May 11, 2015

Number System

In mathematics, an irrational number is any real number that cannot be expressed as a ratio of integers. Irrational numbers cannot be represented as terminating or repeating decimals. As a consequence of Cantor's proof that the real numbers are uncountable and the rationals countable, it follows that almost all real numbers are irrational.

The rational numbers are those numbers which can be expressed as a ratio between two integers. For example, the fractions 1/3 and –1111/8 are both rational numbers. All the integers are included in the rational numbers, since any integer z can be written as the ratio z/1. The Set of all rational numbers are given by Q and the set of all +ve rational numbers are given by Q+

All decimals which terminate are rational numbers (since 8.27 can be written as 827/100.) Decimals which have a repeating pattern after some point are also rationals: for example,

0.083333333... = 1/12.

The real numbers is the set of numbers containing all of the rational numbers and all of the irrational numbers. The real numbers are “all the numbers” on the number line. The Set of all real numbers are given by R and the set of all +ve rational numbers are given by R+

A natural number is a number that occurs commonly and obviously in nature. As such, it is a whole, non-negative number. The set of natural numbers, denoted N, can be defined in either of two ways:
N = {0, 1, 2, 3, ...} if we include 0(even number) they are usually called Whole numbers
N = (1, 2, 3, 4, ...}

The Set of all natural numbers are given by N

The integers are the set of real numbers consisting of the natural numbers, their additive inverses and zero.{..., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5...} Note that 0 is neither +ve or -ve integer.

The set of integers is sometimes written J or Z for short.

A composite number is a positive integer that has at least one positive divisor other than one or the number itself. In other words, a composite number is any integer greater than one that is not a prime number.

Except 1 and itself every natural number divisible by 1 and itself is called a prime number , to test if a number is prime or not, take its nearest square root( if not a perfect square then the number just greater), then divide this number by all the prime numbers below it, if it is not divisible by any one of them it is a prime number.

All the prime numbers between 1 and 100 are 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97

Two numbers are called Co-prime numbers if there is no other common factor other than 1 between them. eg: 15 ,16

Trichotomy Law: If there are any two natural numbers a and b then there exists only relation necessarily:

a < b, 2. a = b, 3. a > b

A series of numbers in which the next number is greater than the previous number by 1 are called Consecutive numbers, e.g 1,2,4,...; 10,11,...

A number is called a perfect number if the sum of all its factors, including 1 but excluding the number itself, is the number, then it is called perfect number

e.g 6 = 3 + 2 + 1. There are only 27 perfect numbers known so far.

64 and 729 are both perfect square and perfect cube numbers.

Triangular Numbers:

Complex number

A complex number can be visually represented as a pair of numbers (a, b) forming a vector on a diagram called an Argand diagram, representing the complex plane. "Re" is the real axis, "Im" is the imaginary axis, and i is the imaginary unit which satisfies i2 = −1.

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, that satisfies the equation x2 = −1, that is, i2 = −1.[1] In this expression, a is the real part and b is the imaginary part of the complex number.