Series 3. What the Hell is Binary, Octal, and Hex?

A SYLVANUS DONAIE PUBLICATION

A NEW APPROACH TO TEACHING GROWN FOLKS

THE DIGITAL CONCEPTS +

3 5 4 12d

3 + 5 4 140

+

3 5 4 Ah

Series Three What the Hell is Binary, Octal, and Hex? Written by Sylvanus Donaie BA, BSEET, MTM, MPM, MBA

Building Bridges

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A NEW APPROACH TO TEACHING GROWN FOLKS

THE DIGITAL PROCESS Written by

SYLVANUS DONAIE About the Author

Since 2001, Professor Donaie has been transforming adult students into frontiers of their career, creating qualified project managers, and electronics engineers and technicians with undergraduate degrees. Having taught at many colleges and universities, he takes into his classrooms, labs and online platforms a broad range of experiences in electronics, computer, networking, refrigeration, and mathematics.

Sylvanus engineered and managed many key projects including designing XM Satellite Radio Terrestrial Repeater System and wireless/land-based system for Level 3, BellSouth, Fluor Daniel, AT&T, Verizon, Sprint and Time Warner. As a Training and Development Engineer, he led the development of the Process Safety Management (PSM) objectives for many refineries, creating training guides, operation manuals, maintenance manuals, operating procedures, self-study questions, PFD, and unit evaluations on crude units, fluid catalytic cracker plants, vacuum units, alkylation units, coker plants, refrigeration units, sulfur recovery and gas plants, and wastewater treaters.

I TRAIN AND DEVELOP PEOPLE Book and Cover design by Sylvanus Donaie

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Twelve Part Series

Series 1. Introduction.

Series 2.

A Paradigm Shift – The Numbering System.

Series 3. 3. What the the Hell Hell is Binary, Octal, and Hex? Series What is Binary, Octal,

and Hex?

Series 4.

Binary in an Electrical Representation.

Series 5.

Having a Logical Mind - Logic Gate.

Series 6.

Marching Lock Step – Counters and Clocks.

Series 7.

A Surprise of Addition – Adders and Comparators

Series 8.

Waving the Flags - Encoding and Decoding.

Series 9.

The Parking Lot - Addressing and Storage.

Series 10. The Bus System – Multiplexing/Communication.

Series 11. The Manipulator – Microprocessor/Microcontroller.

Series 12. What’s Next.

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SERIES THREE

CONTENTS

Series Introduction Binary Number System Octal Numbering System Hexadecimal Numbering System Coming Up in Series Four Series Summary MATHEMATICS FOR GROWN FOLKS Remember the Metric System APPENDIX A - CONVERSION TABLE Decimal - Binary - Octal - Hex

“The ultimate value of life depends upon awareness and the power of contemplation rather than upon mere survival.”

Aristotle Page 4

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SERIES THREE

What the Hell is Binary, Octal, and Hex? In order to achieve a greater understanding of the functionalities of digital systems, it is absolutely necessary to learn binary, octal and hex. Even when it comes to decimal, one must be able to work with it in any numbering system.

Series Two focused primarily on the makeup of the decimal numbering system. We learned that the decimal system consists of ten digits from 0 to 9 (0, 1, 2, 3, 4, 5, 6, 7, 8, 9). It works with the principle of grouping in tens (Base 10). Today, the world accepts and utilizes the decimal system for counting and mathematical calculations.

In this chapter, we will take a look at other systems, which make the world of digitalization not only possible but practical. We will expand our know-how on the binary system as well as octal and hexadecimal (hex). They are the backbone of our modern computerized world.

The parallels of a decimal system are very similar to that of binary, octal, and hex, but with one exception; they operate on different bases: Binary (Base 2), Octal (Base 8), and Hexadecimal (Base 16).

Ending Series Three, we will segue into the metric system. It brings us to understand and define new lines of vocabulary, which simplify the complexity of digitalization. Words, such as terabytes, megahertz, millisecond, and kilovolt, derives from metric. What are the definitions? Stay tuned and learn!

Yes, I know. By now you are questioning yourself? What the hell? Why should I know these numbering systems? Well, it amounts to two things: expertise in the digital world and future dollars in your pocket.

