When we deal with numbers, whether for the purpose of calculation or for other requirements, we are actually dealing with a set of symbols that give a perception about the numerical values or quantity of something by putting multiple such symbols back to back. When it comes to numbers, the number system plays a fundamental role in giving us an understanding of numbers. We, human beings depend on the decimal number system but, it is the use of the binary number system when it comes to our regular computers, smartphones, tablets and other digital electronic gadgets that we all see everywhere.
But, are these the only two number systems that we have in our civilization? Obviously it is not so. Depending upon the requirements, we also have a number of other number systems, which can come in handy and we use them in a number of situations in our everyday life. But, it will be worth knowing what the number system is, and how to define a number system. It might also come to mind, obviously, if you already know about number systems, why we are still using the decimal number system, when there are potentially better number systems available to use for the purpose of calculation and for other requirements. It will also be worth knowing how different other number systems are used for other purposes.
So, without any further delay, let’s get started with the concept of the number system, and later on, I will discuss a few other popular number systems, which we all come across at some points in time.
What is the number system?
A number system defines the number of symbols that we use to represent numbers in that particular system. So, for example in the case of the decimal number system, we have 10 symbols starting from 0 to 9 and in the case of binary number systems, we have only two symbols 0 and 1. So, let’s have a look at the different number systems, starting with the most popular decimal number system that we all use in our everyday lives.
Type of number system
Decimal number system
In the case of the decimal number system, we have 10 symbols, which starts with 0 and ends with 9. However, we can represent any single number with only those 10 symbols. After writing 9, we run out of symbols and even after that we can keep adding symbols to our number system to represent every single number. But even if we keep adding symbols to our number system it will be impossible to represent every single number that we can think of. How can you actually represent 378412 with a single symbol! It is impossible. That’s when we use positional notation.
With the help of position notation, we use another number on the left side, and the complete number is represented by the weightage of the position of the number on the left, multiplied with the number, and the sum of the number on its right. For example, after 9, we use again, but use the number next to 0, i.e. 1 before the zero, and that represents 10. Next time, when we reach 19, we again use 0 on the right. But on the left side, we use the number next 1, which is 2 and this goes on.
For example, if we want to write 247, we get it in the following way.
2 ✕ 100 (as it it is the third number from the left) + 4 ✕ 10 (as it is second from the left) + 7 = 200 + 40 + 7 = 247
It is long since that we are using the base 10 or decimal number system and there are hardly any chances that we will adopt any other number system anytime soon. We use the base 10 number system for an eternity now, possibly because of having 10 fingers in our hands making it easier for us to calculate.
Binary number system
When it comes to computers, data is recognized with electricity. You might have already heard about the binary number system if you have read computers as a subject in your high school or college. When there is electricity within the microprocessor or anywhere else, we consider it to be 1, and the absence of electricity is considered to be 0. With only two symbols, as well, we use the same principle of positional notation, where we again use the number next to 0 on the left to represent the number to be 2 times more than the number on its right.
For example, if we want to represent 1011, a base 2 or binary number, in the decimal number system, it can be converted to decimal in the following way.
1 ✕ 8 (as the first one from the left is the fourth from the right, making it ✕2✕2✕2, or equal to 8) + 0 ✕ 4 (in the same way, ✕2✕2) + 1 ✕ 2 + 1 ✕ 1 = 8 + 0 + 2 + 1 = 11
When it comes to digital number system it is easier to represent, store and process data with only just two symbols, and that’s the way we use our computers and other digital electronic gadgets.
Other number systems based on alphanumeric characters
Apart from the two number systems, which I have discussed, i.e. the decimal and binary number system, any other number system can be represented using fewer symbols like, we can represent numbers using Base 3, Base 4, Base 5 and so on. If we need to represent a number using a number system more than base 10 like Base 12 or Base 15, which are duo-decimal and hexadecimal number system respectively, we use alphabets to represent those numbers after 9. For 10, 11, 12 we use A, B, C respectively and this goes on.
Hexadecimal number system
In the case of the hexadecimal number system, there are 16 symbols or it is the base 16 number system, and we use letters from A to F to represent numbers from 10 to 15. You can find numbers represented in the hexadecimal number system in case of computer memory addresses. There is a reason behind the use of the hexadecimal number system to represent memory addresses in computers.
The number systems, which have fewer symbols, need more symbols to represent a particular number. For example, if we need to represent the number 8, we need only one symbol i.e. simply ‘8‘ to represent the number in the decimal system. However, if we need to represent the number in the binary number system we need 4 binary symbols back to back and the number 8 in decimal number system is equivalent to ‘1000’ in the Binary System.
So, when we deal with physical storage devices like hard drives, RAM modules, which has Gigabytes of storage nowadays, it will require multiple decimal numbers back to back represent one address and that’s why we use the hexadecimal number system to store the memory addresses, as the size of such number will become much less than the same number in decimal number system.
Base 36 and Base 62 number systems
If we keep adding letters after the number 9 starting from A up to Z we can represent numbers from 0 to 35. 35 will be represented by Z. That’s what is called the base 36 number system. However, if we keep adding more letters like a to z, in small, we can have numbers up to 61, where the Number 61 will be equal to z. That is called the base 62 number system.
We often see a combination of numbers, capital letters and small letters in shortened URLs like ‘l2y5uUi’. That is an implication of the base 62 number system. With almost five symbols back to back, we can represent numbers up to 100 million entities or shortened URLs for example. You can also find the same in the case of YouTube videos and the numbers at the end of the link refer to a number which is possibly the ‘nth video upload to YouTube, Where n represents the number at the end of the URL.
‘https://www.youtube.com/watch?v=2wDNUvfhrLc’ is a YouTube video link, which has 11 alphanumeric symbols at the end of the URL, and you can understand what the number will be, ‘2466670926771434792’, to be more precise, if represented in the decimal number system. So, that is the reason why we use the base 62 number system to make the numbers look smaller than the same number when represented in decimal number system or much much longer when represented in the binary number system.
The number system is a fundamental element when it comes to mathematics and computers. It is also necessary at the same time to convert a number from one system to the other for various requirements.
So that was all about number systems. Do you have anything else to say? Feel free to comment on the same below.