Maths Class 11 Chapter 1 Notes
Important Complete Maths Class 11 Chapter 1 Notes of Chapter No.1: Number Systems written by Professor M. Sulman Sherazi Suib. These notes are very helpful in the preparation of Number Systems for the students of Mathematics of the Intermediate and these are according to the paper patterns of all Punjab boards.
Summary and Contents:
Topics which are discussed in the notes are given below:
- Our comprehensive Maths Class 11 Chapter 1 Notes will ensure you're fully prepared for your exams.
- History of number systems
- Number system in civilization
- Civilizations
- Chinese number system
- Egypt civilization
- Mayan number system
- Babylon number system
- Greek number system
- Roman number system
- Indian number system
- Type of numbers
- Natural numbers
- Whole numbers integers
- Odd integers even integers
- Rational numbers
- Irrational numbers
- Terminating decimals
- Recurring decimals
- History of Pi and its formula and value
- Complete solution of the definitions, examples, and exercises solutions.
- We are already familiar with the set of real numbers and most of their properties. We now state them in a uniied and systematic manner. Before stating them we give a prelimi-
- nary deinition. Binary Operation: A binary operation may be deined as a function from A % A into A, but for the present discussion, the following deinition would serve the purpose. A binary operation in a set A is a rule usually denoted by * that assigns to any pair of elements of A, taken in a deinite order, another element of A. Two important binary operations are addition and multiplication in the set of real numbers. Similarly, union and intersection are binary operations on sets which are subsets of the same Universal set usually denotes the set of real numbers. We assume that two binary operations addition (+) and multiplication (. or x) are deined in _.
- Following are the properties or laws for real numbers. 1. Addition Laws: i) Closure Law of Addition ii) Associative Law of Addition iii) Additive Identity (read as zero) is called the identity element of addition. iv) Additive Inverse v) Commutative Law for Addition
- 2. Multiplication Laws
- vi) Closure I.aw of Multiplication
- vii) Associative Law for Multiplication
- viii) Multiplicative Identity
- 1 is called the multiplicative identity of real numbers.
- ix) Multiplicative Inverse
- x) Commutative Law of multiplication
- Multiplication - Addition Law a (b + c) = ab + ac (Distrihutivity of multiplication over addition). (a + b)c = ac + bc In addition to the above properties _ possesses the following properties. i) Order Properties (described below). ii) Completeness axiom which will be explained in higher classes. The above properties characterizes _ i.e., only _ possesses all these properties. Before stating the order axioms we state the properties of equality of numbers.
- Any set possessing all the above 11 properties is called a ield. From the multiplicative properties of inequality we conclude that: - If both the sides of an inequality are multiplied by a +ve number, its direction does not change, but multiplication of the two sides by -ve number reverses the direction of the inequality. a and (-a) are additive inverses of each other. Since by deinition inverse of -a is a.
- The history of mathematics shows that man has been developing and enlarging his concept of number according to the saying that “Necessity is the mother of invention”. In the remote past they stared with the set of counting numbers and invented, by stages, the negative numbers, rational numbers, irrational numbers. Since square of a positive as well as negative number is a positive number, the square root of a negative number does not exist in the realm of real numbers. Therefore, square roots of negative numbers were given no attention for centuries together. However, recently, properties of numbers involving square roots of negative numbers have also been discussed in detail and such numbers have been found useful and have been applied in many branches of pure and applied mathematics. The numbers of the form x + iy, where x, y U_ , and i = ,are called complex numbers, here x is called real part and y is called imaginary part of the complex number. For example, 3 + 4i, 2 - i etc. are complex numbers.