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Binary Number System The binary system functions on a Base 2 system; it uses only two digits: “0” (zero) and “1” (one). The numbers (2, 3, 4, 5, 6, 7, 8, 9) do not exist in binary. Simply put, binary works on a system of grouping in twos where ONLY (0 and 1) happens.

Binary – Grouping in Twos Let me explain. In Figure 1a, we group the peaches in two. The outcome is (1) group of two and (0) remaining. The binary result is 10. Likewise, in Figure 1b, we round (1) group of two and (1) remain. The binary result is 11. The conversion table (See Table 1) shows that the number (3) in decimal equates to (11) in binary, and number (5) in decimal equals (101) in binary.

1 group of two 0 remain Result in Binary = 10 Figure 1a

1 group of two 1 remain Result in Binary = 11 Figure 1b

Decimal

Binary

0

0

1

1

2

10

3

11

4 5

100 Table 1

101

Getting “HIGH” and “LOW” Today, manufacturers are applying digital technologies in a wide range of systems: computers, HVAC, cars, machines, communication systems, military systems, instruments, process controls, and televisions. Digital systems only work in a Base-2 system, which is known as the binary numbering system.

So, why is the binary the backbone of digital technologies? For one reason, binary system uses two digits either “1” or “0,” and electricity exists in two states mode: “ON” and “OFF.” For these characteristics, we are able to use electricity to easily replicate a twostate of “1” or “0.”

Usually, the “ON” mode represents a binary “1,” which is called a HIGH, and the “OFF” mode represent a binary “0,” which is called a LOW. See! Page 6

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CHECKING FOR LEARNING

Suppose an electrical switch turns a lamp on in the following manner (“ON” “OFF” “ON” “ON” “OFF”), how would a receiver interpret the message in binary?

Interpreting a Binary Code Suppose an electrical switch turns a lamp on in the following manner: (“ON” “OFF” “ON” “ON” “OFF”), how would a receiver interpret the message in binary? Simple, it’s (1 0 1 1 0). And each one of these digits is called a bit. Together, we consider the code (1 0 1 1 0) as a 5-bit message.

A sender takes a message, translate into binary codes, and transmit to the receiver. The receiver decodes the message to extract its meaning. However, in order to exert any viable form of communication, the sender and the receiver must predefine their interpretation of the binary codes. In fact, decoding is all about the understanding of what is already known.

Binary codings exist in many standards. Earlier ones include the 5-bit BAUDOT code.

Computers represent printed characters in an 8-bit code called ASCII (American Standard Code for Information Interchange) code. Binary Coded Decimal (BCD) assigns a four-digit binary code to each decimal numbers (0-9). There are also many proprietary ones with only owner’s rights to use. New computers operate on 32-bits and 64-bits message systems.

We shall return to our subject on the binary system. For now, let us all say, “WOW.” Why? I dug and found a 5-bits BAUDOT code chart. For years, telegraph companies transmitted electronic messages using the 5-bit BAUDOT code (see Figure 5) with teletype machines.

The chart in Figure 5 illustrates that the code (1 0 1 1 0) represents “F” in the LETTERS case.

Figure 2: The 5-bit BAUDOT Chart

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Binary Addition Previously, we studied the decimal system, which function on groups of tens. Now let’s examine the paradigm shift. In a very similar manner, the binary system works in groups of twos. Figure 3 illustrates a simple additions of binary numbers. Adding binary “1 + 1” equals “10”. Grouping in two, one would derive at one group of two and zero remaining.

Hence the result is a binary “10”. Likewise, binary “1 + 1 + 1” results in “11” because one group of two and one (1) remaining.

+

1

1

+

1 group of two 0 remain Result = 10

1 10

1 1 11

1 group of two 1 remain Result = 11

Figure 3: Addition of Binary Decimal

Binary

Decimal

Binary

0

0

21

10101

1

1

22

10110

2

10

23

10111

3

11

24

11000

4

100

25

11001

5

101

26

11010

6

110

27

11011

7

111

28

11100

8

1000

29

11101

Table 2: Counting in BINARY

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9

1001

30

11110

10

1010

31

11111

11

1011

32

100000

12

1100

33

100001

13

1101

34

100010

14

1110

35

100011

15

1111

36

100100

16

10000

37

100101

17

10001

38

100110

18

10010

39

100111

19

10011

40

101000

20

10100

41

101001

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Octal Numbering System

The octal uses a system very similar to counting in the decimal except for the number (8) and (9); they are nonexistent. The octal number system operates in Base-8, and consists of eight digits (0) to (7), naming numbers (0, 1, 2, 3, 4, 5, 6, 7).

