Different types of numbers

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Natural Numbers

All countable numbers are called natural numbers. They are denoted by ‘N’, where N = {1, 2, 3…}.


Whole numbers

When zero is included in the set of natural numbers, they are called whole numbers. They are denoted by ‘W’, where W = {1, 2, 3…}.


Integer Numbers

Integer numbers or simply Integers comprise positive natural numbers, their negatives and 0. They are denoted by ‘I’, where I = {…-3, -2, -1, 0, 1, 2, 3…}


Rational Numbers

Any number which can be expressed in the form of p/q, where p and q are integers and q≠0, is called a rational number. The set of rational numbers is denoted by ‘Q’.


Irrational Numbers

Any number which cannot be expressed in the form of p/q, where p and q are integers and q≠0, is called an irrational number. Set of irrational numbers is denoted by ‘R\Q’. They are generally the non-terminating and non-recurring decimal fractions.


Real Numbers

A real number is a value that represents a quantity along a continuous line. The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers such as √2 (1.41421356… the square root of two, an irrational algebraic number) and π (3.14159265…, a transcendental number). Real numbers can be thought of as points on an infinitely long line called the number line or real line, where the points corresponding to integers are equally spaced. Any real number can be determined by a possibly infinite decimal representation such as that of 8.632, where each consecutive digit is measured in units one-tenth the size of the previous one.


Complex numbers

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, which satisfies the equation i2 = −1.In this expression, a is the real part and b is the imaginary part of the complex number.


 Prime Numbers

A natural number which is not divisible by any other number except 1 and itself is called a prime number. E.g. 7 is called a prime number as it is not divisible by any other number except 1 and 7 itself.

Note: Unity (i.e. 1) is not a prime number.


To check whether the number is prime or not

Find the square root of the number and round off the square root to the immediate higher integer. If the number is divisible by any prime number less than the rounded off integer of square root, then it is not a prime number.

E.g., The value of √131 lies between 11 and 12. Prime numbers less than 12 are 2, 3,5,7,9 and 11. Since 131 is not divisible by any of these. Hence, 131 is a prime number.


Perfect Numbers

A perfect number is a positive integer that is equal to the sum of its divisors excluding the number itself.

E.g., 6 = 3, 2, 1 = 3 + 2 + 1, 28 = 14, 7, 4, 2, 1 = 14 + 7 + 4 +2 + 1

Note: Logic of perfect numbers is questioned in programming problems.


Composite Number

A number which has other factors apart from 1 and itself is called a composite number.


Even and Odd Numbers: All integers which are divisible by 2 {…-4, -2, 0, 2, 4, 6…}are even. All other integers are considered odd.

Note: 2 is the only even prime number.


Recurring Numbers

A number in which a pattern of one or more digits is repeated indefinitely, e.g., 7.353535 ... is called a recurring number. 7.353535 can be written as 7. ¯35. ¯(Any Value) shows the recurring part of a number.

 Converting Recurring Number to Fractions:

  1. The first step is to form a simple equation where x= 0.¯78.
  2. By multiplying both sides by 100, we can obtain another equation with 100x =78.¯78.
  3. Now we eliminate the recurring part of the decimal by subtracting x from 100x, where x= 0.¯78 and 100x = 78.¯78 .
  4. 100x – x = 78.¯78 - 0.¯78
  5. 99x = 78 x = 78/99
  6. Thus, the answer of 0.¯78 is 78/99.
  7. The important part to remember is to get two equations in x where the recurring portion after the decimal point is exactly the same.