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# Numbers

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 Sub Topics A number is a mathematical object, which we generally use for counting and measurement purposes in different areas. Numbers are taken as input in different procedures of mathematical operations and their output produced on the other side is also a number.Ten simple digits or numbers (0 to 9) influence our lives in far more ways than we could ever imagine. Birthdays, ages, addresses, telephone numbers, years, station numbers, time, date, account numbers, credit card numbers, factors, squares, accountants, etc. There exists immeasurable variety of hidden wonders surrounding numbers that we use everyday. Different types of operations like unary operation, binary operation and so on, are used on numbers. Unary operations are the operations which take single number as input and produce a single number as output. On the other side, binary operations are the operations which take two numbers as input and produce a single number as output.Notational symbol representing a number is called a numeral. Numbers are classified according to type which can be natural number, integer, rational number, real number, complex number etc.

## Types of Numbers

 Natural Numbers $\mathbb{N}$ Natural numbers are used in everyday life. It consists of positive integers or counting numbers, includes values from one all the way up to infinity. Zero is not included in natural numbers. The sum and product of two natural numbers is a natural number while, the difference between two natural numbers and the division of two natural numbers (occurs only when the division is exact) is not always a natural number. Used for two main purposes: counting and ordering. Examples:Subset of natural numbers is formed by N = {1, 5, 7, 9, 11, 14, 15, 18, ........}Difference: 7 - 9 $\not{\varepsilon}$ $\mathbb{N}$Division: $\frac{4}{8}$ $\not{\varepsilon}$ $\mathbb{N}$ Integers $\mathbb{Z}$ Integers include zero, positive and negative counting numbers. They can be written without a fraction or decimal portion and forms a ring with operations addition and multiplication. Integers vary from minus infinity through zero to plus infinity. Examples: -546846, 5456, -1545, -44, -45, 359, 256, 25, 4, 546 Rational Numbers $\mathbb{Q}$ The word rational comes from the word ratio. Any integer that can be expressed as a fraction as long as the denominator is not 0.P = $\frac{a}{b}$, b $\neq$ 0; where, a and b are integers. In decimal form, they either terminate or begin to repeat the same pattern indefinitely. Examples: $\frac{5}{2}$, $\frac{5}{3}$, $\frac{8}{5}$, $\frac{9}{4}$ Irrational Numbers $\mathbb{I}$ Irrational numbers are the numbers that are not rational. Decimal representation is non-terminating and non-repeating. They cannot be expressed as ratio of two integers. Best known irrational number is $\pi$, that gives the relationship between the perimeter of a circle and its area. Examples: $\sqrt{2}$ = 1.41421.. and $\sqrt{3}$ = 1.7320... Whole Numbers $\mathbb{W}$ Natural numbers including zero. All natural numbers are whole numbers, while all whole numbers are not natural numbers. Examples: 0, 1, 2, 5, 18, 150, 1586, 258, 569. Complex Numbers $\mathbb{C}$ A number that is expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit. It includes real numbers and imaginary numbers. The value of i is $\sqrt{-1}$ Examples: 2 + $\sqrt{-12}$, 5i Real Numbers $\mathbb{R}$ Numbers that can be expressed as the limit of a sequence of rational numbers. It corresponds to a point on the number line. All operations can be performed with real numbers with the exception of the root of an even index and negative radicand, and division by zero. Examples: $\sqrt{5}$, $\sqrt{8}$, $\sqrt{832}$ Imaginary Numbers $\mathbb{Im}$ An imaginary number is denoted by bi, where b is a real number and i is the imaginary unit. It has a negative or zero square. They are the complex numbers whose real part is zero. Useful in the construction of non real complex numbers. Examples: x = $\pm$ $\sqrt{-9}$, x = $\pm$ $\sqrt{-5}$, x = $\pm$ $\sqrt{-25}$ Transcendental Numbers Any real or complex number that is not algebraic. All real and complex numbers are transcendental. Real transcendental numbers are irrational as all rational numbers are algebraic. The name transcendental comes from Leibniz, where he proved sin x is not an algebraic function of x. Examples: e and $\pi$ Perfect Numbers Perfect number is a whole number, whose factors when added give back the number as answer. The factors of the numbers are less than that number. Perfect number is a number that is half the sum of all of its positive divisors (included itself). Examples: 6, 28, 496, 8128 Prime Numbers $\mathbb{P}$ A prime number is a natural number greater than 1, that has no positive divisors other than 1. To check whether a given number is prime or not, we can use trial division, that consists of testing whether 'n' is a multiple of any integer between 2 and $\sqrt{n}$. Example: 2, 3 and 5 are the prime numbers among the numbers 1 to 6.