AAB - Octal and Hex Numbering (Lesson)
Octal and Hex Numbering
Introduction
The other two systems of translation shown in the original table are octal and hex. These interpretations allow for much bigger numbers.
Octal System
he octal system is based on base 8. The table changes to show bases of 8, but the method of use is the same. As you can see, the place value becomes very large quickly. Note that the numbers allowed in the "How Many" row is one less than the base, the natural numbers 0 - 7.
|
|
Octal Numbering System - Base 8 |
|
||||||
|---|---|---|---|---|---|---|---|---|
|
Place Value |
2,097,152's |
262,144's |
32,768's |
4,096's |
512's |
64's |
8's |
1's |
|
Baseexponent |
87 |
86 |
85 |
84 |
83 |
82 |
81 |
80 |
|
How Many |
0 |
0 |
0 |
1 |
0 |
5 |
2 |
6 |
|
Math |
0 * 2097152 |
0 * 262144 |
0 * 32768 |
1 * 4096 |
0 * 512 |
5 * 64 |
2 * 8 |
6 * 1 |
|
Add these up |
0 |
0 |
0 |
4096 |
0 |
320 |
16 |
6 |
The answer is the addition 4096 + 320 + 16 + 1 = 4438 or 4, 438.
Hex System
The hexadecimal numbering system, hex for short, is based on base 16. The table changes to show bases of 16. How, you say, do we get to 16 in one place holding? The letters A through F are used to represent the two digit numbers 10 - 15 as shown below.
|
Decimal |
Letter Used |
|---|---|
|
10 |
A |
|
11 |
B |
|
12 |
C |
|
13 |
D |
|
14 |
E |
|
15 |
Here the place values grow even more quickly.
|
|
Hexadecimal Numbering System - Base 16 |
|
||||||
|---|---|---|---|---|---|---|---|---|
|
Place Value |
268,435,456's |
16,777,216's |
1,048,576's |
65,536's |
4096's |
256's |
16's |
1's |
|
Baseexponent |
167 |
166 |
165 |
164 |
163 |
162 |
161 |
160 |
|
How Many |
0 |
0 |
0 |
0 |
0 |
5 |
D |
E |
|
Math |
0 * 268435456 |
0 * 16777216 |
0 * 1048576 |
0 * 6553 |
0 * 4096 |
5 * 256 |
B * 16 |
E * 1 |
|
Add these up |
0 |
0 |
0 |
0 |
0 |
1280 |
B=11 11*16 = 176 |
E=14 14*1 = 14 |
The answer is the addition 1280 + 176 + 14 = 1470 or 1,470 for readability.
Thus when new symbols or larger numbers were to be transferred, a rewrite of the ASCII codes were not needed. The base on which the original solution was founded, base 2, could be adapted through program interpretation to allow for symbols and letters not initially in the set without. The only change was the adaption of the new decimal number for the new symbol. Abstraction at its best.
The Hexadecimal Numbering System