# Base system

The radix or base is the number of unique numerical digits, including zero, that a numeral system uses to represent numbers. For example, for the decimal system (the most common system in use today) the radix is ten, because it uses the ten digits from 0 through 9.

In any numeral system, the base will always be written as (x)y. For example, (18)10 represents the number eighteen in the decimal system; (10)2 represents the number two in a base two (or binary) system. (Radix is a Latin word for "root". Root can be considered a synonym for base in the arithmetical sense.)

Commonly used numeral systems include:

Base Name Description
10 decimal system the most used system of numbers in the world, is used in arithmetic. Its ten digits are "0–9". Used in most mechanical counters.
12 duodecimal (dozenal) system is often used due to divisibility by 2, 3, 4 and 6. It was traditionally used as part of quantities expressed in dozens and grosses.
2 binary numeral system used internally by nearly all computers, is base two. The two digits are "0" and "1", expressed from switches displaying OFF and ON respectively. Used in most electric counters.
16 hexadecimal system is often used in computing. The sixteen digits are "0–9" followed by "A–F".
8 octal system is occasionally used in computing. The eight digits are "0–7".
60 sexagesimal system originated in ancient Sumeria and passed to the Babylonians. It is still used as the basis of our modern circular coordinate system (degrees, minutes, and seconds) and time measuring (hours, minutes, and seconds).
64 Base 64 is also used in computing, using as digits "A–Z", "a–z", "0–9", plus two more characters, often "+" and "/".
256 byte is used internally by computers, actually grouping eight binary digits together. For reading by humans, a byte is usually shown as a pair of hexadecimal digits.

System 10 2 3 4 5 6 7 8 9
Value 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1
2 10 2 2 2 2 2 2 2
3 11 10 3 3 3 3 3 3
4 100 11 10 4 4 4 4 4
5 101 12 11 10 5 5 5 5
6 110 20 12 11 10 6 6 6
7 111 21 13 12 11 10 7 7
8 1000 22 20 13 12 11 10 8
9 1001 100 21 14 13 12 11 10
10 1010 101 22 20 14 13 12 11
11 1011 120 23 21 15 14 13 12

## Base conversion

To convert a number ‘n’ from a base system of ‘a’ to base system ‘b’, is repeatedly divided by b and all the remainders are arranged in the reverse order to get the desired number 'n' in base 'b'. Consider the following examples where ‘q’ is quotient and ‘r’ is the remainder.

Examples of base conversion

Animation : Decimal to binary

### Relevant exercises

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