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to the computer world.
Whatever form of counting you
use to program your computer,
decimal, hex or binary, it is the
binary form which is important,
and the binary form is the one
stored in the memory. We use
the other forms because binary
is so unwieldly for we humans,
so we let the computer do the
conversion.
As each address, or byte, in
the computer's memory can only
hold 8 binary digits, bits, what
happens if you wish to store a
number like 12345 decimal,
which has more than 8 bits? The
computer takes the 1 6 bit form
of the number, and cuts it in half
ie 00 11000000111001
becomes 00110000001110
01. It then stores each half
separately. The second half, the
low byte, is stored first, and the
first half, the high byte, in the
following address.
This is where decimal, which
is more familiar to humans,
becomes awkward to use in
relation to computers. The low
byte, BIN 0011 1001 , is 57
decimal and the high byte, BIN
001 1 0000, is 48 decimal. At
first glance there seems to be
very little connection between
57 and 48 and 12345. In fact,
you can find the decimal by
multiplying the high byte by 256
and adding the low byte.
256 x 48
ADD
TOTAL
= 12288
57
12345
This lack of an obvious connection
between a number and its
low and high bytes in decimal is
one of the reasons why many
machine code programmers
prefer to work in hex, which is
less unwieldly than binary and
more convenient than decimal
for programming. But before I go
on to talk about hex, I am going
to give you some homework. If
you plan todoany machine code
programming, you will save
yourself a lot of looking up in
tables if you learn by heart the
binary forms of the decimal
numbers 0 to 15, so try to
memorise this little table.
1 100 12
1101 13
1110 14
1111 15
Hex
In hex you have 16 digits
0,1 ,2,3,4,5,6,7 ,8,9,A.B,C,D,
E,F. The column system for
counting is used exactly as
before, but instead of bringing in
a new column after 9, as in
decimal, you carry on until you
reach F, and then introduce the
new column.
HEX
0
1
2
3
4
5
6
7
8
9
A
B
C
D
E
OF
10
BINARY AND HEX
DECIMAL
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Again, the number reached in
hex is not ten, but one-oh hex,
and is equal to 1 6 decimal. Hex
numbers are expressed as 1 OH
or 0010 H,
Each column in hex is worth
1 6 decimal times its right hand
neighbour. The number 12345
decimal is 3039h, and is broken
down as follows:
4096 256
3 0
4096 x3 -
16x3 =
1 x 9 =
4096
2
TOTAL
256
A
16
3
16
C
1
9
12288 +
48 +
9
12345
A hex number containing some
of the letter digits can be converted
as follows:
4096 x2 = 8192 +
BIN DECIMAL 256 x10 = 2560 +
0000 0 16x12 = 192 +
0001 1 1x15= 15
0010 2
001 1 3 TOTAL 10959
0100 4
0101 5 You will remember that we
0110 6 discovered earlier that the high
011 1 7 and low bytes of 12345 were
1000 8 48 and 57 respectively. We
1001 9 have since seen that the hex
1010 10 form of 12345 is 3039h. The
101 1 11 hex form of48dis30hand57d
1
is 39 h. You can see at a glance
what the high and low bytes of a
hex number are, without doing
any calculations. From this, it
follows that there is a direct con
Figure 7. Binary, hex and decimal comparisons
thumb on each hand — and
everyone, including primitive
man who invented counting,
learns to count on their fingers.
If we had three extra fingers per
BIN 001 1 = 03h = 3d
8IN0000 = OOh = Od
BIN001 10000 = 30h = 48d
BIN 001 1 = 03h = 3d
BIN 1001 = 09h = 9d
BIN 001 1 1001 = 39h = 57d
BIN 001 10000001 1 1000 = 3039h = 12345d
nection between the binary and
hex forms of a number which is
much more obvious than the
connection between the
decimal and any other form.
(See figure 1 ).
Clearly, this makes life much
easier for the machine code programmer.
By using hex he can
avoid all the calculations re
quired to find the high and low
bytes when working in decimal.
Also, when he comes to type in
the machine code listing, most
of the numbers are shorter than
the equivalent decimal ones so
there is a saving ifi time too. In
fact, after you have become
familiar with hex you begin to
wonder why decimal is the
universally popular system. I
suspect that is has something to
do with the fact that we are all
born with a built in decimal
calculator four fingers and a
hand I believe that we should all
be working away happily in hex,
and the genius who suggested
that you could have a very good
counting system with only ten
digits would be quietly led away
by men in white coats.
Meanwhile, computer programmers
need to familiarize
themselves with binary and hex.
You simply need to become as
expert with binary and hex as
you are with decimal, and practice
is the only way. You can
help yourself by learning another
conversion table (figure 2). It is
very similar to the first one, but
will allow you to convert between
the three systems. In fact,
you have very little extra to learn
because decimal and hex figures
are the same until you reach 10
decimal.
Can you do sums in hex?
ZX COMPUTING DECEMBER,JANUARY 1985 55