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What Are Rational Numbers?

What Is a Rational Number ?Definition And Examples.

  1. What Is A Rational Number?

Common fractions that are required in daily life are termed as the rational numbers. These are the separate set of numbers that helps in proper assessment of the number system in regular day process. The numbers namely 1/2, 2/ 3, 5/4 are considered as the rational numbers. The rational numbers are termed rational numbers as they are present in the ration format. This is the main reason that they are termed as the rational number. Ration of whole numbers are considered as the rational numbers. Rational numbers are considered as the line of fractional numbers that are based on the sign of the numbers. The fact that the numbers that are present are both positive and negative in nature. The mathematicians are mainly focused on describing the whole numbers that are present in both positive and negative numbers. They are also considered as integers. This is the main reason that the class of rational numbers are considered as the class of integers. 2 = 2/1 = 6/ 3, this is the process that proper processing of rational number can be made. As it can be seen that 2 is a natural number. From 2 it can be written as 2/1. After writing the same as 6/3 it can be stated that the value that is received is similar to that of the natural number. This is the main reason that the functional processing of the integers are considered as the sectional part of the natural numbers as well as the rational numbers. Hence it can be stated that any number that is having a proper assessment of the fractional value of the non-zero denominator is considered. This has been concerned with the quotient and the fraction. It can be stated that the quotient needs to be zero only it is important for the processing of the rational numbers. But again if the rational number is prime in nature it is obvious that the characteristic zero will be present as per the processing of the subfield rational number. It can also be stated that the rational numbers will be the subset of the real numbers. Real numbers can be performed as per the sectioning process of the rational number with the help of Cauchy sequences, Dedekind cuts, or infinite decimals. Rational coordinate section ensures that proper management of the numeric value is preferred over the sectioning of the rational matrix. Usage of polynomials will be beneficial in the sectional processing of the rational expression and the rational function. This is to take into consideration that the accessing of the rational curve is not concerned with the curve that is defined by the rationales whereas the curve is defined via the parameterized rational functions.  The property of the rational number includes presence of the closure property. As per the presence of the closure property the main aspect that is considered is that after performing of the addition, subtraction, division and multiplication of different rational number will generate other rational numbers as well. Implementation of commutative law is also another aspect that is to be considered for the processing of the rational number. As per this law the addition process and the multiplication process is commutative in nature. This is the main reason that the functional process of the rational numbers can be depicted as a+b = b+a in case of the addition process and a×b = b×a in case of the multiplication process. But this can be stated that the division process of the rational numbers are not commutative in nature. That is the result that is received after the division is not commutative in nature. Associative law implementation will also be getting benefitted as per the sectional processing of the rational numbers. As per the associative law the main aspect that is considered is that in case the three numbers x, y and z are considered in the system, addition is performed then x+(y+z)=(x+y)+z will be the output of the project. Similarly in case of the multiplication section the main aspect that is considered is that x(yz)=(xy)z. this proves that the associative law will be benefitting the functional process of the addition and the multiplication process. In case of considering the associative law, 0 is also considered as the rational number.    

  1. Rational Number Examples

Examples of Rational Numbers

 1/2, 2/ 3, 5/4

  1. What’s an Irrational Number?

Irrational numbers are the real numbers that are not rational in nature. These are the numbers that are formed from the ratios. In case the reaction of the two segments that are irrational in nature, the main aspect that is considered is that the segment is considered to be incommensurable in nature. They are considered as the integer multiple of itself. This is the major aspect that is needed to be considered for the processing or the numeric assessment. As per the irrational numbers the main concern that is taken into consideration is that ration pie of the circle’s circumference and as per the diameter. This is the major aspect that is needed to be performed for better analysis of the irrational number is that the sectional integration of the natural numbers along with the perfect squares are made. These are the categories when the number is considered as the irrational number. This numbers do not contain sub sequential digits. This is the main reason that the functional processing of the irrational numbers will be made in decimal manner. Pie can be considered as a prime example of irrational number. In the case of pie the main aspect that is considered is the number is present in the decimal format and this will ensure that the entire value can never be detected. In case of Pie, 3.14159 is the original value that is generally considered for performing numerical calculations. In this case the main aspect that can be stated that despite the decimal points that are considered, there can never be an end to the number system that can be stated. It is considered that the square root value of the number is the irrational number. The numbers are again classified and calculated as the cubic irrationals that provides irrational magnitudes. As per the properties of the irrational number the main aspect that is considered for the processing of addition of an irrational numbers, after addition the result that will be received is irrational in nature. For example the number x which is rational and y that is irrational, the result that was received is irrational in nature. Another property that is considered is that the multiplication of the irrational number with a non zero number will give a number that is irrational in nature. For example, x being an irrational number and y being a non zero number the product of x and y i,e. xy will be an irrational number. It is also stated that LCM of more than one irrational number is not available. However it is stated that product of 2 irrational number might be producing rational number.

  1. Famous Irrational Numbers
  • The famous irrational numbers that are considered is that the Euler number. Euler number is considered as a famous irrational number with number set being 7182818284590452353602874713527.
  • Pie is also a famous irrational number

From the above discussion it can be stated that the rational numbers that include integers which include integers and whole numbers along with their negative counterpart. The negative counterpart also ensures that the sign of the number does not play any major role in the detection of the rational and irrational numbers. In case of the rational numbers it is expressed in ratios. This is the main reason it can be stated that the number is a part of the whole number. Rational numbers are the fractions that have numerators which depicts the part that is taken from the whole sectioning. After this basic evaluation it can be stated that rational numbers are the numbers that are expressed in fractions. Another aspect that is to be considered is that the denominators of the fraction is a non zero number, This is the  main aspect that differentiation in between the rational number and the irrational number gets easier. In case a and b both are integers, a/b, b≠0. From this equation it can be stated that the integers along with the characteristics of the finite decimal process, the numbers are considered as the rational numbers. Again in case of the irrational number the main aspect that is considered is that the numbers are the real numbers that ensures proposing of the simple fraction that will be performing the fractions like 6/1. Pie is considered as an irrational number. Despite the fact that pie is a real number the main aspect that is considered is that it is infinite in nature. Hence due to the infiniteness of the number system of pie, it is considered as the irrational number.  

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