Binary

 (also known as base 2) is a positional number system structured around the number 2 and its powers. Numbers are expressed as a sum of powers of 2.

Usage
Binary is one of the most useful number bases, as it's the core base to store data in programming and computers. Almost all computers use it internally. Everything that has two possible states can be considered binary - the most common example is ON/OFF.

However, it is not a very practical base for human usage, as the numbers grow to massive sizes really quickly (for example, 8 digits can only express a number up to 28-1=255.) This is why compact representations of binary, most notably Octal and Hexadecimal are commonly used instead to express binary numbers in short form.

Mulitplication table
The binary multiplication table is the simplest possible, as it doesn't contain a single non-trivial product. However, while usually the alpha number (11) follows a special pattern in multiplication (11, 22, 33, 44, 55...) and exponentation (11, 121, 1331, 14641, 15AA51...) until the digits run out, this does not work in binary since by the time we get to 11 we have already ran out of digits. Therefore, operations with 11 (310) are much harder than they look.

Patterns
Binary contains a number of patterns that can help translating a number from another base into binary:

Fractions
A binary fraction will only terminate if the denominator is a power of 102.

Related number systems
Binary is the base with the highest amount of related number systems.
 * The Fibonacci system is a non-standard positional system related to binary.
 * Location numerals are a non-positional number system closely related to binary.
 * Negabinary is the negative base corresponding to binary.
 * Bijective binary is the bijective base corresponding to binary.
 * The Non-adjacent form is the balanced equivalent of binary.
 * Quaternary digits correspond to 2 binary digits.
 * Octal digits correspond to 3 binary digits.
 * Hexadecimal digits correspond to 4 binary digits.
 * Duotrigesimal digits correspond to 5 binary digits.
 * Tetrasexagesimal digits correspond to 6 binary digits.