Base (numeral system)
From Halbeeg, the open encyclopedia · Af-Soomaali
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A base (or radix) is the number of distinct symbols used in a positional numeral system. It is also the value by which we multiply when moving from one place to the next as we move from right to left. For example, a numeral system with base 10 uses ten symbols: 0 through 9.
The concept of a base operates on the principle of positional notation: each position in a number has a value determined by its place. As a result, the same digit can have different values when the base is changed.
How it works
In a numeral system with base b, each digit is a value between 0 and b−1. The overall value is calculated by summing each digit multiplied by b raised to the power of its position. For example, in base 10, the number 342 equals 3×100 + 4×10 + 2×1.
Common bases
Base 10 (decimal) is the most widely used in everyday life. Base 2 (binary) uses only two symbols, 0 and 1, and forms the foundation of computing. Base 8 (octal) and base 16 (hexadecimal) are used in programming because they convert to and from binary readily. Base 60 is linked to ancient Mesopotamian mathematics and can still be seen in the division of hours and minutes.
Base conversion
Converting a number from one base to another typically involves repeated division: the number is divided by the new base, the remainder is kept, and the process is repeated with the quotient. The remainders, taken from last to first, form the number in the new base.