How was the mathematical number system formed? How was this "odd" & "even " concept started in mathematics?

The number system is the numeral representation of numbers in It is formed by using digits or various Mathematical symbols.Now there are three types of numeral systems.

• Decimal Numeral/Number System- It is the system used for notifying integers or non integers.This system is also referred to as the base ten positional numeral system.Decimals are mainly identified having a decimal separator which is a '.' . For example 3.14 ,7.00 etc.The number representation in the decimal system are in the form of decimal fractions (p/(10^{q}) ,where p is an integer and q is a non negative integer.

Decimal Numbers are counted this way

000 , 001 , 002 , 003 , 004 , 005 , 006, 007 , 008 ,009 , 010.............100.........1000 and so on .

• Binary Number System - This is a base 2 number system usually used in Mathematics as well as Computers. It comprises only two symbols 0 and 1 . It has very direct flow of representations and is widely used in logic gates in digital electronics. Binary number counting only includes zeros and ones . For example binary number for 2 is 10. On diving 10 by 2 we get the remainder 0 and the quotient 5 and on dividing 5 by 2 again we get the remainder 1 and quotient 2. Now rearranging the remainders in reverse order we get 10 .
• Unary Numeral System - This is a base one numeral system.This is the simplest way for representing natural numbers. The way of representation of N is representation of a symbol exactly N number of times. For example the unary system for 5 is 11111 and 4 is 1111 if 1 is the chosen symbol for the representation.

The most common representation of numbers is the Hindu Arabic Numeral System developed by Aryabhata and Bramhagupta. In Mathematics there is a term called the Decimal place Value System also called Positional Notation. This concept of place value notation was developed by Aryabhata and is the backbone of the number system because without place value notation we cannot represent numbers and Bramhagupta first used the symbol for Zero .Again without Zero number theory is not possible. These two concepts although developed India later was widely spread-ed in other countries and the Arabs modified it.

As mentioned above the simplest numeral system is the Unary numeral system used for tallying marks and scores and coding purposes.Lets use the unary representation for larger numbers. For example 543 if - stands for units / for tens and + for hundreds then the representation will be +++++////--- for 543. This system is also referred to as the sign modification System and the Roman Numeral System is a modification of this concept.

Place value System or Positional Notation System is a very useful and demanding concept of Number System.This means that the value of a number is determined by the position of the digit that is the value of a number is actually the product of the digit by a factor which is determined by the position of the digit.For example lets take three identical digits 999. Here the interesting part is in words the number is written as nine hundred and ninety nine . The hundreds tens and the units here are being determined by the position of the digits that is digit at the first place represents the units, second place represents the tens and the third place represents hundreds. Similarly any digit at the fourth place shall reprimand thousands.

In computers the mainly used number systems are the binary number system(base 2) , Octal Number System(grouping of binary digits by 3) and hexadecimal number system( grouping of binary digits by 4 ).

The Hindu texts on numerals were translated by the Arabs and was spread throughout the Western World. Further modifications were done by the Western Culture and were called the Arabic Numerals. This widely helped in spreading the Mathematical Number System throughout the World.

Now this idea of dividing numbers into two kinds odd(not divisible by 2) , even (divisible by 2) is called Parity. The generalized way of representing odd numbers is 2k+1 , where k is an integer and even numbers is 2k , where k is an integer. Some examples of even numbers are 2, 4, 6, 8 ,54 ,78 ,-2, -18 and some examples of odd numbers are 3 ,9 ,17, 43, 19, -5.

Some basic rules for odd and even numbers include -

• Addition - Addition of two even numbers is even and odd numbers is even. for example 2+4=6 (both 2 and 4 are even numbers and the resultant 6 is also an even number) . Similarly 13+15 = 28 ( both 13 and 15 are odd numbers and the result is an even number) where as addition of an even and odd number is odd. Example 17+2= 19 (17 is an odd number and 2 is an even number but the result is odd)
• Subtraction- The rules for subtraction is same as in addition.
• Multiplication- Product of two even numbers is even, two odd numbers is odd and product of an even and odd number is even.
• Division- In this case if the quotient is an integer and the and the dividend has more factors of two than the divisor then only the result will be even.

0 is considered to be an even number.

Disclaimer : The history of odd and even numbers that is their origin is not properly known to me