Octal – Grouping in Eights In the octal system, it simply means that we are grouping in eights. Let’s see. In Figure 4, there are eight apples (based on the decimal) If we group them in eight, the result equals (1) group of eight and (0) remain. In octal, the number is (10).

1 group of eight 0 remain Result = 10 Figure 4: Grouping in Eights

Figure 5 shows another example. This time, there are twelve apples (based on the decimal system) If we group them in eight, the resulting number equals (14): (1) group of eight and (4) remaining. The number (12) in decimal equates to (14) in the octal system.

1 group of eight 4 remain Result = 14

Decimal

Table 3: Counting in OCTAL

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Figure 5: Grouping in Eights Octal

Decimal

Octal

0

0

11

13

1

1

12

14

2

2

13

15

3

3

14

16

4

4

15

17

5

5

16

20

6

6

17

21

7

7

18

22

8

10

19

23

9

11

20

24

10

12

21

25

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Hexadecimal Numbering System

So far so good: we can certainly consider ourselves to have some expertise in decimal system (Base 10), binary (Base-2), and octal (Base-8). And now we are on track to study hexadecimal; it is shortened to just hex.

Hex works on a Base-16 system, which means that it represent numbers using a set of sixteen symbols. But, how is this accomplish? Let see! If I start to count in hex, it goes like:

(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, _, _, _, _, _, _, 10, 11, 12…)

Notice the blank spaces; six symbols are missing. Unlike binary, octal, and decimal, hex makes use of the alphabet (A, B, C, D, E, F) to fill the spaces. As a result, counting in hex appears in this order:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, 10, 11, 12 13, 14, 15, 16, 17, 18, 19, 1A, 1B, 1C, 1D, 1E, 1F, 20, 21, 22…

The sequence may appear strange, but it’s just a new concept. In fact, computer programmers find programming most microprocessors or microcontroller in machine language (We know it as the binary number system) a daunting task. They used hex because it is more efficient to use. In the Series 4, I will show you the easiest way to navigate between these systems.

Convert Decimal (16) to Hex Let’s take a turn into the makeup of hexadecimal. Hex is a Base-16 system. As its base implies, it simply mean working in groups of sixteens. Figure 6 shows apples in 1 group of sixteen and 0 remaining. Therefore, the number of apples is 10 in hexadecimal.

1 group of sixteen 0 remain Result = 10

Figure 6: Grouping in Sixteen - Hex

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Convert Decimal (24) to Hex Figure 7 illustrates another example of counting in hexadecimal. This time, there are 24 apples (based on the decimal system). Grouping in sixteen, the resulting number equals (18) in hex. There is (1) group of sixteen and (8) remain. Therefore, the number (24) in the decimal system equates to an (18) hex.

1 group of sixteen 8 remain Result = 18

Figure 7: Grouping in Sixteen - Hex

Convert Decimal (28) to Hex Analyzing the layout in Figure 8 is a little tricky since it requires counting in hexadecimal.

Now, remember the count in the hex system (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F). So, let’s explore! We have (1) group of sixteen and (C) remain. Note: (C) in hex is equivalent to (12) in decimal. Then, the number is (1C) in hex.

1

5

8

2

6

A

3

7

B

4

8

C

Figure 8: Grouping in Sixteen - Hex

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1 group of sixteen C remain Result = 1C

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Identification for Numbering Systems

It is important to know that (10) in decimal is not equal to (10) in binary, nor (10) in octal, nor (10) in hex. Moreover, it is equally important to know the applicable numbering system of a number: its interpretation is critical. For this reason, engineers and technicians add a letter to the number to indicate its numbering system.

A letter (either D, B, O, or H) is placed after a number to identify its numbering system. Use as follows: decimal - (D), binary - (B), octal - (O), and hexadecimal - (H). If the number has no letter, the user understands it as a decimal number. The letter works in either upper or lower case. Later in this publication, you will learn that to program microprocessors, the programmer must identify number as either decimal, binary, octal or hex.

