A numeral system is a writing system for expressing numbers using digits or other symbols in a consistent manner.

Value

In general the value of a number in any numeral system can be expressed using the equation:

Where is the value of the digit at position and is the base of the numeral system.

Alternatively this can be written as a sum:

Decimal Values

A number system can be extended with fractional values by introducing a decimal indicator (usually a "."). When the decimal is included the value is extended into negative exponents as follows

Where is the value of the digit at position to the left of the decimal indicator, is the base of the numeral system, is the value of the digit at position to the right of the decimal indicator.

Alternatively this can be written as two sums:

Counting

Counting begins with the incrementing the least significant digit (usually the rightmost digit). When the available symbols for this position are exhausted, the least significant digit is reset to the zero symbol, and the next digit of higher significance (one position to the left) is incremented, and the incrementing of the least significant digit resumes.

Examples

In Decimal (Radix 10)

In Binary (Radix 2)

Conversion from One Numeral System to Another

When converting between two different numeral systems where one is not decimal (base 10) it is usually easiest to first convert from the first base to decimal and then to the second base.

Converting to Smaller Radix

To convert to a smaller base a succession of Euclidean divisions by the smaller base needs to be performed. The remainder becomes the value of the right-most digit and the quotient is used in the next Euclidean division, this process is performed until the quotient is .

Example

Convert the number A10B in hexadecimal (base 16) to decimal (base 10).

Assembling all these numbers from right to left we get the final value of in decimal

Converting to Larger Radix

To convert to a larger base the equation the equation is used.

Example

Convert the number in decimal (base 2) to decimal (base 10).