Concept