And All about Understanding Number Systems for Digital Computing
We start counting from 1 and go up to ten, and then suddenly the mind goes into an auto mode (subconsciously of course) and increases the count in another set of ten numbers, i.e. 11 to 20, 21 to 30, and so on so forth. If you would notice, numbers are a universal phenomenon. They do not change. Nine will always come after eight, and 13 will always come before fourteen, whichever number system you use to represent them. Then why do we have so many different number systems? On close observation, it will occur to you that different number systems represent the same numbers in different notations or symbols or digits. So all the confusion is about how to represent a particular number. The number in itself is always the same, irrespective of the number system.
Since the advent of electric signals in computers, computers have started thinking in terms of 0 and 1. They create all the numbers from a repository of these two numbers (0 and 1) only. And hence, they have a number system that is based on two digits, hence the base is 2. And as computing is now based on these two digits, hence the computers are DigitalComputers now.
Number Systems
The most commonly used number system is the Hindu-Arabic number system. This is the number system that we use in our day to day life, for counting, calculating, and mathematical operations.
India’s first satellite named after Aryabhata. By ISRO — https://imagine.gsfc.nasa.gov/science/toolbox/missions/aryabhata.html, Public Domain, https://commons.wikimedia.org/w/index.php?curid=10082533Two Indian mathematicians, separated by a century, Aryabhata (476–550 CE) and Brahmagupta( (c. 598 CE — c. 668 CE). Aryabhata developed the place value notation system and Brahmagupta introduced the symbol for Zero. This number system along with the concept of Zero developed by Hindus of India spread to the nearby regions thru commercial and military activities. The merchants started trading using this system, as it was simpler, more scientific, and stable. It reached Europe from Arabia and they modified the symbols a bit and started calling it, the Arabic Numerals. They are very similar to the Sanskrit-Devanagari numbers notations, which are still used in India and Nepal.
But this is not the only number system in use. Various civilizations had contributed in this regard and had their own number system with different notations. They all can be divided into two broad groups.
Sign Value Notations
Place Value Notation
Sign Value Notation As the name suggests, these numbers have “Signs” and those signs have “Values”.
A typical clock face with Roman numerals in Bad Salzdetfurth, Germany. By JuergenG — made by JuergenG, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=832860
In Sign Value Notations, the signs represent value, that added together to equal the number represented. Roman numerals are one such example. In Roman numerals, CCLVII means 257 (100+100+50+5+1+1). Tablet with proto-cuneiform pictographic characters (end of 4th millennium BC), Uruk III. By MbztOwn work, CC BY 3.0, https://commons.wikimedia.org/w/index.php?curid=25834613The sign value system may appear to be cumbersome now, as we are used to the more efficient number systems, but they were the first step towards numbers in general by mankind. Let us understand this by this ancient story. The tribe wanted to count their sheep, and also record the number of sheep owned by different people in the tribe. They made a sheep-like figurine on a clay tablet to show one sheep, and two sheep figures show two sheep. But then one person with 24 sheep came, and it was not possible to record his number by marking sheep figures like before. So they created another symbol that represents 10 Sheep(Why 10? We will answer that for sure). And similarly as and when the number of the herd of sheep kept increasing, they kept inventing new symbols for 50 sheep, 100 sheep, and so on. This was perhaps the first sign value notation used as numbers, about 4500 to 400 BC in Mesopotamian civilization(Present-day Iraq-Syria-Iran region). Limestone Kish tablet from Sumer with pictographic writing; may be the earliest known writing, 3500 BC. Ashmolean Museum. By José-Manuel Benito — Own work, Public Domain, https://commons.wikimedia.org/w/index.php?curid=944897Now after reading this, it sounds only logical to have such an approach. But as human’s across the globe kept evolving, so did their intellectual needs and so did the number system. Place Value Notations (Also known as positional notation) As evident from the name, the “Place” of a number notation determines its “Value”. Let us see two examples.
