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October 1974 Popular Electronics
 Table of Contents
Wax nostalgic about and learn from the history of early electronics. See articles
from
Popular Electronics,
published October 1954 - April 1985. All copyrights are hereby acknowledged.
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This first of a three-part
digital electronics series run by Popular Electronics magazine begins with
introducing binary (base 2), octal (base 8), and decimal (base 10) number systems,
along with conversions between the types. Sure, this is probably old-hat to most
RF Cafe visitors, but there is always a new cadre of electronics enthusiasts entering
the field who appreciate the instruction. No matter how advanced digital electronics
gets, a fundamental understanding and fluency in binary arithmetic is essential
to success whether as a hobbyists or as a professional. If you cannot move effortlessly
between the various number formats (which also include
hexadecimal
and Gray code),
then continuing to Boolean algebra and
combinational
logic and sequential logic circuits will be as futile as hoping to study
circuit theory with knowing Ohm's law.
See Part 1,
Part 2,
and Part 3.
Basic Digital Logic Course - Part 1: Number Systems
 By Forrest M. Mims
The emergence of experimental digital IC projects has been so rapid that many
people tend to get lost amid strange-sounding names like "quad 2-input positive
NAND gate" and "BCD to 7 segment decoder/driver." Such terms describe the building
blocks of digital electronics. To provide an introduction to logic for beginners
and refresher information for more advanced experimenters, here is the beginning
of a Digital Logic Course series.
This first installment describes number systems and provides important background
information for the next two installments.
Number Systems. Early man was forced to count with small pebbles or knots on
a string when he wanted to inventory his possessions. As time went on, and perhaps
because he found it convenient to count with his fingers, man eventually devised
a number system with ten digits. This provided a far more convenient and versatile
counting system since, for example, the number 16 could be represented with merely
two digits rather than 16 pebbles or knots.
The comparatively recent development of electronic digital computers has revived
interest in number systems based on something besides the decimal A system based
on two digits is of particular importance in electronic digital computers. The reason
for this is that an electronic circuit can be made to occupy only one of two states:
on or off (saturated or cut off). Of equal significance is that any form of logic
statement can be reduced to contain only true and false assertions.
Since electronic circuits required to impalement true and false logic statements
are very simple, a computer can be designed based on a two-digit number system,
in which one digit corresponds to "true" and the other to "false ". The two-digit,
or base-two, number system is called the binary system and its digits, called bits
(for binary digits), are 1 and 0.
The Binary System
The easiest way to understand the binary system is to learn to count in binary
fashion. One basic rule governs counting in any number system: record successive
digits for each count until the count exceeds the total number of available digits;
then start a second column to the left of the first and resume counting.
Since the binary system leas only two digits, counting is very easy. You can
prove this to yourself by counting to the equivalent of the decimal number 10 in
binary. The binary of decimal 0 is 0. The binary of 1 is 1. Here the similarity
ends. To express 2 in binary, you must start a new column since both binary, bits
have been used in the first column. Hence, the binary of 2 is 10 (read one-zero
- not ten). Three is expressed as 11 (one-one) in binary, which uses up both binary
bits for the first two columns. So, a new column must be started for binary 4, which
becomes 100, while 5, 6, and 7 become 101, 110, and 111. With 8, represented by
the binary 1000, we must once again start a new column.
Binary Arithmetic
By learning how to count in binary, we have also derived three basic rules for
addition: (1) 0 + 0 = 0; (2) 0 +1 = 1; and (3) 1 +1 =10(1 + 1 = 0, carry 1). These
rules can be used to add any two binary numbers. For example, let us add 12 and
9 in binary:
Start with the right-hand column and add the least significant digit. Then continue
adding each successive column, working from right to left, finishing up with the
most significant bit:
Note in the above example that the addition of the two most significant bits
yielded a 0 with a 1 carry. A carry can also occur within the addition as in:
1011 + 1101 = 11000.
Converting Binary to Decimal
Binary numbers are fundamentally easy to work with. But how do you convert a
string of 1's and 0's to easily recognized decimal numbers? The process is simple,
using a technique known as "expansion." Each digit column of a decimal number corresponds
to a power of the base-10 to which it must be raised. Let us use the number 846
as an example:
A binary number can be expanded in the same way and converted into a decimal
number. For example, let us expand (10111)2. The subscript denotes the
base of the number system -in this case, 2 or "binary" - and helps in preventing
confusion. The expansion is as follows:
Since the position of each digit in a binary number determines the power of 2
invoked, it is easy to convert binary to decimal simply by assigning the decimal
equivalent to each column. A 0 in a column means that the column's power-of-2 decimal
equivalent is not invoked. Therefore, the decimal equivalents of all columns containing
a 1 are added to find the total decimal equivalent. Let us convert 10011 to decimal:
Manual binary arithmetic involving numbers containing more than three or four
bits is both tedious and cumbersome when you are accustomed to counting in a decimal
system. But an electronic computer can perform thousands of lengthy binary additions
in fractions of a second. This ability is vital to the success of digital computers
and calculators, since all arithmetic operations can be performed by addition or
its variations. Subtraction is the inverse of addition, while multiplication is
simply repeated additions and division is the inverse of multiplication.
These facts about addition are important because they mean that even the most
complicated arithmetic operations can be solved by addition. In practice, manual
arithmetic rarely invokes this process. After all, you would find it inconvenient
to multiply 641 by 197 if you had to write 197 times the number 641 and add the
columns. But an electronic computer does the equivalent of this in only a few milliseconds.
The Octal System. Sometimes binary numbers are condensed into other number systems
to further simplify computer processing. Since the binary system has only two digits,
it does not take long to accumulate a string of seemingly endless 1's and 0's. A
decimal number with only two digits, for example, requires five binary bits. A six-digit
decimal number requires 19 bits.
 Complicated binary numbers can be simplified
by dividing them into groups of three or four bits and encoding the results in other
number systems. Since the binary numbers for the decimal digits 0 through 7 form
groups of no more than three binary digits each, a long binary number can be reduced
to a third of its length by converting it to a base -8, or octal, number system.
You can use a table of octal numbers and their binary equivalents to convert
a long binary number such as 11101100001101 into octal. First, divide the number
into groups of three bits each, beginning with the least significant bit:
11 101 100 001 101.
Then assign the octal equivalent to each three-bit group, using the octal-to-binary
equivalents given in the table:
Hence, (11101100001101)2 equals (35,415)8. It is obvious
that the latter number is easier to process than the former.
Sometimes the base-16 (hexadecimal) number system is used to further simplify
long binary numbers. The hexadecimal technique requires that the binary number be
subdivided into groups of four bits each, again starting with the least significant
digit. The result is a hexadecimal number that is only a fourth the length of the
original binary number.
Next month, we will discuss logic concepts and circuits and demonstrate how logic
circuits can be combined to make a binary adder.
Posted September 16, 2019
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