Information and Communication Technology English Version chapter 3: Number Systems and Digital Devices

Chapter Three

Number Systems and Digital Devices

Figure: Bangladeshi students participating in international robotics competitions

Science and technology always had the greatest impact in the history of human civilization. Modern science and the modern world owes most of their debt to computers and electronic gadgets we hold so dear nowadays. Not so long ago we needed a whole building complex to reside a computer. But now we carry a mobile phone in our pockets with vastly more computing power than all of those early computers combined. These computers and electronic gadgets are based upon a field of study we call Digital Electronics. This branch of study relies on a mesmerizingly simple mathematical structure called binary numbers and Boolean Algebra. In this chapter you will be introduced with the fundamentals of this system.

At the end of this chapter students will:

• Learn the history of discovering the numbers
• Be able to explain number systems
• Know the classifications of number systems
• Determine the interrelations between the number systems
• Be able to add and subtract in binary
• Determine $2$’s complement
• Learn the concept of coding
• Compare different codes
• Know the concepts of Boolean Algebra
• Prove Boolean Theorems
• Use logic operators in practical scenarios for Boolean Algebra
• Explain inner workings of digital devices tied with Boolean Algebra

Printed page 80

3.1 History of Inventing Numbers

We use both languages and numbers in every sphere of our daily lives. Necessity drove us to discover numbers alongside languages. Even some animals and birds can count up to an extent. It is strange to us but in the far reaches of the world there are many aboriginal groups who do not know how to count because they do not need numbers in their day to day lives. There is an indigenous tribe called Piraha in Brazil who can count only upto $2$. More than two of something and they describe it as ‘many’.

In ancient times when the humans were hunter gatherers, they did not have much use of numbers and counting. But when human civilization learned agriculture, farmed animals, built cities and settled down - then the need for accounting crept up on them. That is why the history of human civilization and the history of numbers are interconnected in so many ways. Necessity made us learn how to count huge numbers, made us learn what to do with numbers and how to process them.

Prehistoric people used scratches on bones, tree branches and stones or seashells or small stones to keep track of numbers. But eventually the ancient human beings felt the need for much larger numbers that can not be kept track with just scratches or seashells. That's why they invented symbols to represent numbers. About five thousand years ago the Sumerian-Babylonian civilization and the Egyptian civilization were booming and in both cases we found that they had invented number systems using written symbols. The Sumerian-Babylonian number system was $60$ base and the Egyptian number system was $10$ based or decimal number system. We still use the Egyptian decimal system in current day and time and we have traces of the $60$ based number system in our counting of time and angles. Seconds, minutes, hours are $60$ based so are the angles we calculate in a plain. Both these ancient systems used different symbols to represent nothingness but they had not invented the number zero.

Three more civilizations played important roles in developing the number system - the Mayan Civilization, Chinese civilization and the Indus river valley civilization. Mayan number system was $20$ based and Chinese and Indian number systems were decimal. Traces of the $20$ based number system can still be found in our villages while counting weights or lengths. Both the Mayan and Indian number systems used local values of the numbers. Indians were the first people to use zero as a number in their system, this was a revolutionary discovery in the history of mathematics. Mayans and Chinese number systems used two or three symbols to express all the numbers, whereas the Indian system used nine separate symbols to express numbers from $1$ to $9$ and one different symbol for $0$ - a total of ten symbols. These symbols are known as digits.

Printed page 81

The Greeks developed a decimal number system based on the Sumerian-Babylonian and the Egyptian number systems. The Greek revolution in the field of mathematics was cut short by the Roman conquest of the Greek empire. The Roman empire did not feel the need for much mathematics in their lives. They did not have separate symbols for all the digits. Numbers were expressed via Roman numerals. That unscientific and unnecessarily complex system is still alive as we use them in clock faces and sometimes marking literary topics.

Figure: Figure 3.1: Ancient numerals

Labels: Hindu-Arabic, Roman, Greek, Egyptian, Babylonian, Chinese, Mayan

Hindu-Arabic	Roman	Greek	Chinese
○	(blank)	(blank)	○
$1$	I	A	I
$2$	II	B	II
$3$	III	Γ	III
$4$	IV	Δ	IIII
$5$	V	E	IIIII
$6$	VI	F	T
$7$	VII	Z	TT
$8$	VIII	H	TTT
$9$	IX	Θ	TTTT
$10$	X	I	—
$50$	L	N	≡
$100$	C	P	$100$

Description: Table of ancient numerals for the values $0$ to $100$ in seven columns. The Hindu-Arabic, Roman, Greek and Chinese columns are transcribed; the Egyptian, Babylonian and Mayan columns are drawn symbols (Mayan: dots and bars), not text, and are not transcribed.

With the expansion of Islamic civilization, the Indian decimal number system found their way into Europe. The symbols we use now $1,2,3 \ldots$ are all Arabic numerals. Funny part here is, the christian rulers resisted the concept of adopting the number zero by spreading propaganda that it is the devil’s number.

