What is number system what are various number system explain with a suitable example?

Answer 1

A number system is simply a way to record numbers. Humans have used a variety of numbering systems over the years, but the decimal system is by far the most prevalent today. This system uses the ten Arabic symbols, 0123456789, to represent the digits from zero to nine, and is known as base 10 for this reason. Digits are aligned on columns, with units on the right, 10s to their left, and 100's to their left, and so on. Each column is therefore 10 times the value of the column to its right. In other words, each column is an increasing power of 10, beginning with 10^0 on the right, then 10^1 and so on.

You are undoubtedly familiar with base 10, however the above is relevant when discussing other number systems as the same principals apply.

Computers use base 2 (binary), which is the lowest base of all. It uses the 2 Arabic digits, 0 and 1. Since it is base 2, the columns represent powers of 2. So the rightmost column represents 2^0, then 2^1, 2^2, 2^3, and so on the further left we go. So the number 4 in decimal would be represented by 100 in base 2. That is, 1 * (2^2), which is 4 (all other columns are zero, so they evaluate to zero).

Computer programmers use base 16 (hexadecimal). This is because binary numbers, despite their apparent simplicity, are incredibly difficult for humans to work with. One digit out of place could be disastrous. Converting them to decimal is clearly an option, but hexadecimal is a lot simpler to work with because base 2 and base 16 are interchangeable and align with each other more closely than decimal.

Four binary digits gives us 16 possible combinations. 0000, 0001, 0010, 0011, 0100, 0101, 0110, 0111, 1000, 1001, 1010, 1011, 1100, 1101, 1110 and 1111 (decimal zero to decimal 15, respectively). With only 16 combinations to consider, each of these can be represented by a single hexadecimal symbol. There are only 10 Arabic symbols for numbers, so we must use 6 additional symbols for the numbers 10 to 15. By convention we use the letters a through f. Thus each of the binary combinations above can be represented by 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, a, b, c, d, e and f, respectively.

Binary digits (bits) are usually combined into groups of 8 bits, known as bytes. 8 is a multiple of 4, so we need 2 hexadecimal digits to represent a full byte. To do this we simply divide the byte into two half bytes (known as nybbles), then convert each nybble to its hexadecimal form. Thus the byte 01101101 is represented as nybbles 0110 and 1101, which is 5d in hexadecimal (often denoted as 0x5d). This equates to (5 * (16^1)) + (13 * (16^0)), which is 93 decimal. So 01101101 is binary for 93 decimal, or 0x5d hexadecimal.

Regardless of the length of a sequence of bits, breaking them into groups of 4 allows them to be translated directly into hexadecimal. So a 32-bit number requires 8 hexadecimal digits. Reading and writing 8 digits is clearly a lot simpler than deciphering 32 bits of 1s and 0s, and because binary and hexadecimal have a consistent alignment (4 bits equals 1 hex digit), they are much easier to deal with than decimal which has a more variable alignment with binary (4 bits could be 1 or 2 decimal digits).

Other bases that are in common use today include base 60, which is the basis for our clocks. 60 seconds is 1 minute and 60 minutes is 1 hour. Then we switch to base 12 for the hours (or base 24 if using a 24-hour clock). You may ask why we never "decimalised" our time-keeping (dividing the day into 10 or 20 longer hours, each with 100 minutes, each with 100 seconds, for instance). The main reason is that 60 is evenly divisible by 2, 3, 4, 5 and 6, whereas 100 is evenly divisible by just 2, 4 and 5, and a 12-hour period (which is also division of 60) is evenly divisible by 2, 3, 4 and 6 whereas 10 is evenly divisible by just 2 and 5.

Inches and feet are also base 12. So while we are quite familiar with base 10, we actually use other bases without realising it. Of course we don't symbolise numbers greater than 9 with letters like we do in hexadecimal, but the principal is the same.