Numeration Systems and Codes

8/6/2019 Numeration Systems and Codes

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Digital Electronics_Jean-Paul NGOUNE 1

Courses In

Electrical

Engineering

Volume II

DIGITAL ELECTRONICS

CHAPTER ONE : NUMERATION SYSTEMS AND CODES

By

J-P. NGOUNE

DIPET I ( Electrotechnics), DIPET II (Electrotechnics)

DEA ( Electrical Engineering)

Teacher in the Electrical Department, GTHS KUMBO, Cameroon.


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Chapter One

NUMERATION SYSTEMS

AND CODES

1.0 Specific objectives:

At the end of this chapter, the student will be able to:

- know binary, hexadecimal and octal numeration systems;

- Know Gray, BCD and ASCII codes;

- Master the principle of conversion from each numeration system to another.

1.1 Introduction:

Numbers are used to express quantities. There are many numerations

systems used in the field of digital electronics, one of the most important being the

binary system of numeration on which is based the computer science. Each of the

various numerations systems and codes has its advantages but also inconvenient.

The aim of this chapter is to present and explain the most common numeration

systems and codes used in the conception of digital circuits.

1.2 Digital versus Analogue representation:

There are two basic ways we can represent quantities: Analogue

representation and digital representation. With analogue representation, the quantity

is symbolised in a way that is infinitely divisible. With digital representation, the

quantity is symbolised in a way that is discretely packaged.

Example 1.1:

The height of the red column which indicates the temperature measured by a

thermometer is an analogue representation.

An electronic watch whose digits changes second after second, minute afterminute, shows a digital representation.


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The comparison between digital and analogue representations can be given as in the

following chart:

Analogue representation Digital representation

Infinitely divisible Discrete (Step by step)

Prone to errors of precision Absolute precision

1.3 Systems of numeration:

To represent quantities in the different systems of numeration, specific

symbols are used, which are also called ciphers.

1.3.1 Decimal numeration system:

Decimal system is the most common numeration system for daily uses. It is

constituted by 10 symbols or ciphers: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Each cipher

represents an integer quantity and each place from right to left in a decimal notation

represents a weight for each integer quantity.

Example 1.2:

Lets consider the decimal notation 1253. This number can be broken into its

constituent weight-products as such:

01231031051021011253

131051002100011253

35020010001253

+++=

+++=

+++=

We can easily notice that the cipher 1 is more weighted than the cipher 2

which in his turn is more weighted than the cipher 5. The cipher 3 is the lessweighted.

In the decimal numeration system, each cipher is called a digit. Each weight or

place value is ten that of the one to the immediate right. The less weighted cipher

carries the One place, the cipher at the immediate left carries the Tens place, the

follower carries the Hundreds place, thousands place, and so on


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1.3.2 Binary numeration system:

The binary numeration system uses only two ciphers instead of ten as the

decimal numeration system. Those two ciphers are and . In binary system of

numeration, ciphers are called bit (Binary Digit). Cipher are arranged right to left in

doubling values of weight ( instead of multiplying the weight by 10 as in the case of

decimal system).

Example 1.3:

Lets consider the following binary number

A = 1 0 1 1 0 12

10

012345

45

148032

212021212021

=

++++=

+++++=

A

A

A

Each weight is 2 that of the one in the immediate right. The less weighted

cipher carries the Ones place (20), the cipher at the immediate left carries the twos

place (21

), the following cipher carries the fourth place (22

Exercise 1.1:

Convert the following binary numbers to decimal numbers:

A = 110101 C = 11110111101

B = 100110101 D = 101100001111

5 4 3 2 1 0

Base 2

Weights


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1.3.3 Binary versus decimal numeration system:

Let us count from 0 to 15 using binary and decimal systems of numeration

Binary

D(MSB) C B A(LSB) Decimal

0 0 0 0 0

0 0 0 1 1

0 0 1 0 2

0 0 1 1 3

0 1 0 0 4

0 1 0 1 5

0 1 1 0 6

0 1 1 1 7

1 0 0 0 8

1 0 0 1 9

1 0 1 0 10

1 0 1 1 11

1 1 0 0 12

1 1 0 1 13

1 1 1 0 14

1 1 1 1 15

It is obvious that the representation of a quantity in binary numeration system

takes mores ciphers than in decimal system. We can therefore ask ourselves why the

binary system is preferred to decimal system in computer sciences. The reason is

that in electronics, it is easier to materialise two quantities- and -(by two

different voltages for example) than to materialises 10 different quantities ,

, and- (by 10 different voltages). In fact, in digital circuits, 0

and 1 are materialised by specific ranges of voltages or current; this will be discussed

later.

