Mathematics in thr Modern World, Schemes and Mind Maps of Mathematics

Download Mathematics in thr Modern World and more Schemes and Mind Maps Mathematics in PDF only on Docsity! LET REVIEW MATERIALS IN MATH  Counting numbers/ Natural numbers or positive integers: 1, 2, 3, and so on.  Integers: positive integers, negative integers and zero ( neither (+) nor (-) E.g. 5, -19, 0  Rational number: any number that can be represented by the division of one integer by another nonzero integer; are used to specify parts of a quantity E.g. 5, -19, 5/8, -11/3  Irrational numbers: cannot be written as the division of one integer by another. E.g. √2, ᴫ  Real Numbers: The integers, the rational numbers, and the irrational numbers, including all such numbers that are (+), (-), or zero.  Imaginary number: the square root of a negative number; not real numbers  Fraction - may contain any number or symbol representing a number in its numerator or in its denominator; Indicates the division of the numerator by the denominator ; may be a number that is rational, irrational, or imaginary. Examples:  Absolute value- the absolute value of a positive number is the number itself, and the absolute value of a negative number is the corresponding positive number. E.g. The absolute value of 6 is 6, and the absolute value -7of is 7. We write these as | 6 | = 6 and | -7| = 7.  SIGNS OF INEQUALITY : On the number line, if a first number is to the right of a second number, then the first number is said to be greater than (>) the second. If the number is to the left of the second, it is less than (<) the second  RECIPROCAL NOTE: Every number, except zero, has a reciprocal. The reciprocal of a number is 1 divided by the number.  DENOMINATE NUMBERS: Numbers that represent a measurement and are written with units of measurement Examples: To show that a certain HDTV set weighs 28 kilograms, we write the weight as 28 kg. To show that a giant redwood tree is 110 metres high, we write the height as 110 m. To show that the speed of a rocket is 1500 metres per second, we write the speed as 1500 m/s. To show that the area of a computer chip is 0.75 square centimetre, we write the area as 0.75 cm 2 . To show that the volume of water in a glass tube is 25 cubic centimetres, we write the volume as 25 cm 3.  LITERAL NUMBERS: It is usually more convenient to state definitions and operations on numbers in a general form. To do this, we represent the numbers by letters, called literal numbers. Variables- literal numbers that may vary in a given problem Constants- literal numbers that are held fixed ALGEBRA AND ARITHMETIC Arithmetic: only numbers and their arithmetical operations (such as +, −, ×, ÷) occur Algebra: also uses variables such as x and y, or a and b to replace numbers. The major difference between algebra and arithmetic is the inclusion of variables. Variable - a letter or symbol used in algebra to represent numbers;  allows the making of generalizations in mathematics  allows arithmetical equations (and inequalities) to be stated as laws (such as a + b = b + a for all a and b), and thus is the first step to the systematic study of the properties of the real number system.  allows reference to numbers which are not known  may represent a certain value which is not yet known, but which may be found through the formulation and manipulation of equations.  allows the exploration of mathematical relationships between quantities (e.g.: "if you sell x tickets, then your profit will be 3x − 10 dollars"). Expressions  (In elementary algebra), may contain numbers, variables and arithmetical operations.  conventionally written with 'higher-power' terms on the left  Examples: Properties of operations Operation Is Written commutative associative identity element inverse operation Addition a + b a + b = b + a (a + b) + c = a + (b+ c) 0, which preserves numbers: a + 0 = a Subtraction ( - ) Multiplication a × b or a•b a × b = b × a (a × b) × c = a × (b× c) 1, which preserves numbers: a × 1 = a Division ( / ) Exponentiation a b or a^b Not commutative a b ≠b a Not associative 1, which preserves numbers: a 1 =a Logarithm (Log) and nth root Properties of Numbers PROPERTY EXPLANATION EXAMPLE COMMUTATIVE The word "commutative" comes from "commute" or "move around" For addition: a + b = b + a For multiplication: ab = ba 2 + 3 = 3 + 2 2×3 = 3×2 ASSOCIATIVE The word "associative" comes from "associate" or "group"; the rule that refers to grouping. For addition: a + (b + c) = (a + b) + c For multiplication: a(bc) = (ab)c 2 + (3 + 4) = (2 + 3) + 4 2(3×4) = (2×3)4 DISTRIBUTIVE "multiplication distributes over addition; Either takes something through parentheses or else factors something out. a(b + c) = ab + ac 2(3 + 4) = 2×3 + 2×4 2(x + y) = 2x + 2y IDENTITY For addition: the sum of a number and 0 a+ 0= a Note: Never divide by zero. Division by zero is the only undefined basic operation. All the other operations with zero may be performed as for any other number.  