Number bases

A number base of a numeral system defines the symbols and notations used to represent a value.

For example, base 2 provides two distinct symbols that are then combined to represent larger quantities.

Convert to decimal

Given some numeric representation with digits $a_n{a}_{n-1}{a}_{n-2}\dots a_0$ in base $b$, the decimal conversion can be found via $$a_n\times b^n+a_{n-1}\times b^{n-1}+\dots +a_0\times b^0$$

This may also be extended to cover non-integers with a negative exponent for the fractional digits. $${(a_n{a}_{n-1}\dots a_0.a_{-1}{a}_{-2}\dots a_{-m})}_b=\sum _{i=-m}^n{a}_i\times b^i$$

For example, ${16.4}_{16}$ is equivalently expressed as $1\times {16}^1+6\times {16}^0+4\times {16}^{-1}$ or ${22.25}_{10}$.

Convert from decimal

To convert a decimal value to an alternate base, $b$, repeatedly divide by $b$ and build the value right-to-left with the remainder until the quotient is $0$.

For example, to express $52$ in base 2: $$\begin{array}{rlr}52÷2& =26R0\to & 0_2\\ 26÷2& =13R0\to & {00}_2\\ 13÷2& =6R1\to & {100}_2\\ 6÷2& =3R0\to & {0100}_2\\ 3÷2& =1R1\to & {10100}_2\\ 1÷2& =0R1\to & {110100}_2\end{array}$$

Repeated division by base for integer component. Repeated multiply by base for fractional component.

Conversion from binary to $2^n$

Segment into groupings of size $n$.

$$\begin{array}{llllllll}& & & 1& 1& 1& 1& 1_2\\ \times & & & & & & 1& 1_2\\ \\ & & & 1& 1& 1& 1& 1\\ +& & 1& 1& 1& 1& 1& 0\\ & 1& 0& 1& 1& 1& 0& 1_2\end{array}$$

Historic notation

Additive systems - Bakairi - Attic numerals

vs positional