Remark 1.1:

With n bits we can represent n2 different binary numbers. The higher H number is

given using the following formula.

12 =n

H (1)


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Example 1.4:

With 4 bits we can represent 24 = 16 different binary numbers (from 0 to 15), and the

higher number is H = 24 1 = 15.

Remark 1.2: Conversion from binary to decimal

To convert a number written in binary numeration system to its equivalent in

decimal, we just have to calculate the products of the bits with their respective

weights, as in example 1.3 above.

For binary numbers with binary point (equivalent of decimal point for decimal

numbers), the conversion is done as follow.

A = 1 0 1. 1 0 1

10

321

321012

625.5

2

1

2

0

2

1104

212021212021

=

+++++=

+++++=

A

A

A

Exercise 1.2:

Convert from binary to decimal:A = 10110.01 C = 11110111.1011

B = 111.111 D = 10110101101.111101

1.3.4 Octal numeration system:

The octal numeration system is a place weighted system with a base of eight.

Valid ciphers include the symbols, andf.

To convert from binary to octal numeration system, we just have to divide thenumber into groups of binary numbers having 3 bits each. And each group of 3 bits is

replaced by its equivalent in octal.

Example 1.5:

Lets convert the following binary numbers in octal:

A = 10110101

B = 11010111.01

2 1 0 -1 -2 -3


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A = 010 110 101

The bits are grouped from the right to the left. A zero has been added to the

two firs bits to form a group of 3 bits. That zero is called an implied zero.

B = 011 010 111 . 010

Two implied zeros have been added to the number to form groups of 3 bits.

1.3.5 Hexadecimal numeration system:

The hexadecimal numeration system is a place weighted system with a base

of sixteen. Valid ciphers include the normal decimal symbols

;9 plus six alphabetical characters A, B, C, D, E, and

F. The following table summarises the equivalence between decimal, binary, octal

and hexadecimal systems.

Decimal Binary Octal Hexadecimal0 0000 0 01 0001 1 12 0010 2 23 0011 3 34 0100 4 45 0101 5 56 0110 6 67 0111 7 78 1000 10 8

9 1001 11 910 1010 12 A11 1011 13 B12 1100 14 C13 1101 15 D14 1110 16 E15 1111 17 F

To convert from binary to hexadecimal numeration, we group bits in fours.

Each group of four bit is replaced by its hexadecimal equivalent.

2 6 58

101101012 = 2658

3 2 7 . 28

11010111.012 = 327.28


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Example 1.6:

Convert the following binary numbers in hexadecimal.

A = 1101011101

B = 11101011101.11

As explained above, we just have to group the binary number in groups of four bits

each:

A =0011 0101 1101

The binary number has been grouped is groups of four bits each, from the right to the

left two implied zeros have been added at the extreme left. In the same way the

number B can also be converted.

B =0111 0101 1101 . 1100

1.4 Changing of base:

We have already seen in the previous section how to change from binary to

decimal, octal or hexadecimal systems of numeration. The present section is

intended to show how to move from a given system of numeration to any other

system.

1.4.1 From octal and hexadecimal to binary and decimal:

The octal and hexadecimal systems are actually used by computer engineer

just to obtain a shorthand representation of binary numbers (because octal and

hexadecimal representations take a few numbers of ciphers or symbols as compared

to binary system). It should therefore be understood that only binary system is

implemented in the electronic circuits of digital systems (through two levels of

voltages or currents: high (1) and low (0)), the others systems being used by

engineers just for simplification issues.

However, we sometimes have the need to convert either of those systems to

binary or decimal forms.

3 5 D16

11010111012 = 35D16

7 5 D C16

11101011101.112 = 75DC16


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1.4.1.1 Octal and hexadecimal to binary:

It is obvious that, to convert from octal to binary, we just have to convert each

octal cipher to its binary equivalent in 3 bits. In the same way, to convert from

hexadecimal to binary, we should convert each hexadecimal symbol into its binary

equivalent in 4 bits.

Example 1.7:

a) Convert the following octal number to digital 5238.

b) Convert the following hexadecimal number to binary 4DC216.

5238 = 101 010 0112

4DC216 = 0100 1101 1100 00102

1.4.1.2 Octal to decimal:

Because octal is a base of eight numeration system, each place weight value

differs from either adjacent place by factor of eight.

