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BINARY AND HEX
Binary and Hex
The Arithmetic of Computing
by
Carol Brooksbank
It might seem odd to start an article
about binary and hex by
talking about decimal arithmetic,
but the processes involved
in all three systems are the
same. Only the number of digits
used is different. The trouble is
that we are all so expert in using
decimal counting that we no
longer think about the
mechanics of what we are doing.
Almost everyone can count
from 1 to 1 00 without worrying
about the processes involved,
and even someone with no head
for figures can add 1 to 99 and
get the right answer without a
second thought about what
they did. They just know how to
doit. But if you understand what
is happening when you count in
decimal, you are half way to
becoming fluent in binary and
hex, so t am going to analyse
decimal counting. Please don't
skip this bit just because it
seems very elementary. You will
get to grips with the other
systems much more easily if you
read on.
Decimal
In decimal arithmetic, there are
1 0 digits which are always used
in the same order when counting:
0123456789
Because of the need to use
numbers higher than 9, a col
umn system is used, Imagine the
digits arranged in a vertical column.
When 9 is reached, we
have run out of digits, so a second
column is brought in to the
left of the first one, and this new
column is assumed to hold 0. To
progress beyond 9,1 is added to
the amount in the new column,
and the original one is returned
to 0 . This gives the number we
call ten.
0
1
2
3
4
5
6
7
8
09
10
You can then count on, until you
reach 9 again in the right hand
column, when you repeat the
process, adding 1 to the left column
and returning the right one
toO.
19
20
This process can continue until
99 is reached, when you need to
bring in another column to the
left.
98
099
Now, as before, you add 1 to the
leftmost column and return all
the others to 0.
099
100
You can go on like this for as
long as you need. Every time you
add 1 to a column, you return all
those to the right of it to 0, and
every time all the columns in use
reach 9, you bring in another to
the left. In this way, any
number, as big or small as you
wish, can be expressed.
Because there are 1 0 digits in
use, each new column is worth
1 0 times the one on its right.
The number 548,672 is actually
made up like this:
100,000 x 5 500,000 +
10,000 x 4 = 40.000 +
1,000 x 8 = 8,000 +
100 x 6 = 600 +
10x2= 20 +
1x7= 7
TOTAL 546827
This columns system is exactly
the same in binary and hex, so
the only new factor is the
number of digits.
Binary
In binary arithmetic, there are
only 2 digits: 0 and 1 .As before,
you count in one column until
you run out of digits, when you
introduce a new one to the left.
The difference is that you run
out of digits much more quickly
in binary.
Binary Decimal
0 0
01 1
10 2
The number reached at this
point in binary, 10, is not ten. It
is one-oh binary. Ten is a name
reserved for one-oh decimal,
and in order to avoid any confusion,
you should always think of
the binary numbers as one-oh
binary, one-one binary and so
on.
You carry on counting in
binary in exactly the same way
as in decimal, introducing new
columns as they are needed.
11 3
100 4
101 5
110 6
111 7
1000 8
You will see by now the great
disadvantage of binary. With so
few digits the numbers quickly
become longer, so that the
number expressed as '8' in
decimal is a four figure number
in binary. In fact, because the
numbers are long and in order to
reduce confusion between
systems, it is usual to express all
binary numbers up to 255
decimal as eight figure numbers:
8 decimal BIN 00001000
255 decimal - BIN 111 11111
As there are ohly 2 digits, each
column in binary is worth twice
the column to its right.
128 64 32 16 8 4 2 1
1 1 1 1 1 1 1 1
To convert a BIN number to
decimal, you work as follows:
BIN 1 111 1 11 1 -
128 X 1 =
64 X 1 =
32 X 1
128 +
64 +
32 +
16 X 1 = 16 +
8 X 1 = 8 +
4 X 1 = 4 +
2 X 1 = 2
1 X 1 = 1 +
TOTAL 255
In binary, just as in decimal, you
can introduce as many columns
as you need to express a number
of any size. Take the decimal
number 1 2345, for instance. In
binary, this is
001 10000001 1 1001
Since binary is such a cumber
some system with long awkward
numbers, what use is it?
Well, in ordinary everyday life,
not a lot, but in the world of electronics,
it has one great advantage,
Because there are only
two digits, it is possible to express
the numbers electronical-
iy-
Imagine that you have a bank
of eight electric wires, each
equipped with a switch. If you
switched the switches so that 0
was represented by an inert wire
(switch off), and 1 was
represented by a live wire
(switch on), you could express
the binary form of any number
between 0 and 255 decimal.
Put very crudely, this is what
happens inside your computer.
You can think of each address in
your computer's memory as a
minute bank of 8 electric wires
and switches. When you poke a
number into an address, the
switches are manipulated so
that the pattern represents the
binary form of your number. The
particular combination of live
and inert wires causes the computer
to perform some operation,
It is beyond the scope of
this article to go any further into
the workings of the computer,
but I want you to understand the
importance of binary arithmetic
54 ZX COMPUTING DECEMBER/JANUARY 1985