In the place value system, there are number systems with different bases, which we will look into shortly. But let us first understand what the base is. The base is nothing but a “number of unique symbols” that a particular number system has. So for the number system with base 10, there are ten unique symbols (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9). And hence deriving from the word Deca (which means 10), this system is called Decimal system. Similarly, for a Binary system, the base is 2 and hence the unique symbols are only two (0 and 1). Depiction of a Number with base 10 and showing various aspects like Index, digits and base. Source: Created by the Author.The Hindu-Arabic Number system (Decimal system) is the most commonly used number system with base 10, but it is not the oldest one. The first positional number system was the Babylonian number system, base 60. Its influence is felt in many aspects even today. The 60 minutes in an hour, 360 Degrees of angle in a circle are from that system with a base of 60. That brings back us to the original question we started with. Why do we count in Tens? Or why the Decimal number system has a base of 10? Einstein counting on fingers. https://martinlakewriting.files.wordpress.com/2013/11/einstein-counting-on-fingers.jpgRecall the most common way of counting we follow, when we do not have any other apparatus. Our fingers. And how many fingers do we have in our hands? Ten… Now you get it. It’s not a random thing that most of the world’s number systems, many of which developed independently, have base 10. To read a place value number system, the following method is applicable. It holds true for the number system of any number as a base. Formula to calculate value of any number with any base. Source: Created by Author.This will become clearer once we see simple examples of conversion for the different number systems shortly. List of numeral systems. In Wikipedia, The Free Encyclopedia. Retrieved 10:22, February 10, 2021, from https://en.wikipedia.org/w/index.php?title=List_of_numeral_systems&oldid=1005801737For digital computing, there are four main number systems. All of them are Positional number systems or Place Value number systems. They are 1. Binary Number System (Base 2) 2. Octal Number System (Base 8) 3. Decimal Number System (Base 10) 4. Hexadecimal Number System (Base 16) Binary Number System (Base 2) In digital electronics and mathematics, a number system with base two is known as a binary (bi means 2) number system. This is the most commonly used number system in all modern computers, because of its simple implementation of digital electronic circuitry using the logic gates. This base-2 number system is a positional notation with two symbols, typically “0” (Zero) and “1” (One). Each digit is referred to as a bit (which is short for binary digit). Numbers in Decimal and their equivalent Binary Numbers. Source: Created by author. This counter shows how to count in binary from numbers zero through thirty-one. By Ephert — Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=38752676A binary number can be converted to a decimal number and vice versa. The counter on the left showcases how Decimal number is being represented in Binary format. Take your time to understand this. Notice how the numbers on the middle row correspond to the value of decimal numbers, when the corresponding binary numbers in the third row are being added. For example, 14 can be written as 1110 in binary. Let’s see how its value is arrived at using the formula we saw above. Why is Binary Important in Digital Computing? Since the days of electromechanical computers to today’s electronic computers, all of them operate on electrical signals. The electrical signal has two parameters, Voltage (Volt) and Current (Ampere). There can be thousands of values of these two, and hence it will become very complex to assign numbers to different values of Volt and Ampere for the computer to understand. To make it simpler, computers use only two values, ON and OFF. ON is 1 and OFF is 0. Hence, binary number system suits computers or for that matter any digital machine to store, communicate and process data. The word ‘Wikipedia’ represented in ASCII binary code, made up of 9 bytes (72 bits). By User:Atyndall (See account on English Wikipedia) — Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=3904637Not only numbers, the text, and special characters are also represented in binary for the computer to understand. Octal Number System (Base 8) As it’s a base-8 number system, it has 8 digits or symbols (0, 1, 2, 3, 4, 5, 6, and 7). Octal numbers are commonly used in computer applications. In the early days of computers, the octal numbering system was very popular for counting inputs and outputs because as it works in counts of eight, inputs and outputs were in counts of eight, a byte (one byte = 8 bits) at a time. The octal number system provides a convenient way to convert very long binary number strings into more compact and smaller groups. However, nowadays the octal number system has almost disappeared and given way to a more efficient hexadecimal number system. Hexadecimal Number System (Base 16) By now you are familiar with the base. Hexadecimal is a 16 base number system. Unlike the decimal number system which we are used to in our day to day life, it has 16 distinct symbols or digits. Hexadecimal numbers are not used in common mathematics as frequently, but they are the absolute darling of computer system designers and programmers. This is simply because they provide a human-friendly way to represent the otherwise cumbersome binary numbers. Each hexadecimal digit represents four bits (binary digits). These 4 bits are called a nibble, and 8 bits make a byte. Let us have a look at how binary to hexadecimal makes things simpler. Conclusion Mathematics is a universal language and it is independent of the number system in use. This sounds great in principle, but humans couldn’t become so advanced in Physics and other applied sciences if there was no “Zero” (0). Today we take Zero for granted, but it was not always the case. The Sign Value number systems do not have zero. Even the value-based number systems existed for centuries without Zero. For digital computers, Binary is the preferred number system. It fits so well that though the computer systems have been changed in all other aspects, from electromechanical to electrical (Vaccum-tube-based) to electronic (semiconductor-based), but for the basic language of the computer. The basic language the first digital computers understood decades ago, is still the same, a Binary number system. Hence it becomes imperative to study and know binary. Decimal is the language of choice for humans for its simplicity and stability and clubbed with Zero, it has outlived all other number systems (there were many which we didn’t discuss here). Now it is so widely used that it has become synonymous with “numbers” and most people do not realize that it is only one of the many number systems. Where do the Octal and Hexadecimal fit in then? Both octal and hexadecimal were the bridge between the computers used number system (Binary) and the human used number system (Decimal). For the large numbers in the decimal system, the binary representation becomes too big to be remembered or reproduced by humans. Hence, to shorten them by half, the octal system comes in handy. And going even further, the Hexadecimal system reduced four bits to one digit of hexadecimal, reducing the size to one-fourth of the binary number. This is interesting to note that there are arithmetic operations for all the number systems, just like we are used to for decimal ones. But let us conclude it with this thought, that we need to understand why other number systems apart from decimal system exist, and how are they useful, as we saw just now. One last thought, I was wondering, what if we had only eight fingers? Then octal would have been the most commonly used number system. It is interesting to note, how biological evolution shapes the direction of so unrelated fields, of human invention and progress in Mathematics and that too, Numbers. There might be some alien race, intelligent enough to understand the secrets of nature, like mathematics and physics. What if they have only two fingers? Will they be able to talk to computers in their most common language? I leave you with that thought…