Example 1.8:

Let us convert the following octal number to decimal: A = 264.748

A = 2 6 4. 7 48

10

21012

9375.180

6414

8171486642

8487848682

=

++++=

++++=

A

A

A

Exercise 1.3:

Convert the following octal number to decimal:

A = 4562.368 C = 264.3658

B = 523411.2328 D = 4516328

Is the number 12586 an octal number?

5 2 3

4 D C 2

5238 = 1010100112

4DC216 = 1001101110000102

2 1 0 -1 -2


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1.4.1.3 Hexadecimal to decimal:

The technique for converting hexadecimal notation to decimal is the same as

the one used above, except that each successive place weight changes by a factor

of sixteen.

Example 1.9:

Let us convert the following hexadecimal number to decimal: A = 34DF.AC216

A = 3 4 D F.A C 216

10

3210123

67236.13535

000488281.0046875.0625.015208102412288

1621612161016151613164163

=

++++++=

++++++=

A

A

A

Exercise 1.4:

Convert from hexadecimal to decimal.

X = A23C.DF16

Y = 7D3E16

Z = D96EC.FA16

1.4.2 Conversion from decimal numeration system to others systems:

The conversion from decimal numeration system to others systems of

numeration is an important task for everyone dealing with computer science, because

it permits to move from daily world to digital world.

1.4.2.1 General method:

To convert a number from decimal numeration system to binary, octal or

hexadecimal, we use repeated cycles of divisions to break the decimal numeration

down into multiples of binary, octal or hexadecimal place weight values.

In the first cycle of division, we take the original decimal number and divide it

by the base of the numeration system that we are converting to: It meant that for

binary, we should divide by 2, for octal we should divide by 8, for hexadecimal we

should divide by 16. Then we take the whole number portion of the division result and

divide it by the result again, and so on, until we end up with a quotient of less than

the base value.

3 2 1 0 -1-2-3


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1.4.2.1 Decimal to binary conversion:

Let us convert the decimal number 8710 to binary, using the principle

described above. It meant that the decimal number should be repeatedly divided by

2.

87 2

43 2 1

21 2 1

10 2 1

5 2 0

2 2 1

1 0

The coloured ciphers are the reminders of repeated division of the decimal

number by 2. To obtain the binary number, we just have to take those reminders,

beginning with the last one, as indicated by the arrow. Then we have:

210 101011187 =

In short, the binary bits are assembled from the reminders of the successive

division steps, beginning with the LSB (Least Significant Bit) and proceeding to the

MSB (Most significant Bit).

Exercise 1.5:

Convert the following decimal numbers to binary

A = 15310 C = 4610

B = 25510 D = 3810

1.4.2.2 Conversion of decimal numbers less than 1 to binary:

For converting a decimal number less than 1 to binary, we use repeated

multiplication by 2, taking the integer portion of the product in each step as the next

digit of our converted number. Let us convert the decimal number 0.37510 to binary:

0.375x2 = 0.75 Integer portion of the product = 0


0.5x2 = 1 Integer portion of the product = 1

(we stop when the product is a pure integer)


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Each step gives us the next bit further away from the binary point, so the

binary number is obtained taking the bits from up to down.

0.37510 = 0.0112

Remark 1.3:

With integer division, worked from the LSB to the MSB (down to up), but with

repeated multiplication, we worked from up to down.

Exercise 1.6:

Convert from decimal to binary:

A = 0.812510 C = 0.87510

B = 0.62510 D = 0.4062510

Remark 1.4:

To convert a decimal number greater than 1 with a less than 1 component, we

should use both techniques, one at time. Let us convert the decimal number 23.12510

to binary.

Step one: repeated division for the integer portion 2310.

23 2

11 2 1

5 2 1

2 2 1

1 0

Partial answer:

2310 = 101112

Step two: repeated multiplication for the less than 1 portion 0.12510.



0.5x2 = 1 Integer portion of the product = 1

Partial answer:

0.12510 = 0.0012

Complete answer:

101112 + 0.0012 = 10111.0012


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Exercise 1.7:

Convert from decimal to binary

A = 17.37510 C = 27.87510

B = 43.62510 D = 49.4062510

1.4.2.3 Decimal to octal conversion:

Let us convert the number 12310 from decimal to octal numeration system. As

explained before, we just have to divide the decimal number successively by 8.

