IntroductionNumbers have a great role in Mathematics and they can be sub-classified in further groups of number based on certain properties and it is fascinating to observe those groups of numbers. We will learn about these specific groups of the numbers in this article.
System of numbersWe all know that basic number digits with us for making a number are 0, 1, 2 ….. 8, and 9. Using these 10 symbols or digits we can make any number and in Mathematics, we call it as numbers in a decimal system. Numbers could be of different types like positive numbers, negative numbers, whole numbers, fractional numbers, rational numbers, irrational numbers, real numbers, imaginary numbers, decimal numbers, binary members, natural numbers etc and now in this article, we will learn further grouping and categorisation within the main group known as natural numbers. We will be focussing on the set of natural numbers consists of 1, 2, 3, 4, 5 ……. and so on.
Number theoryIn number theory we learn about the various groups which are formed within natural numbers and have some peculiar properties or even strange sequence. Let us now see one by one what are those groups and what are their characteristics-
Even numbers are known as those numbers which are divisible by 2 and hence the series of even numbers which starts with 2 is represented by 2, 4, 6, 8, ………. and so on. One interesting point about the even numbers is that any number whose right most digit is zero or divisible by 2 is an even number otherwise it will be an odd number. For example, 774 is an even number because the right most digit that is 4 is divisible by 2. Similarly 930 is an even number as the right most digit is 0.
The series of odd numbers start from 1 and they are not divisible by 2. Obviously, the series of odd numbers will be 1, 3, 5, 7, ………. and so on. Odd and even numbers alternate which means that after every odd number one even number comes after which again an odd will be there. In a month of 30 days if the local police ask us to park our vehicles right side on the odd days and left side on the even days then we will be parking our vehicle 15 times in the right side and 15 times in left side. In many cities such practice exists so that the shops on both the sides get fair treatment for visibility of their shops to the customer.
These are the numbers whose square roots are natural numbers and the series of such square numbers is represented by 4, 9, 16, 25, ………… and so on. The interesting thing is that the separation or difference between the successive numbers increases as we go ahead.
Prime numbers are very interesting and they can not be divided fully by any number except 1 and themselves. For example 13 is a prime number as it cannot be divided by any number. The series of prime numbers is 2, 5, 7, 13, 17, ………….. and so on. Searching the bigger prime numbers is a Mathematical challenge as we go ahead, it becomes more and more difficult. Computer programs are used to find newer and newer prime numbers. It is a very interesting project to find the newer and newer prime numbers.
This is another interesting group in which the successive numbers are related to the earlier numbers in such a way that every number is the sum of earlier two numbers. This group is -
1, 1, 2, 3, 5, 8, 13, 21, 34, …… and so on. It is interesting to note that in nature surrounding us certain items mimic this sequence like the number of leaves propagating to all sides in a branch of some plants. Some special patterns which look repetition of the basic pattern but in a different size. These are used in fashion fabric designs.
These are the numbers which have some factors which mean that they are created by the multiplication of some smaller numbers. For example 4, 6, 8, 9, 10 ….. all are composite numbers.
Practical ApplicationNumber theory has many practical applications. In Arithmetic, it is used for finding the factors, Greatest Common Divisor (G.C.D ), Common factors, Coding a sequence in a particular fashion etc.
For example let us find out the G.C.D. of three numbers 12, 36, and 96. For that we would find out with which numbers they are divisible. As these are composite numbers so let us see with which numbers they are divisible.
12 is divisible by 1, 2, 3, 4, 6, and 12.
36 is divisible by 1, 2, 3, 4, 6, 9, 12, 18, and 36.
96 is divisible by 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, and 96.
From the above three observations, we can easily see that 12 is the number by which all the three numbers are divisible so the G.C.D. of these three numbers is 12.
Apart from the applications belonging to an above Arithmetic category, the understanding of number theory has a pivotal role in many complex mathematical applications in areas like Security System in banking securities, E-commerce websites, Coding theory, Barcodes, Making of modular designs, Memory management system, Authentication system, Hash functions, Linear congruences, Pseudorandom numbers etc. It is expected that the number theory would be utilised in more and more innovative ways in the coming times.
ConclusionUnderstanding the number theory helps us in understanding the various sequences in which numbers can fall and can be used to create logic and sequence progression to solve or code a particular mathematical process. Today's advanced computer industry uses them in various configurations in the computer programs to take advantage of their peculiar properties.