Dr Prya Mathew SJCE Mysore

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Gillian Ellis
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4 The word Mathematics derived from two Greek words Manthanein means learning Techne means an art or technique So Mathematics means the art of learning related to disciplines or faculties disciplines = mental training 4

5 Mathematics reveals hidden patterns that helps us to understand the world around us. Mathematics is a study of patterns and order dealing with numbers, geometrical objects, forms, algorithms, chance and change. Mathematics is either the science of number and space or the science of measurement, quantity and magnitude - Dictionary meaning It is a systematised, organised and exact branch of science. 5

6 It is a diverse discipline and It deals with measurements, data analysis, observations from various fields of knowledge, inductive generalizations, proofs, logical deductions, etc Mathematics relies on logical reasoning rather than observation as its measure of validity of truth; Yet it employs observation, simulation and even experimentation as means of discovering truth. 6

7 In addition to theorems and theories mathematics offers distinctive models of thought which are versatile and powerful, including abstraction, optimisation, generalisation, logical analysis, inference and use of symbols. 7

8 Mathematics is the queen of sciences and arithmetic is the queen of all mathematics - Gauss Mathematics is a science of order and measure Descartes Mathematics is the indispensible instrument of all physical resources- Kant Mathematics is the gate way and key to all sciences Bacon 8

9 Mathematics is the study of quantity Aristotle Mathematics is a subject identical with logic Bertrand Russel Mathematics is a way to settle in the mind a habit of reasoning- Locke 9

10 What is the origin of the mathematical concepts which is used in the school curriculum?????? Arithmetic : need for counting Basic algebra : need for simple operations in daily life Geometry & Trigonometry : human quest for possession of land other properties needed, curiosity & quest for the understanding of the universe & Happenings around us. Calculus : understaning the processes in physics New branches of Maths: problems faced by scientists, social scientists, commerce and trade organisations 10

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13 Some terms cannot be defined precisely in mathematics The members of a minimal set of mathematical terms whose meaning we take for granted without explaining in terms of other mathematical terms, and give meaning of other mathematical terms in terms of these mathematical terms are known as undefined terms. Start with some terms which are taken as undefined The choice of undefined terms is completely arbitrary and generally to facilitate the development of the structure. Example: Point, line, plane, number, natural number, etc. They are only abstract ideas. Eg: Define rational numbers in terms of integers in terms of natural numbers what about natural numbers?????????? 13

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15 of a term is a characteristics explanation involving terms which are either undefined or terms which have been already defined. Characteristics explanation of a term is an explanation which characterizes the given term i.e the explanation which is true for and only for the given term. Definition 15

16 In case of triangle, if A, B, and C are three noncollinear points, then triangles ABC is the union of line segments AB, BC, and AC. here we used the terms collinear, point, union, line segments An equilateral triangle is a triangle in which all sides are equal. Common divisor of two integers is a number which divides both the given integers. 16

17 A postulate or axiom is an accepted statement of fact. They are the self- evident truth Basic building blocks of the logical system in mathematics Euclid is the father of geometry Euclidean geometry is the work of Euclid which is deduced from a small number of Axioms and Geometrical Postulates 17

18 The general statements which are accepted without question and which are applicable to all branches of science Accepted as true because of their conformity with common experience and sound judgment. Example: (Common Notions -5 axioms) Things which are equal to the same thing are equal to one another. If equals are added to equals, the wholes are equal If equals are subtracted from equals, the n the remainders are equal The things which coincide with one another must be equal to one another. The whole is greater than its part. 18

19 Axioms are propositions which are assumed and accepted without any evidence to prove it. Axioms are consistent in the sense that by using other axioms and applying rules of logic, we should not be able to arrive at anything contrary to any of the axioms Axioms are not superfluous: we should not be able to prove any axioms using other axioms. Axioms are adequate: it should be possible to prove any known results of the Mathematical theory with the help of the set of axioms. 19

20 The statements which are particular to Geometry and accepted without questions are the statements which specify relationship among the basic geometric facts and concepts which are assumed to be true without logical proof. Example: 1. one and only one straight line can be drawn through two points 2. any number of lines can be drawn through a point. 3. A straight line may be produced to any length on either side. 20

Mathematical theorem is a logically valid conclusion drawn from a set of premises, axioms, and already established theorems of Mathematical system.

21 Mathematical theorem is a logically valid conclusion drawn from a set of premises, axioms, and already established theorems of Mathematical system. Theorems are usually in the form of an implication (or a biconditional) with premises followed by connective if then (or if and only if) Eg. If p is a prime number and p divides ab then either p divides a or p dived b If in a quadrilateral the diagonals bisect each other then the quadrilateral is a parallelogram. 21

FACTORIZATION AND THE PRIMES

I FACTORIZATION AND THE PRIMES 1. The laws of arithmetic The object of the higher arithmetic is to discover and to establish general propositions concerning the natural numbers 1, 2, 3,... of ordinary