Unit 1 Chapter 2 (Number systems) PDF

Preview

Summary

IFTE SIR'S CLASS NOTES...

Description

Unit-1 Chapter 2 Number Systems 2.1 Sets of number Important sets which we meet in elementary mathematics are sets of numbers. Of particular importance, especially in analysis, is the set of real number which is denoted by R. The set of real numbers and its properties is called the real number system. 2.1.1 List of numbers: Take a straight line (which is extended to infinites on both sides) and fix a point 0 on it. Choose a suitable segment as unit length. To the right side of 0 we mark off points corresponding to positive integral multiple of unit length, to the left side, those corresponding to negative integral multiple. This line is known as the line of numbers. -∞

+∞

-2 -1 0 1 2 2.1.2 Real number R: Every point on this line corresponds to a real number and every real number can be represented by a point on this line. 2.1.3 Integer Z: We denote the integers by Z and, Z = {……-3, -2, -1, 0, 1, 2, 3……} The integers can be also referred to as the whole numbers, & Z R 2.1.4 Rational Number Q: The rational numbers are those real numbers which can be expressed as the ratio of two integers. We denote the set of rational numbers by Q. Accordingly a Q = {x│x = where a, b Z, b ≠ 0} and Z  Q  R b The sum, product, difference and quotient of two rational numbers are again a rational number. 2.1.5 Natural Number N: The natural numbers are the nonnegative integers. So, N = {0, 1, 2, 3…….} The natural numbers are closed only under the operations of additions and multiplication. N  Z  Q  R. 2.1.6 Prime Numbers P: The prime numbers are those natural numbers P, excluding 1, which are divisible by 1 and P itself. e.g.: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31….

1

2.1.7 Irrational numbers: A real number which is not rational is called irrational. e.g. 2 , Π etc. are irrational numbers. The irrational numbers are non-terminating decimals and these non-terminating decimals can also be classified into, i. Repeating decimals: example: 1/3 = 0.3333333…, 1/7 = 0.142857142857142... ii. Non-repeating decimals: example: π = 3.14159265358979323... , √2 = 1.41423562373… 2.2 Properties of Real numbers: If a, b and c are real numbers then, 1. 2. 3. 4. 5.

Only one of the relations ab or a =b is true for a and b If a...