123 8

15 8 3

1 7

12310 = 1738

The octal digits are determined by the reminders left over by each division

step. These reminders are between 0 and 7.

Exercise 1.7:

Convert the following numbers from decimal to octal:

A = 32310 C = 12810

B = 45210 D = 9910

1.4.2.4 Decimal to hexadecimal conversion:

Let us convert the number 45616 from decimal to hexadecimal. This

conversion is obtained by repeated division of the decimal number by 16.

456 16

28 16 8

1 12

(C16)

45616 = 1C816


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Exercise 1.8:

Convert from decimal to hexadecimal:

A = 452310 C = 99710

B = 86710 D = 123810

1.5 Codes:

A code is a system of letters, numbers or symbols that represent information.

We have seen in previous sections that every decimal number can be converted in

binary; by so doing, we can say that we are achieving a pure binary codification.

There are many codes used in computer science to facilitate the operation of certain

digital circuits. Some of those codes are: BCD code, Gray code, and alphanumerical

codes.

1.5.1 Binary coded decimal (BCB) code:

The BCD code of a decimal number is obtained by replacing each digit of the

number by its equivalent in four bits, within the interval 0000 to 1001. Because of the

fact that the maximal digit of the decimal numeration system is 9, the allowable codes

goes from 0 (0000) to 9 (1001). So, the BCD code does not use the codes 1010,

1011, 1100, 1101, and 1111.

Let us convert the number A = 45610 to BCD.

A = 4 5 6 Decimal

A = 010001010110BCD

Example 1.10:

Convert the following BCD number in decimal: X = 0110100000111001. Can the

following series of bits be the BCD code of a decimal number? Y = 011111000001.

X = 0110 1000 0011 1001

X = 683910

Y =0111 1100 0001

0100 0101 0110 BCD

6 8 3 9

7 ? 1


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The second group of bit (1100) is not allowable in the BCD code, so the number Y

cannot be the BCD code of a decimal number.

Exercise 1.9:

Determine the equivalent BCD code of the following decimal numbers:

A = 153610 C = 5698910

B = 8975610 D = 23569810

Determine if possible the decimal numbers corresponding to the following BCD

numbers:

A = 100101110110

B = 110111100111

Remark 1.5: Difference between BCD code and binary number

It is important to realise that the BCD code is not a numeration system as

binary, octal or hexadecimal numerations systems. In fact, it is just a decimal system

whose digits have been replaced by their binary equivalent in four bits. On the other

hand it should be noticed that a BCD number is not a binary number. When we are to

convert a decimal number to binary, the whole number is taken into consideration

meanwhile to convert from decimal to BCD, each individual digit is replaced by its

binary equivalent in four bits.

For example, let us convert 1910 to binary and to BCD:

3510 = 1000112

3510 = 0011 0101 (BCD).

It is obvious that the conversion from binary to BCD takes more bits than the

conversion from decimal to binary. So the BCD code is not as efficient as the binary

system. The advantage of the BCD code is just the fact that it is very easy to convert

from decimal to BCD and vice versa.

The BCD code is found in digital systems using 7 segments displays like

digital voltmeters, digital watch

1.5.2 Gray code:

The Gray code is a non weighted code in which each coded representation

differs from the previous representation only by one bit. It is not the case for binarysystem where many bits can change when we move from a number to the following


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number. For example, when we move from 0111 (710) to 1000 (810), all the four bits of

the representation are changed. The Gray code is not suitable for arithmetical

calculations (because it is not weighted); it is used in the determination of outputs

equations of digital circuits ( Karnaugh mapping) and in the design of Analog

Digital Converters.

The following table gives us the equivalence between binary representation

and Gray code.

Decimal Binary Gray

0 0000 0000

1 0001 0001

2 0010 0011

3 0011 0010

4 0100 0110

5 0101 0111

6 0110 0101

7 0111 0100

8 1000 1100

9 1001 1101

10 1010 1111

11 1011 1110

12 1100 1010

13 1101 1011

14 1110 1001

15 1111 1000

Remark 1.6: How to generate a Gray sequence

If you observe attentively the Gray sequence above, you will notice that:

For the first column of ciphers (coming from the right to the left), the first zero

is followed by two ones, which are followed by two zeros, two ones, two

zeros

For the next column of ciphers you can notice that the two first zeros are

followed by four ones, which are followed by four zeros, four ones


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For the third column of ciphers, the four first zeros are followed by eight ones,

which are followed by eight zeros, eight ones

This is the principle to be used in order to generate a Gray sequence.

