Sunday, December 15, 2019

Classification and applications of Real Numbers

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A mathematical object used to measure, count, and label is called a number. Mathematically we can split the numbers into two types; real and complex numbers. Real numbers can be further divided into two parts namely rational and irrational numbers. If a number has a terminating or non-terminating recurring (repeating) decimal expansion, then the number is called real number. If a number has non-terminating non-recurring (non-repeating) decimal expansion, then the number is called an irrational number.


Rational numbers can be written in the form a/b; b is not equal to 0, and they are further classified into three types; natural numbers, whole numbers and integers.

     Natural numbers: The smallest natural number is 1 and the largest natural number cannot be determined. Consecutive natural numbers differ by 1 and these are positive,i.e. 1, 2, 3, 4,....
     Whole numbers: The smallest whole number is 0 and the largest whole number cannot be determined. Unlike natural numbers, consecutive whole numbers also differ by 1. Except 0, every whole number is a natural number, i.e. 0, 1, 2, 3, 4,....
     Integers: The combination of natural numbers, zero and the negative of natural numbers constitutes integers. Negative numbers are left of zero on the number line whereas the positive numbers on the right side of zero. We cannot write the smallest and the largest integer since there are an infinite number of negative and positive natural numbers, i.e. ….,-3, -2, -1, 0, 1, 2, 3,....

As we know, a rational number a/b; (b is not equal to 0) is called an irrational number if its decimal expansion is non-terminating and non-repeating. For example root 2 is an irrational number because the value of root 2 is 1.41421356…, which is a non-terminating and non-repeating decimal expansion.

Few mathematicians study the real numbers with other logical foundations of Mathematics. In particular, the real numbers are also studied in constructive mathematics and in reverse mathematics. A basic fact is that, numbers play a significant role in our day-to-day life for existence. Real numbers are used in order to count and to measure out quantities of different items or shapes. For example, to find the number of toys in the house we use real numbers or to measure how much portion of milk is required to prepare tea, etc.

Adding to the above applications, real numbers are also used in geometry while calculating the areas, perimeters, volumes, etc., For example, to find sphere volume we need radius in the form of a real number to get an exact result. Other applications such as performance efficiency, Year on Year projections will help to calculate the amount of work done by a certain number of people in an organisation.

There are subdivisions in integers namely even numbers, odd numbers, prime numbers, composite numbers, Fibonacci numbers and so on. Furthermore, a number which is exactly divisible by 2 is called an even number, otherwise it is an odd number. Prime number is an integer greater than 1, which does not have any other factors except 1 and the number itself.

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