Numbers and Their Types in Mathematics

Digits: -           The 10 symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 are called digits. It used to create numbers in the base 10 decimal number system.

Numerals संख्यांक: -    The symbols used to denote the numbers are called unmerals.

                        The Arabic numerals are  0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ...........

                         The roman numerals are I, II, III, V, X, L, C, D, M,  

 Numbers:-      A number is a mathematical tool which is used in counting individual quantities, calculating and quantifying.

In general, decimal number system is used which consists of 10 digits, from 0 to 9.

What are various operations that can be performed on numbers?

Mathematical or Arithmetic operations used on numbers are four add (+) , sub (-) , Mul ( × ) and div (÷ )

Addition:-   It is the process of finding out single number or fraction equal to two or more quantities taken together.

Subtraction:-          It is the process of finding out the quantity left when a smaller quantity (number/ fraction) is  reduced from a larger one.

Multiplication:-      It signifies repeated addition. If a number has to be repeatedly added then that number is multiplicand.

                              The number of multiplicands considered for addition is multiplier. The sum of  the repetition is the product.

Division:-    It is a reversal of multiplication. In this we find how often a given number called divisor is contained in another given number called dividend.

                   The number expressing this is called the quotient and the excess of the dividend over the product of the divisor and quotient is called remainder.

Classification of numbers:

Real number : -         Numbers which can be quantified and represented by a unique point on the number line are called real numbers.

                              OR          The set of all the rational and irrational numbers is called real numbers.

             Ex- {..., ^-3, ^-2, ^-1, 0, ⁺1, ⁺2, ⁺3, .........  , , π }

   Rational numbers include the whole numbers (0, 1, 2, 3, ...), the integers (..., - 2, - 1, 0, 1, 2, ...), fractions, and repeating and terminating decimals. Irrational numbers ( , , π ) 

Complex number : -   Complex numbers are the numbers which have both real and imaginary part.

 

The Complex Numbers:- The complex numbers are the set {a + bi | a and b are real numbers}, where i is the imaginary unit,  –1.

The complex numbers include the set of real numbers.  The real numbers, in the complex system, are        written in the form a + 0i = a. a real number. This set is written as C for short. The set of complex numbers is important because for any polynomial  p(x) with real number coefficients, all the solutions of p(x) = 0 will be in C.

Rational number : -   Numbers of the form ^p/_q, where p and q are integers and q≠0 are called rational numbers.

• They can be expressed as a ratio between two integers. Ex 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.
• All terminate decimals are rational numbers (since 8.27 can be written as 827/100.)
• Decimals repeating which have a pattern after some point are also rational. Ex: 0.083333333... = 1/12.

Irrational numbers:-           Numbers which are not rational but can be represented on the number line are called irrational numbers.

               OR Any number that cannot be expressed by an integer or the ratio of two integers.

               Irrational numbers are expressible only as decimal fractions where the digits continue forever with no repeating pattern.                        

  Ex-  , , π . 

• They cannot be written as a ratio (or fraction). 
• In decimal form, it never ends or repeats.

Integers:-      The group of positive and negative whole numbers, ..., ^-3, ^-2, ^-1, 0, ⁺1, ⁺2, ⁺3, ... is called integers.

The set of all integers is usually denoted by Z or Z+  OR

The integers are the set of real numbers consisting of the natural numbers. {..., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5...}

• The set of integers is sometimes written J or Z for short.
• The sum, product, and difference of any two integers is also an integer. But this is not true for division just try 1 ÷ 2.
• The sum their additive inverse is zero. Ex {1+ (-1) = 0, 12 + (-12) = 0 }

Fraction:-   Fractions are a type of rational numbers, which are of the form ^p/_q, where p and q are integers and q≠0.

Whole numbers:-   Whole numbers are the set of positive integers along with (from) 0.

EX- {0, 1, 2, 3, 4, 5, 6, 7, 8......}

They do not have any decimal or fractional part.

• The set of whole numbers is denoted by W = {0, 1, 2, 3, 4, 5, ...}

Negative integers:-             Negative integers are the set of negative numbers before 0.

EX- {... .,-8, -7, -6, -5, -4, -3, -2, -1}

They do not have any fractional or decimal part.

Natural numbers:-     The set of counting numbers are called natural numbers.

EX- {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ..... ∞}

The natural numbers are called counting numbers and the positive integers. OR

Natural numbers are the set of positive integers, that is, integers from 1 to ∞.

• The set of natural numbers is denoted by N = {1, 2, 3, 4, 5, ...}
• The sum of any two natural numbers is also a natural number (for ex 4 + 2000 = 2004), and the product of any two natural numbers is a natural number (4 × 2000 = 8000). This is not true for subtraction and division, though.

Even numbers:-         Numbers divisible by 2 are called even numbers.

Recognize- It’s unit place digit be always 0, 2, 4, 6 or 8.                        

EX- {2, 4, 6, 8, 10, 12, ..... ∞}

• The sum of any two even numbers is also a even number (for ex 4 + 2 = 6), and the product of any two even numbers          is a even number (4 × 12 = 48). This is not true for subtraction and division, though.(may be & may not be)

Odd numbers:-         Numbers which are not divisible by 2 are called odd numbers. Odd numbers leave 1 as the remainder when divided by 2.

Recognize- It’s unit place digit be always 1, 3, 5, 7 or 9.

EX- {1, 3, 5, 7, 9, 11, ..... ∞}

• The sum of any two odd numbers is a even number (for ex 5 + 3 = 8), and the product of any two odd    numbers is a odd number (7 × 9 = 63).

This is not true for subtraction and division, though.(may be & may not be)

• The sum of odd and even numbers is a odd number (for ex 17 + 12 = 29), and the product of odd and even numbers is a even number (17 × 2 = 34).
• This is not true for subtraction and division, though.(may be & may not be)

Prime numbers:-        Any number other than 1 which does not have any factor except 1 and itself (the number) is called a prime number. EX- {2, 3, 5, 7, 11, 13, 17, 19 ..... ∞}

Twin Prime numbers:-          Two prime numbers are called twin prime number if there is one composite number between them.           

                     EX- (3 and 5), (5 and 7), (11 and13) and (17 and 19)

Co-prime/relatively prime numbers:-            Two numbers are said to be co-prime or relatively prime if they do not  have any common factor except (other than) one.

                  EX- {2 & 3, 3 & 4, 4 & 5, 5 & 6, 6 & 7, …...... ∞}

Composite numbers:- A number that has more than two distinct factors is called a composite number.

EX- {4, 6, 8, 9, 10, 12, 14, 15, 16, 18, ..... ∞}

Perfect numbers:-      A number is said to be a perfect number if the sum of all its factors including 1and itself is equal to the double of number.

                       EX:- 6 and 28 ( 1+2+3+6= 2×6 And  1+2+4+7+14+28= 2×28)

Transcendental numbers:-  Any number that cannot be the root of a polynomial equation with rational coefficients.

                                               They are a subset of irrational numbers examples of which are Pi = 3.14159... and e = 2.7182818..., the base of the natural logarithms.