1.5.3 Alphanumerical codes

A computer would have been useless if it wasnt able to treat non numerical

information. In fact, a computer should be able to recognize codes corresponding to

numbers, letters or some special characters. Such codes are called alphanumerical

codes. Generally the keyboard of a computer should contain the following symbols:

The 26 letters of the alphabet (capital and small letters);

The 10 ciphers of the decimal numeration system,

Almost 25 special characters like +, /,>,


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L 100 1100

M 100 1101

N 100 1110

O 100 1111

P 101 0000

Q 101 0001

R 101 0010

S 101 0011

T 101 0100

U 101 0101

V 101 0110

W 101 0111

X 101 1000

Y 101 1001

Z 101 1010

0 011 0000

1 011 0001

2 011 0010

3 011 0011

4 011 0100

5 011 0101

6 011 0110

7 011 0111

8 011 1000

9 011 1001

Blank 010 0000

. 010 1110

( 010 1000

+ 010 1011

$ 010 0100

* 010 1010

) 010 1001

- 010 1101


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/ 010 1111

, 010 1100

= 011 1101

Exercise 1.10:

Give the octal and hexadecimal equivalent for all the ASCII codes given

above.

The following instruction coded in ASCII is composed on the keyboard of a

computer. Give its signification: 101 0011, 101 0100, 100 1111, 101 0000.

Answer: Using the table above we find that the instruction is STOP.

1.6 Conclusion

This chapter has permitted us to study and to master (I hope so) the most

common numeration systems and codes. We have also studied methods of

conversion from each numeration system to another. In the next chapter, we will

study the behaviour of logic gates, which can be considered as elementary bricks

used in the construction of any digital circuit.

REVIEW QUESTIONS

1. Give the difference between analogue and digital representations.

2. Convert from binary to decimal:

A = 110112 D = 10010.0112

B = 10111012 E = 101001111112

C = 10111112 F = 1110111.00012

3. Convert from binary to octal:

A = 111011.011012 D = 111011111012

B = 101101112 E = 10011101.1102

C = 11011110.01012 F = 100111101011112


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4. Convert from binary to hexadecimal.

A = 1111010.11102 C = 111010111.112

B = 101111101111012 D = 10110110.1111012

5 Convert from octal to binary:

A = 1238 C = 3578

B = 6538 D = 5478

6 Convert from hexadecimal to binary.

X = F47B16 Z = 8CE016

Y = 5FD316 P = FFFC16

7 Convert from octal to decimal :

A = 1258 C = 5638

B = 2568 D = 4538

8. Convert from hexadecimal to decimal:

X = F47B16 Z = 8CE016

Y = 5FD.316 P = FFF.C16

9. Convert from decimal to binary:

A = 2310 C = 5310

B = 25.37510 D = 101.2510

10. Convert from decimal to octal:

A = 42310 C = 43810

B = 126410 D = 342310

11. Convert from decimal to hexadecimal:

A = 126210 C = 256310

B = 356210 D = 56423610

12. Convert from octal to hexadecimal:A = 123.628 B = 432.58


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13 Determine the highest decimal number that can be represented using 8bits, 16

bits.

14 In most of the computers, the addresses of memory locations are expressed in

hexadecimal. Those addresses are sequential numbers that identify each

memory location.

a) A computer can store data of 8 bits (1byte) in each of his memory

location. If the addresses of the memory locations run from 000016 to

FFFF16, then give the number of memory locations of that computer.

Deduce the capacity of its memory.

b) Another computer has 4096 memory locations. Give the interval of their

addresses starting from 000016.

15 Determine the number of bits to be used to represent the decimal numbers

from 00010 to 99910: a) using pure binary code b) using BCD code.

16 Express in ASCII the following instruction: X = 25/Y

17 Convert from BCD to binary: A = 01110100 (BCD).

References:

1. Digital systems, principles and applications, Ronald J.Tocci, 3rd edition,

Prentice-Hall inc., Englewood Cliffs, New Jersey , USA,1985.

2. Lessons In Electric Circuits Volume IV Digital, Tony R. Kuphaldt, Fourth

Edition, 2007, www.allaboutcircuits.com. www.ibiblio.org/obp/electricCircuits.
http://www.allaboutcircuits.com/http://www.allaboutcircuits.com/http://www.ibiblio.org/obp/electricCircuitshttp://www.ibiblio.org/obp/electricCircuitshttp://www.allaboutcircuits.com/

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