Introduction Mathematics is a very interesting subject and though some people feel it as dry and boring one but the fact is that once the fundamental concepts are understood and made clear it evokes a lot of interest and curiosity. In this article, we would try to understand the basics of the number system and its various types and also the fundamental concepts behind a particular number system.
Understanding numbers or digitsThe basic purpose of the numbers in any number system is for counting and any such system would be useful if we can count in it, add in it, subtract in it, multiply in it etc. Before going to the other types of number systems first we will try to understand the system which we are using in our day to day life. Presently, we are using the decimal system.
Decimal SystemWe are using decimal number system in which we have ten symbols with the help of which we can denote any number and these ten symbols are nothing but the usual digits we use that is 0, 1, 2 …. 9. So these are the only ten digits or basic decimal number symbols that are available in the decimal system and are sufficient to represent any number small or big or very small like the size of an atom or very big like the size of the universe. The total number of digits is ten so the system is called as 'base to 10' system. When we say base to 10 it means that we can represent the numbers in that fashion. Let us go through one example and let us say we want to represent 638 in base to 10 form. So what we would do is break this number in parts like -
638 = 600 + 30 + 8
Now we would bring various powers of 10 in this simple summation so that the number is understood in that fashion. Let us recall that 10^0 = 1, 10^1 = 10, and 10^2 = 100. Using these simple conversions we can write the number 638 as -
(6 x 10^2) + (3 x 10^1) + (8 x 10^0)
So from the above example, it is now clear that in the decimal system we can represent any number in the base to 10 form.
At this stage, someone might ask that how can we represent small numbers in this system. The answer is that there we have to use the negative powers of 10 to represent it. Let us represent a small number say 0.287. How can we break this number? We can break it like this -
0.287 = 0.200 + 0.080 + 0.007, so easy, it is a sum only.
To represent the above summation in base to 10 form, we have to take help of negative powers of 10 in order to represent this small number. How we do it? As we know that 10^(-1) = 0.1, 10^(-2) = 0.01, and 10^(-3) = 0.001, so using these conversions we can write as -
0.287 = [2 x 10^(-1)] + [8 x 10^(-2)] + [7 x 10^(-3)]
Representing a small number in base to 10 form might look complicated but in principle, it is same as that of representing a big number, only thing is that it is represented in negative powers of 10.
Types of numbersNumbers can be negative ( -4, -395, -3.63 etc) or positive ( 34, 7, 823 etc) or fractional (3/4, 14/729, 1/4 etc) and they are categorised under some groups to understand them more clearly. These groups are -
Real and imaginary numbersNumbers are broadly categorised in two categories, one is real numbers as we have seen in the preceding paragraph and other is imaginary numbers. Imaginary numbers are having two parts and one of the parts is real while other is imaginary. Imaginary numbers are obtained when we try to take the under root of some negative numbers which of course is an unreal thing but the concept is useful in some advanced mathematical application.
Small and big numbersIt is interesting to note as to how big and how small a number could be. Actually, there is no limit to that. For example, if we say that the approximate age of the universe as per scientific calculations is 14 billion years then we cannot write it as a number as it would be too long to write it and hence we have to depict it in positive powers of 10. So we say that Universe is 14 x 10^9 years old. But this is not the end of the big number. It could be still bigger tending to infinity. Similarly when we say that radius of a particular atom of an element is 175 trillionth of a meter then it is difficult to write it and hence we say that it is 175 x 10^(-12) meter. But that is not the end of smallness, as the nucleus of the atom is still approximately 1000 times smaller than the size of the atom. So a small number can be infinitesimally small tending to zero but it never reaches zero. Please note that zero is only a reference point and however small be a small number, it cannot become zero.
Other number systemsIn the decimal system we have 10 symbols available or say 10 digits are available that is 0 to 9. Suppose our ancestors who invented these digits had invented only eight of them that is 0 to 7 then what would have happened. Now there is no eight (8) and no nine (9). What to do now. Let us try. We have only 0, 1, 2, 3, 4, 5, 6, and 7. So how to make the next number. Same way as we make in the decimal system. The next numbers would be 10, 11, 12 …. 17 and then 20, 21, 22 … and so on. This system is known as the octal system. Hence we can very well observe that -
8 in the decimal system is 10 in octal system.
9 in the decimal system is 11 in the octal system.
10 in the decimal system is 12 in the octal system and so on.
Let us think another possibility that our ancestors were liberal and instead of making 10 symbols what would happen if they had made 16 symbols that are 6 more than the existing. Let us denote those 6 symbols by A, B, C, D, E, and F. If it was so then how the counting would happen? This system is known as the hexadecimal system and the sequence of numbers will be like this -
0, 1, 2, …… 8, 9, A, B, C, D, E, F, 10, 11, 12, ……… 1D, 1E, 1F, 20, 21, 22, ………. 2D, 2E, 2F, 30, 31, 32 and so on.
Now assume a totally different scenario. There are only two symbols available with us that is 0 and 1. What to do now? No problem, as we would go by the same logic and write the sequence of numbers as -
0, 1, 10, 11, 100, 101, 110, 111, 1000, …. and so on. This is known as the binary system. This is the one which is used in computers for all the work and is the very basic mathematical ingredient in computer architecture and computer processing. In computer circuits, everything flows in electrical pulses. An electrical pulse may be there or it may not be there. These are the only two stages and they are depicted with 1 and 0 and all the calculations and data processing is done in the binary system. It is no surprise that we use binary Algebra (scientifically known as Boolean Algebra) in computers rather than the usual school Algebra.
For example, if we want to add two binary numbers then we will have to remember only three basic things in our mind that in binary numbers 0 + 0 = 0, 0 + 1 = 1, and 1 + 1 = 10. Keeping these three in mind we can add any binary number to any other binary number.
ConclusionSo in this article what we learnt is that it is the number of symbols (number of basic digits) we have, that decides about the sequence of numbers in that system and then we do all the mathematical operations as per that number system. As we are familiar with the decimal system only we might not get conversant with other systems so easily but once we learn the basic symbols in it and the way a mathematical operation is carried out, it becomes as simple as our known decimal system.