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The hexadecimal ( not well known as sedecimal ) notation is used by
most machine language programmers when they talk about a number or address
in a machine language program. The hexadecimal numerical system is one
of all value-positional numerical systems.
Some assemblers let you refer to addresses and numbers in decimal (
base 10 ), binary ( base 2 ), or even octal ( base 8 ) as well as hexadecimal
( base 16 ) or just "hex" as most people say. These assemblers
do the conversions for you.
Each position within a hexadecimal value refers
to a base of sixteen.
Hexadecimal probably seems a little hard to grasp at first, but like
most things, it won't take long to master with practice.
The hexadecimal numerical
sytem uses 16 numbers.
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By looking at decimal ( base 10 ) numbers, you can see that each
digit falls somewhere in the range between zero and a number equal
to the base less one ( e.g. 9 ).
This is true of all number bases.
Binary ( base 2 ) numbers have digits ranging from zero to one
( which is one less than the base ). Similarly, hexadecimal numbers
should have digits ranging from zero to fifteen, but we do not
have any single digit figures for the numbers ten10
to fifteen10, so the first six letters of the alphabet
are used instead ( A, B, C, D, E, and F ).
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|
| Decimal |
Hexadecimal |
Binary |
|
0
|
0
|
00000000 |
|
1
|
1
|
00000001 |
|
2
|
2
|
00000010 |
|
3
|
3
|
00000011 |
|
4
|
4
|
00000100 |
|
5
|
5
|
00000101 |
|
6
|
6
|
00000110 |
|
7
|
7
|
00000111 |
|
8
|
8
|
00001000 |
|
9
|
9
|
00001001 |
|
10
|
A
|
00001010 |
|
11
|
B
|
00001011 |
|
12
|
C
|
00001100 |
|
13
|
D
|
00001101 |
|
14
|
E
|
00001110 |
|
15
|
F
|
00001111 |
|
16
|
10
|
00010000 |
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Let's look at it another way; here's an example
of how a base 10 ( decimal number ) is constructed:
| Base 10 decreasing powers: |
103
|
102
|
101
|
100
|
|
| Equals to: |
1,000
|
100
|
10
|
1
|
|
| Consider 4,569 ( base 10 ) |
4
|
5
|
6
|
9
|
|
|
Multiply each decimal digit in the 3rd row by
its positional value in the 2nd row, and put each multiplication
result into the rightmost column. At last sum up all multiplication
results within the rightmost column.
|
|
|
|
9
|
|
| |
|
6
|
* 10=
|
60
|
| |
5
|
*100=
|
500
|
|
4
|
*1,000=
|
4,000
|
|
|
= 4,569
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Now look at an example of how a base 16 ( hexadecimal
number ) is constructed:
| Base 16 decreasing powers: |
163
|
162
|
161
|
160
|
|
| Equals to: |
4,096
|
256
|
16
|
1
|
|
| Consider 11D9 ( base 16 ) |
1
|
1
|
D
|
9
|
|
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Multiply each hexadecimal digit in the 3rd row
by its positional value in the 2nd row, and put each multiplication
result into the rightmost column. At last sum up all multiplication
results within the rightmost column.
|
|
|
|
9
|
|
| |
|
13
|
*16=
|
208
|
| |
1
|
*256=
|
256
|
|
1
|
*4,096=
|
4,096
|
|
|
= 4,569
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Therefore, 4,569 ( base 10 ) = 11D9 ( base 16 ).
The range for addressable memory locations of a MZ normally is 0 -
65,535. This range is therefore $0000 - $FFFF in hexadecimal notation.
Usually hexadecimal numbers are prefixed with a dollar sign ( $ ).
This is to distinguish them from decimal numbers.
Let's look at some "hex" numbers, using the monitor subprogram
of the S-BASIC, by displaying the contents of some memory by typing:
B. ( to invoke the monitor subprogram of the S-Basic ) and next
type in:
D6BCF 6BE6. This will display the contents of the memory from
$6BCF to $6BE6 ( you can do this by the monitor program 1Z-013A too
but in this case omit the command "B.").
You'll see 3 rows of 9 hex numbers. The first 4-digit is the address
of the first byte of memory being shown in that row, and the other eight
numbers are the actual contents of the memory locations beginning at
that start address.
You should really try to learn to "think" in hexadecimal.
It's not too difficult, because you don't have to think about converting
it back into decimal. For example, if you said that a particular value
is stored at $10F0 instead of 4,336, it shouldn't make any difference.
