# Definition:Hexadecimal Notation

## 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

**Hexadecimal notation** can also be referred to as **hexadecimal representation**.

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

## Examples

### Example: $4 \mathrm B$

The integer expressed in decimal as $75$ is expressed in hexadecimal as $4 \mathrm B$.

### 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 $11 \, 134$.

## Also see

- Results about
**hexadecimal notation**can be found**here**.

## 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.

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $16$ - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**hexadecimal** - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $16$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**hexadecimal system** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**hexadecimal system** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**hexadecimal representation**