Decimal number system was made popular due to the fact that we have ten digits on our hand. But later the $2$, $8$ and $16$ based number system gained popularity in the modern age.

3.2 Number System

A number system is defined as a system of writing to express numbers. We use many different notations and symbols to denote numbers. These symbols are used in two different ways.

3.2.1 Classification of Number System

Number systems can be classified into two types: non positional and positional system.

Non Positional Number System: In a non positional system, the symbol will maintain its value no matter where it is placed or positioned. Roman numbers are examples of non positional number systems. For example - In Roman numerals the symbol V is used to express the

Printed page 82

number $5$. But in VI, VIII, XV although the symbol V is in three different positions but all of them express the value $5$. Non positional numbers were popular in ancient times.

Positional Number System

In this system the value of the number is dependent upon the position of the symbols. With the invention of the modern number theory positional number system was made popular. The decimal number system we use in our daily lives are an example of the positional number system. For example, the number $555$ has three $5$ symbols but each has different values. The leftmost $5$ means $500$ while the middle $5$ means $50$. Each position has a local value in this number system. The rightmost digit has the local value of $1$, the next digit has the value $10$ and so one. Each position from left to write increases the local value with the multiples of $10$ as it is a decimal or $10$ based number system. If it was a $4$ based number system, each position's local value would increase by the multiples of $4$. If it was a $16$ based system, each position from left to write would increase the local value with the multiples of $16$.

Let us see some examples of different positional number systems.

Binary Numbers

We are very familiar with the $10$ based decimal number system as we use it in our daily lives. But for digital devices $10$ based numbers are not very realistic as you would have to use $10$ different values of voltage to represent $10$ numbers. While using two different voltages for two numbers seems quite easy and cost effective. This is why digital electronics rely on the binary number system.

As we know, in the decimal number system we have ten different symbols or digits - $0,1,2,3,4,5,6,7,8,9$. Similarly the binary number system has only two symbols or digits, $0$ and $1$. To express larger numbers in the binary number system we have to use comparatively more digits than the decimal system, this is the only limitation of the binary number system. Any number can be expressed using the binary number system. Any type of mathematical operations can be performed using the binary number system too.

As it is a positional number system, each position has a local value. In the decimal number system the local value increases in multiples of $10$s like $100$, $101$, $102$, $103$ and such. Similarly in the binary number system the local values increase with the multiples of $2$, like $20$, $21$, $22$, $23$ and such. To express fractions, decimal numbers decrease in $10$s multiples like $10-1$, $10-2$, $10-2$ and such. Similarly binary numbers decrease after the radix point like $2-1$, $2-2$, $2-3$ $\ldots$. As such to compare let us observe the following table of decimal and binary numbers–

Printed page 83

Figure

Labels: Decimal Number, Binary Number, Decimal Point, Binary Point, MSD, LSD, MSB, LSB Description: Two place-value diagrams side by side. Decimal number: digits 2 3 5 0 1 . 2 3 7 with powers of 10 above them (10^4, 10^3, 10^2, 10^1, 10^0, 10^-1, 10^-2, 10^-3); the decimal point lies between 10^0 and 10^-1, MSD is under the leftmost digit and LSD under the rightmost digit. Binary number: bits 1 1 0 0 1 . 1 1 0 with powers of 2 above them (2^4 to 2^-3); the binary point lies between 2^0 and 2^-1, MSB is under the leftmost bit and LSB under the rightmost bit.

Here MSD and LSD means Most Significant Digit and Least Significant Digit. Similarly MSB and LSB means Most and Least Significant bits. Like in decimal numbers, to determine a binary numbers value we have to multiply each bit with their local value and add them up.

$(11001.110)_2 = 1\times2^4 + 1\times2^3 + 0\times2^2 + 0\times2^1 + 1\times2^0 + 1\times2^{-1} + 1\times2^{-2} + 0\times2^{-3}$

$= 16 + 8 + 4 + 0 + 0 + 1 + 0.5 + 0.25 + 0$

$= (25.75)_{10}$

Here the subscript with 2 and 10 are there to represent the base of the numbers so we can know which is a binary number and which is a decimal number. The number of digits needed in a number system is referred to the base. That is why the decimal system has a base of 10 and the binary system has a base of 2. Similarly the octal system has a base of 8 and the hexadecimal system has a base of 16. Generally we do not write the base of the system alongside the numbers unless we are using more than one number system in conjunction. This is done to avoid confusion.

In this chapter we are aiming to learn the number systems needed for operating digital devices where fractions are not required. This is why further discussions about number systems will revolve around whole numbers only.

Figure: Table: 3.1

Labels: 2^3 = 8, 2^2 = 4, 2^1 = 2, 2^0 = 1 (local values); Decimal Number

2^3 = 8	2^2 = 4	2^1 = 2	2^0 = 1	Decimal Number
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

Description: A table of 4-bit binary values with their decimal equivalents, serially arranged from 0 to 15.

3.1 table displays the some serially arranged values of decimal and binary numbers to compare.

Printed page 84