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We've converted a hex number into a decimal number. Now we'll
convert a decimal number into a hex number.
To this, use the table at the right. The table shows the result
for each positional value.
E.g. $C000 is to compute by:
$C * 164 ( base 16 )
is adequate to
12 * 65,536 = 786,432 ( base 10 ).
You find the base 10 multiplication result 786,432 by crossing
the row for $C and the column for its positional value 164.
|
|
| Decimal |
Hex |
powers of 16
|
|
164
65,536
|
163
4,096
|
162
256
|
161
16
|
160
1
|
|
1
|
1
|
65,536
|
4,096
|
256
|
16
|
1
|
|
2
|
2
|
131,072
|
8,192
|
512
|
32
|
2
|
|
3
|
3
|
196,608
|
12,288
|
768
|
48
|
3
|
|
4
|
4
|
262,144
|
16,384
|
1,024
|
64
|
4
|
|
5
|
55
|
327,680
|
20,480
|
1,280
|
80
|
5
|
|
6
|
6
|
393,216
|
24,576
|
1,536
|
96
|
6
|
|
7
|
7
|
458,752
|
28,672
|
1,792
|
112
|
7
|
|
8
|
8
|
524,288
|
32,768
|
2,048
|
128
|
8
|
|
9
|
9
|
589,824
|
36,864
|
2,304
|
144
|
9
|
|
10
|
A
|
655,360
|
40,960
|
2,560
|
160
|
10
|
|
11
|
B
|
720,896
|
45,056
|
2,816
|
176
|
11
|
|
12
|
C
|
786,432
|
49,152
|
3,072
|
192
|
12
|
|
13
|
D
|
851,968
|
53,248
|
3,328
|
208
|
13
|
|
14
|
E
|
917,504
|
57,344
|
3,584
|
224
|
14
|
|
15
|
F
|
983,040
|
61,440
|
3,840
|
240
|
15
|
|
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| The decimal number 1,999 is to convert into its hex value. To
this we paint a table like this and compute each digit step by step. |
|
|
Decimal
number
|
Hexadecimal number
|
|
163
4,096
|
162
256
|
161
16
|
160
1
|
|
1,999
|
|
|
|
|
|
| |
| Now find a value in the first table that is less or equal to 1,999
( I've coloured all values to find during our conversion yellow
). You'll get 1,792 = 7 * 256 = 7 * 162. That means we
do not need the column 163 in our table for the result
of the conversion but we can put "7"
into the column 162. |
|
|
Decimal
number
|
Hexadecimal number
|
|
163
4,096
|
162
256
|
161
16
|
160
1
|
|
1,999
|
|
7
|
|
|
|
| |
| We've used 1,792 from 1,999, that means the remainder is 1,999
- 1,792 = 207. Now again we've to find a value in the first table
that is less or equal to 207. It is 192 = 12 * 161. We
put "C" into the column
161. |
|
|
Decimal
number
|
Hexadecimal number
|
|
163
4,096
|
162
256
|
161
16
|
160
1
|
|
1,999
|
|
7
|
C
|
|
|
| |
| The remainder is 207 - 192 = 15. 15 * 1 = 15 * 160.
We put "F" into the
column 160. $7CF is the hex value of 1,999 ( base
10 ). |
|
|
Decimal
number
|
Hexadecimal number
|
|
163
4,096
|
162
256
|
161
16
|
160
1
|
|
1,999
|
|
7
|
C
|
F
|
|
50,860
|
C
|
6
|
A
|
C
|
|
| |
|
|
| I've put one more conversion into the
table for the decimal number 50,860. Try to do the conversion for
this number, if you want. |
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You can test the correct value by converting back the hex value
into a decimal value: |
|
|
C
|
==>
|
12
|
*
|
4,096
|
==>
|
49,152
|
|
6
|
==>
|
6
|
*
|
256
|
==>
|
1,536
|
|
A
|
==>
|
10
|
*
|
16
|
==>
|
160
|
C
|
==>
|
12
|
*
|
1
|
==>
|
12
 |
| |
|
|
|
|
|
50,860
|
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It's important to know how to do the conversions like our knowledge
about multiplication, addition, substraction, and division. Sure, we
do all mostly by an electronic calculator. All these conversions can
be done by an electronic calculator too.
Be sure your browser is able to run JavaScript and Java and then
click here to try an electronic calculator written in the Java language.
It takes some time until all Java classes will be transferred to your
PC, please wait.
You can download this electronic calculator
now ( 72KB ) if you want.
That's it. Next will be to convert a binary number into a hex number
and vice versa.
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