A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner.

The value of each digit in a number can be determined using −

• The digit
• The position of the digit in the number
• The base of the number system (where the base is defined as the total number of digits available in the number system)

## Number system in computer system

• Binary number system
• Octal number system
• Decimal number system

### Decimal number system

• The number system that we use in our day-to-day life is the decimal number system. The decimal number system has a base of 10 as it uses 10 digits from 0 to 9. In the decimal number system, the successive positions to the left of the decimal point represent units, tens, hundreds, thousands, and so on.
• This computer understands everything in the multiple of 10.
• Like 721, seven in the first position of a three-digit number. So it will translate 7*103

### Binary number system

• Characteristics of the binary number system are as follows −
• Uses two digits, 0 and 1
• Also called a base 2 number system
• Each position in a binary number represents a 0 power of the base (2). Example 20
• The last position in a binary number represents the x power of the base (2). Example 2x where x represents the last position – 1.

### Octal number system

• Characteristics of the octal number system are as follows −
• Uses eight digits, 0,1,2,3,4,5,6,7
• Also called a base 8 number system
• Each position in an octal number represents a 0 power of the base (8). Example 80
• The last position in an octal number represents the x power of the base (8). Example 8x where x represents the last position – 1

• Characteristics of hexadecimal number system are as follows −
• Uses 10 digits and 6 letters, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
• Letters represent the numbers starting from 10. A = 10. B = 11, C = 12, D = 13, E = 14, F = 15
• Also called as base 16 number system
• Each position in a hexadecimal number represents a 0 power of the base (16). Example, 160
• Last position in a hexadecimal number represents a x power of the base (16). Example 16x where x represents the last position – 1

### Decimal to Other Base System

Step 1 − Divide the decimal number to be converted by the value of the new base.

Step 2 − Get the remainder from Step 1 as the rightmost digit (least significant digit) of the new base number.

Step 3 − Divide the quotient of the previous divide by the new base.

Step 4 − Record the remainder from Step 3 as the next digit (to the left) of the new base number.

Repeat Steps 3 and 4, getting remainders from right to left, until the quotient becomes zero in Step 3. •

The last remainder thus obtained will be the Most Significant Digit (MSD) of the new base number.

### Other Base System to Non-Decimal System

Step 1 − Convert the original number to a decimal number (base 10).

Step 2 − Convert the decimal number so obtained to the new base number.