RATIO AND PROPORTION Ratio: The quotient is a/b is also called the ratio a of to b. Proportion: An equation stating that two ratios are equal; If the ratio of to 8 equals the ratio of 3 to 4, we have the proportion: Since a proportion is an equation, if one of the numbers is unknown, we can solve for its value as with any equation. Usually, this is done by noting the denominators and multiplying each side by a number that will clear the fractions. WORD PROBLEMS Age  1. Father is aged three times more than his son Ronit. After 8 years, he would be two and a half times of Ronit's age. After further 8 years, how many times would he be of Ronit's age? A. 2 times B. 2 1 times 2 C. 2 3 times 4 D. 3 times Let Ronit's present age be x years. Then, father's present age =(x + 3x) years = 4x years. (4x + 8) = 5 (x + 8) 2 8x + 16 = 5x + 40 3x = 24 x = 8. Hence, required ratio = (4x + 16) = 48 = 2. (x + 16) 24 2. A father said to his son, "I was as old as you are now when you were born". If the father's age is 38 years now, the son's age five years back was: A. 14 years B. 19 years C. 33 years D. 38 years Answer: Option A Let the son's present age be x years. Then, (38 - x) = x 2x = 38. x = 19. Son's age 5 years back (19 - 5) = 14 years. 3. The sum of the present ages of a father and his son is 60 years. Six years ago, father's age was five times the age of the son. After 6 years, son's age will be: A. 12 years B. 14 years C. 18 years D. 20 years Answer: Option D Let the present ages of son and father be x and (60 -x) years respectively. Then, (60 - x) - 6 = 5(x - 6) 54 - x = 5x - 30 6x = 84 x = 14. Son's age after 6 years = (x+ 6) = 20 years.. Work A can do a work in 15 days and B in 20 days. If they work on it together for 4 days, then the fraction of the work that is left is : A. 1 4 B. 1 10 C. 7 15 D. 8 15 Answer:Option D A's 1 day's work = 1 ; 15 B's 1 day's work = 1 ; 20 (A + B)'s 1 day's work = 1 + 1 = 7 . 15 20 60 (A + B)'s 4 day's work = 7 x 4 = 7 . 60 15 Therefore, Remaining work = 1 - 7 = 8 . 15 15 NUMBERS Three times the first of three consecutive odd integers is 3 more than twice the third. The third integer is: A. 9 B. 11 C. 13 D. 15 Answer: Option D Explanation: Let the three integers be x, x + 2 and x + 4. Then, 3x = 2(x + 4) + 3 x = 11. Third integer = x + 4 = 15. Trigonometry-a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves. In this right triangle: sin A = a/c; cos A = b/c;tan A = a/b Trigonometric function Definition Sine function (sin) the ratio of the side opposite the angle to the hypotenuse. Cosine function (cos), the ratio of the adjacent leg to the hypotenuse. Tangent function (tan of the opposite leg to the adjacent leg. Cosecant Secant Cotangent Hypotenuse- the side opposite to the 90 degree angle in a right triangle; it is the longest side of the triangle, and one of the two sides adjacent to angle A. adjacent leg- the other side that is adjacent to angle A. opposite side- ithe side that is opposite to angle A. The reciprocals of these functions are named the cosecant (csc or cosec), secant (sec), and cotangent (cot), respectively: Calculate the length of the side x, given that tan θ = 0.4 Solution: ANALYTIC GEOMETRY In analytic geometry, the plane is given a coordinate system, by which every point has a pair of real number coordinates. The most common coordinate system to use is theC artesian coordinate system, where each point has an x-coordinate representing its horizontal position, and a y-coordinate representing its vertical position. These are typically written as an ordered pair (x, y). This system can also be used for three-dimensional geometry, where every point in Euclidean space is represented by anordered triple of coordinates (x, y, z)