## Definition

Hexadecimal notation is the technique of expressing numbers in base $16$.

Every number $x \in \R$ is expressed in the form:

$\ds x = \sum_{j \mathop \in \Z} r_j 16^j$

where:

$\forall j \in \Z: r_j \in \set {0, 1, \ldots, 15}$

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:

 $\ds 10$ $:$ $\ds \mathrm A$ $\ds 11$ $:$ $\ds \mathrm B$ $\ds 12$ $:$ $\ds \mathrm C$ $\ds 13$ $:$ $\ds \mathrm D$ $\ds 14$ $:$ $\ds \mathrm E$ $\ds 15$ $:$ $\ds \mathrm F$

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 does not look as good in proportional font.

## Also known as

The abbreviation hex is common, particularly in the field of computer science.

## Examples

### Example: $\mathrm C 8$

The integer expressed in decimal as $200$ is expressed in hexadecimal as $\mathrm C 8$.

### Example: $12 \mathrm C$

The integer expressed in decimal as $300$ is expressed in hexadecimal as $12 \mathrm C$.

### Example: $2 \mathrm C 8$

The integer expressed in decimal as $712$ is expressed in hexadecimal as $2 \mathrm C 8$.

### Example: $6 \mathrm C 5$

The integer expressed in hexadecimal as $6 \mathrm C 5$ is expressed in decimal as $1733$.

### Example: $2 \mathrm B 7 \mathrm E$

The integer expressed in hexadecimal as $2 \mathrm B 7 \mathrm E$ is expressed in decimal as $11 \, 134$.

## Historical Note

Hexadecimal has often been suggested as a base for a new cunting system.

Augustus De Morgan, in his $1872$ work A Budget of Paradoxes, reported that John Williams Nystrom proposed in $1862$ a completely new number system where the digits from $0$ to $15$ be:

Noll, An, De, Ti, Go, Su, By, Ra, Me, Ni, Ko, Hu, Vy, La, Po, Fy

while $16$ be given the name Ton.

It was to follow that Ton-an, Ton-de, etc. were to be used for $17$, $18$, etc.

The number which in the system has the symbol:

$28(13)5(11)7(14)0(15)$

was to be pronounced:

Detam-memill-lasan-suton-hubong-ramill-posanfy.

David Wells also mentions this proposed system in his Curious and Interesting Numbers of $1986$.

The system was cumbersome, and too arbitrary to catch on, and little note was taken of hexadecimal notation until the age of computing, at which time the current more streamlined and intuitive convention was adopted.

Hexadecimal notation, like binary notation, has 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.