The hardware - The Computing Universe: A Journey through a Revolution - page 25

expressions in Boolean algebra. Relays are switches that can be closed or open
corresponding to a state being logical true or false, respectively. This means
that any function that can be described as a correct logical statement can be
implemented as a system of electrical switches. Toward the end of his thesis,
Shannon points out that true and false could equally well be denoted by 1 and
0, respectively, so that the operation of the set of switches corresponds to an
operation in binary arithmetic. He wrote:

Fig. 2.2. Two relay switches in series
serving as an AND gate. The current can
only flow if both relays are closed.

It is possible to perform complex mathematical operations by means of
relay circuits. Numbers may be represented by the position of relays and
stepping switches, and interconnections between sets of relays can be made
to represent various mathematical operations. 6

As an example, Shannon showed how to design a relay circuit that could add
two binary numbers. He also noted that a relay circuit can make comparisons
and take alternative courses of actions depending on the result of the compar-
ison. This was an important step forward, since desktop calculators that could
add and subtract had been around for many years. Shannon's relay circuits
could not only add and subtract, they could also make decisions.
Let us now look at how Shannon's insights about relay circuits and Boolean
algebra transformed the way early computer builders went about designing
their machines. Computer historian Stan Augarten says:

Shannon's thesis not only helped transform circuit design from an art into
a science, but its underlying message - that information can be treated like
any other quantity and be subjected to the manipulation of a machine - had a
profound effect on the first generation of computer pioneers. 7

B.2.3. Augustus De Morgan (1806-71)
is known for his pioneering work in
logic, including the formulation of
the theorem that bears his name.

But before we take a deeper look at relay circuits and Boolean algebra, we
should say a few words about binary arithmetic.

Binary arithmetic, bits, and bytes

Although some of the early computers - like the ENIAC - used the familiar
decimal system for their numerical operations, it turns out to be much simpler
to design computers using binary arithmetic. Binary operations are simple, with
no need to memorize any multiplication tables. However, we pay a price for
these easy binary operations by having to cope with longer numbers: a dozen is
expressed as 12 in decimal notation but as 1100 in binary ( B.2.4 , Fig. 2.4 ).
Mathematics in our normal decimal system works on base 10. A number is
written out in “positional notation,” where each position to the left represents
10 to an increasing power. Thus when we write the number 4321 we under-
stand this to mean:

4321 = (4 × 10 3 ) + (3 × 10 2 ) + (2 × 10 1 ) + (1 × 10 0 )

Here 10 0 is just 1, 10 1 is 10, 10 2 is 100, and so on; the numbers of powers of ten
in each field are specified by digits running from 0 to 9. In the binary system,
we use base 2 instead of base 10, and we specify numbers in powers of two and
use only the two digits, 0 and 1. Thus the binary number 1101 means:

Fig. 2.3. Two relay switches in parallel
equivalent to an OR gate. The current
can flow if at least one of the relays is
closed.