Definition:Hexadecimal Numbers
Definition
Hexadecimal numbers are numbers expressed in base $16$ notation.
That is, every number $x \in \R$ is expressed in the form:
- $\displaystyle x = \sum_{j \in \Z} r_j 16^j$
where $\forall j \in \Z: r_j \in \left[{0 .. 15}\right]$.
In order to be able to represent numbers in such a format conveniently and readably, it is necessary to render the digits $10$ to $15$ using single characters.
The convention is for the following:
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle 10\) | \(:\) | \(\displaystyle \mathrm A\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | |||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle 11\) | \(:\) | \(\displaystyle \mathrm B\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | |||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle 12\) | \(:\) | \(\displaystyle \mathrm C\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | |||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle 13\) | \(:\) | \(\displaystyle \mathrm D\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | |||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle 14\) | \(:\) | \(\displaystyle \mathrm E\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | |||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle 15\) | \(:\) | \(\displaystyle \mathrm F\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) |
Thus $\mathrm{FFFF}_{16} = 15 \times 16^3 + 15 \times 16^2 + 15 \times 16 + 15 = 65\,535_{10}$.
Their lowercase renditions can equally well be used, e.g. $\mathrm{ffff}_{16} = 65\,535_{10}$, but it doesn't look as good in proportional font.
Hexadecimal numbers, like binary numbers, have particular relevance in the field of computer science.
In that context, a number is usually indicated as being hexadecimal by subscripting $\mathrm H$ or $\mathrm h$ rather than $16$.
That is, $\mathrm{FFFF}_{16}$ would be rendered $\mathrm{FFFF_H}$ or $\mathrm{ffff_h}$, and so forth.
