Rational and Irrational Numbers

In the previous post we have discussed about How to tackle Irrational Number
and In today's session we are going to discuss about Rational and Irrational Numbers, Our number line consists of various types of numbers which play different types of roles in different areas. These numbers are real numbers, natural numbers, complex, whole, prime, rational and irrational numbers.
But here we are going to discuss about only rational and irrational numbers.
Rational and irrational numbers are totally opposite of each other. Because rational numbers are those which can be expressed in terms of a fraction or quotient like a/b, where a and b are integers and b is not equals to zero. On the other side irrational numbers cannot be represented in the form of a simple fraction like a/b.
In rational numbers, the decimal expansion of the number either terminates after some finite sequence of digits or repeats same finite sequence of digits over and again.
Whereas, in irrational numbers, the decimal expansion of the number continues forever without even repeating same finite sequence of digits again and again.
Rational numbers are dense in nature, by saying that we mean that in between any two integers on the real number line; there exists many rational numbers in between them.
Examples of rational numbers are: 1.333333….., 2.5, 4/5, etc.
And examples of irrational numbers are: e (Euler’s number), π (pi),   √2 (square root of 2), φ (golden ratio), etc.
If we look towards the Cantor’s proof: it says that on the real number line almost all the numbers are irrational in nature, because as we know that our number line is a mixture of rational and irrational numbers, but real numbers are uncountable or infinite and rational numbers can be counted, so finally we can say that the remaining irrational numbers cannot be counted or are uncountable. Hence, the Cantor’s proof is verified.
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How to tackle Irrational Number

In the previous post we have discussed about Is 0 a Natural Number and In today's session we are going to discuss about Irrational Number, We have various types of numbers on our number line in mathematics, like, real numbers, whole numbers, rational, irrational number, complex, natural, integers, etc. But now, our most priority is irrational number which is nothing but a number which cannot be written in the form of a/b or fraction or a quotient, where a and b are integers and b is not equal to zero.

The decimal expansion of an irrational number neither ends or terminates itself nor it repeats some same sequence of digits over and again.

Since irrational number cannot be represented in the form of a simple fraction, so we can say that it is not a rational number, which is just the opposite of rational number as it can be expressed in the form of a quotient or a fraction like a/b.

Irrational numbers consists of numbers like e (Euler’s number), π (pi), √2 (square root of 2), φ (golden ratio), etc.

Whereas irrational numbers cannot be represented like 5/2, 2.14, 1/10, which rational numbers can.

According to the Cantor’s proof: as we that our number line on the coordinate plane is full of real numbers, which means they are uncountable and since we can count rational numbers, so we can finally conclude that almost all the real numbers are irrational in nature and not rational. But on the other side we have to say that rational numbers are dense in nature, which means that between any two integers there are many rational numbers but still they can be counted.

There are many facts which irrational numbers give, like when we calculate the ratio of lengths of two line segments and if it comes out to be an irrational number, then those line segments are known as incommensurable, which means that they have no common measures to share.   

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Is 0 a Natural Number

In the previous post we have discussed about List of Prime Numbers and In today's session we are going to discuss about Is 0 a Natural Number. Before discussing about the question: is 0 a natural number, let us first discuss about natural numbers. A natural number can be simply defined as a number which occurs very commonly and obviously in nature.
We can easily get the answer of the question: is 0 a natural number, by this definition, the natural number is a number in whole or non-negative number. We denote the set of natural numbers by N and it can be defined in either of the two ways: Positive numbers or non-negative numbers.

N = 0, 1, 2, 3, 4 ...

N = (1, 2, 3, 4, 5,….

So, the answer of the question: is 0 a natural number, is that there is no universal agreement on including zero in the set of natural numbers: many of us define the term natural number as positive integers only and for others natural numbers are the non-negative numbers. (know more about Natural numbers, here
Another explanation to the question: is 0 a natural number, goes like, the set of natural numbers, no matter it includes zero or not, is a denumerable set which means no matter how many elements  are there in a set, every element is denoted by the list which leads to the identity of the element itself in the set.
Now we are going to give an example in the question: is 0 a natural number, the example is no matter which of two lists of natural numbers we select ( 1, 2, 3, 4, ... or the list 0, 1, 2, 3, ...), the numbers 356,834,252 will always be a natural number, but 356,834,252.5, 3/4, and -32 can never be natural numbers.
So finally, the answer to the question: is 0 a natural number, is , that it is not certain that 0 is not a natural number, some people do consider it as a natural number.a
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List of Prime Numbers