Number System

Identification

Base

Decimal

d

X10

10d

10d

Octal

o

X8

10o

8d

Binary Hex

b

h

X2

X16

INTERPRETATION

10b

10h

Table 4: Identify Number Systems

2d

16d

APPENDIX A contains a conversion table of decimal, binary, octal, and hex to help navigate between the numbering systems.

Note: It’s a good idea to memorize the conversion of decimal to binary between (0) to (16). A quick reference in digital analysis!

Coming Up in Series 4

Yes, we have completed another milestone. Hopefully, you are able to understand the theories behind the numbering systems. Believe me! Binary, octal, and hex are the foundation of digital electronics.

In Series 4, you will learn how to interact within the framework of octal and hexadecimal as relating to binary. Octal and hex represent a compressed form of binary. You will also learn my simple routines of converting numbering system without dipping in difficult mathematical calculations.

I will introduce new words such as bits, nibble, bytes, word, double-word, and quadword. These terminologies will begin to take you into the underground world of digital technologies, one bit at a time.

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SUMMARY The similarities in the foundation of a decimal system are similar to that of binary, octal, and hex, but with one exception. They operate in different base: Binary (Base 2), Octal (Base 8), and Hexadecimal (Base 16).

These digital systems only work in a Base 2 system, which is known as the binary numbering system.

Binary system uses two digits either “1” or “0,” and electricity exists in two states mode: “ON” and “OFF.” For these characteristics, we are able to use electricity to easily replicate a state of “1” and another for “0.”

The octal number system operates in Base-8, and consists of eight digits (0) to (7), naming numbers (0, 1, 2, 3, 4, 5, 6, 7). Number (8) and (9) are nonexistence in octal.

Hex works on a Base-16, which means that it represents numbers using a set of sixteen symbols. Six symbols are added: (A, B, C, D, E, F).

Counting in hex: (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, 10, 11, 12…)

Email your questions; Ask me anything about DIGITAL ELECTRONICS sylvanus.donaie@gmail.com I TRAIN AND DEVELOP PEOPLE

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MATHEMATICS FOR GROWN FOLKS Remember the Metric System

Who remembers the catastrophic days when the United States attempt to implement the metric system? If you do, you just revealed your age. But, I will stick to this; I was a long time ago. But better yet, do you know that the metric system has been legal in this country since 1866? Implementation of metric is a battle yet to be won.

The U.S. is the only industrialized nation that has not fully converted to the metric system. However, many industries in the scientific, manufacturing, and medical world use metric units almost exclusively.

In the electrical and electronics fields, all measurements are taken using metric, both analog and digital system. A major advantage of the metric system in our field is a simplified method to work with very large numbers and very small fractions and decimals.

For instance, instead of 400,000 volts, we use 400 kilovolts.

Let take some time to learn and review the basics of metric as it relates to electronics, beginning with scientific notation and engineering notation. These are the best means to express numbers that are either too small or too big.

Scientific notation 1240

1 . 2 4 0.

3

2

1

(Moving decimal point 3 place to the left)

1240

1 2 . 4 0.

2

1

(Moving decimal point 2 place to the left)

Scientific notation = 1.240 x 103 Engineering notation = 1.24E3 Scientific notation = 12.40 x 102 Engineering notation = 1.24E2

al cim 3 delace p al cim 2 delace p

Working with Large Numbers In electrical and electronics, it is a common practice to move the decimal point in multiple of 3 to the left: 103, 106, 109, and 1012.

Number

8,344 68,329 4,200,000 7,620,000,000

Scientific Notation

8.344 x 103 68.320 x 103 4.2 x 106 7.62 x 109

Engineering notation

8.344E3 68.320E3 4.2E6 7.62E9

Table 5: Metric System - Large Number

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Working with Small Numbers Move the decimal point in multiple of 3 to the right: 10-3, 10-6, 10-9, and 10-12.

Number

0.853 0.000074 0.000000002 0.1

Scientific Notation

853 x 10-3 74 x 10-6 2 x 10-9 100 x 10-3

Engineering notation

853E-3 74E-6 2E-9 100E-3

Table 5: Metric System - Small Number

We will continue with the metric system in Series 4.

PRACTICE SCIENTIFIC AND ENGINEERING NOTATION Number

83,201 0.0000638

Scientific Notation

3.868×109 670,400 7,070,000,000

4 × 103 5.06 × 10-6

Page 15

Engineering Notation

9.46E-6

Scientific Notation

83.201x103 63.8x106 3.868x109 9.46x10-6 4 × 103 0.6704x106 7.07x109 5.06 × 10-6

Engineering Notation

83.201E3 63.8E6 3.868E9 9.46E-6 4E3 0.6704E6 7.07E9 5.06E-6

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Number

83,201 0.0000638 3,868,000,000 0.00000946 4000 670,400 7,070,000,000 0.00000506

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APPENDIX A

CONVERSION TABLE: DECIMAL, BINARY, OCTAL, AND HEX. Decimal

Binary

Octal

Hex

Decimal

Binary

Octal

Hex

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

0 1 10 11 100 101 110 111 1000 1001 1010 1011 1100 1101 1110 1111 10000 10001 10010 10011 10100 10101 10110 10111 11000 11001 11010 11011 11100 11101 11110 11111 100000 100001 100010 100011

0 1 2 3 4 5 6 7 10 11 12 13 14 15 16 17 20 21 22 23 24 25 26 27 30 31 32 33 34 35 36 37 40 41 42 43

0 1 2 3 4 5 6 7 8 9 A B C D E F 10 11 12 13 14 15 16 17 18 19 1A 1B 1C 1D 1E 1F 20 21 22 23

36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71

100100 100101 100110 100111 101000 101001 101010 101011 101100 101101 101110 101111 110000 110001 110010 110011 110100 110101 110110 110111 111000 111001 111010 111011 111100 111101 111110 111111 1000000 1000001 1000010 1000011 1000100 1000101 1000110 1000111

44 45 46 47 50 51 52 53 54 55 56 57 60 61 62 63 64 65 66 67 70 71 72 73 74 75 76 77 100 101 102 103 104 105 106 107

24 25 26 27 28 29 2A 2B 2C 2D 2E 2F 30 31 32 33 34 35 36 37 38 39 3A 3B 3C 3D 3E 3F 40 41 42 43 44 45 46 47

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A TRAINED WORKFORCE

“A company with a robust training program converts good employees into great ones. By investing in their continued knowledge of your product and services, you are ensuring a greater likelihood of your employee delivering the best of a company’s offering with each customer interaction.” Attorney Latania Mena A trained workforce has long been a mark of distinction for the superiority of a successful business.

Employees are the frontline leaders for a business similar to soldiers first in line for battle. Whether dealing directly with the customer, producing the goods, or delivering the services, they are the most important representative of the business. The quality of the employee interactions and production solidly translate to the success of the business.

If you don’t believe, try the alternatives: bitchy employees, daunting processes, and intimidating supervisors.

These are tantamount to high turnover rate, low-value products, and services that suck -- the signature of hidden high expenses. Customers move to competitors resulting in loss of revenues and eventually business shutdown. Now, ask yourself, “Is training and development a good trade-off?”

In the current climate, many employers find the training and development of their workforce expensive. In fact, they budget little on training. Now think about it. A trained workforce has long been a mark of distinction for the superiority of a successful business.

• • • • • •

Compare the significant benefits to the marginal costs.

Build strength within the organization.

Address workforce weaknesses.

Ensure consistency and compliance.

Reduce customer complaints and return goods.

Increase accuracy as well as productivity.

Develop the quality brand standards.

• • • • • •

Increase innovation and creativity.

Cut employee turnover.

Enhance company brand.

Improve employee performance.

Improve employee satisfaction and morale.

Result in happy people.

Today, technologies are ingrained into our lives; it’s just the era that we live in. It changes so fast that continuous training must be part of the job to keep up. I design these series in Digital Electronics to help grown folks who see digital a threat to their livelihood. Others can benefit as a refresher or new information.

Email your questions; Ask me anything about DIGITAL ELECTRONICS sylvanus.donaie@gmail.com I TRAIN AND DEVELOP PEOPLE