In the previous post we have discussed about How to Solve Fraction Problems and In today's session we are going to discuss about List of Prime Numbers. The definition of a prime number is that it is the number which is not divisible by any number except 2 numbers which are 1 and the number itself. List of prime numbers starts with the number 2 and then keeps going on because the list of prime numbers is infinite. Generation of the subsets of the list of prime numbers can be carried out by the different formulae of the primes. The list of prime numbers in which first 100 prime numbers are given is as follows. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397,401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523 and 541.  
List of prime numbers by type of some important primes is as follows. Prime numbers in which the addition of their digits also gives a prime number are called the additive primes. Some of its examples are 2, 3, 5, 7, 11, 23, 29, 41, 43, 47, 61, 67, 83, 89, 101, 113, 131, etc. The prime numbers to which if we add 2 also give a prime or a semi prime are called chen prime numbers. Some of its examples are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 47, 53, 59, etc. The prime numbers of the 2n form are called the even prime numbers. There is only one even prime number that exists which is 2.
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How to Solve Fraction Problems

In the previous post we have discussed about how to compare fractions and In today's session we are going to discuss about How to Solve Fraction Problems. If we talk about the fraction numbers, we say that the numbers which can be expressed in the form of  a/ b , where a and b are whole numbers and b <> 0. Here we say that a is the numerator and b is the denominator. Thus to learn about how to do fractions, we must look into the different mathematical operators and the logical operators of the fraction numbers.
By the logical operators we mean comparing of two fraction numbers. If two fraction numbers are to be compared, we first check the numbers are proper fraction or not. Every time a proper fraction is less than the improper fraction number. As proper fraction number is always less than 1 and the improper fraction number is greater than 1.
If both the fractions are proper or improper, then we will check if the denominators of the two fraction numbers are same or not. In case the two denominators are same, it means that the smaller numerator represents the smaller fraction. In other situation we will look if the numerators of the two fraction numbers are same or not. If the numerators are same, then the smaller numerator represents the larger number.
 In another situation we find the fraction number where neither the numerator is same nor the denominator is same. In such cases we will convert the two fractions into their equivalent form such that the denominator of both the fraction numbers becomes same and it is done by first finding the LCM of the two denominators and then converting the fraction into their equivalent fraction numbers such that their denominator becomes equal to the LCM of the two denominators.
We can visit online math tutor which can help us to learn about what are prime numbers. In Gujarat state education board prime numbers are in the syllabus of grade 4 mathematics.

how to compare fractions

In mathematics, fraction is defined as the number of parts chosen to the total number of parts. Fraction is in the form of P/Q. Where P and Q are the two whole numbers. P is known as numerator and Q is known as denominator of the fraction. It means that the upper part of the fraction is known as numerator and the lower part is known as denominator. Fraction is written as:
                                                       1/3
Where 1 is P i.e. numerator or upper part and 3 is Q i.e. denominator or lower part of the fraction.
Comparison of fraction is an important topic in mathematics because sometimes it is required to compare the fraction. There are three ways to compare the fraction:
1. by decimal method.
2.  by same denominator method.
3. by different denominator method.

·         Decimal method is done by the use of calculator. Just convert the fraction into decimal then you can easily compare the smaller and bigger fraction. Just as (1 ÷4) = 0.25 and (2 ÷5) = 0.4. Therefore 0.4 is the bigger one so, 2/5 is greater than ¼.

·         In same denominator method, you have to just compare the numerator part i.e. the upper part of the fraction and write the result.

·         In different denominator method we have to do the cross multiplication. In this method first we have to multiply the numerator of the first fraction to the denominator of the second fraction and then multiply the denominator of first fraction to the numerator of the second fraction. At last compare the cross multiplication values and write the result i.e. bigger one.

For example:
Compare the fraction 3/5 and 5/6?
Solution:
Step1: first we have to multiply the numerator of the first fraction to the denominator of the second fraction i.e. 3 x 6 = 18
Step2: then multiply the denominator of first fraction to the numerator of the second fraction i.e. 5 x 5 = 25
Step3: compare the cross product values i.e. 18 and 25, here 25 is greater than 18 so 5/6 is greater than 3/5.
At last how to compare fractions and T Distribution Table are also described in west Bengal board of primary education.