density property of real numbers

In the previous post we have discussed about Decimal Place value and In today's session we are going to discuss about  density property of real numbers. As we all are very well aware from the concept of number system. Number system is a collection of all kinds of numbers that are part of mathematics. From those numbers, real number is one of them that deal with irrational numbers, fractional numbers and integers values. Here we are going to discuss about the density property of real numbers. Before understanding the concept of density property of real number we need to understand why we use it.

So answer is that properties with real numbers play an important role for solving various kinds of mathematical problems. Like in algebra real numbers property helps in solving algebraic equations in a understandable and easier manner.

Density property of real numbers are kind of property that tell us that there are infinite numbers are exists between two real numbers. In the simple mean we can say that density property of real numbers is a kind of property which shows that between any two real numbers there is some other numbers exists.

According to the standard definition we can say that density property can finds the infinite numbers that are lies between any two real numbers. Suppose there are two real numbers given that is 2 and 3. Now by using the concept of density property of real number we can say that there are infinite numbers are lies between them. Like 2.1, 2.2, 2.3, 2.4 and so on. (know more about density property of real numbers, here)

The concept of density property of real number also varies with the fractional numbers. It means to say that if we have two fractional value then there are infinite numbers also exists between the fractional value.

The concept of Van Der Waals Forces is used in physical chemistry which calculates the sum of attractive or repulsive force between the molecules. Students who are appearing in board exam then icse sample papers helps them for their exam preparation.

Decimal Place value

Decimal can be define as a mathematical fundamental concept that deals with the numbers of numerals system. Decimal can be defined as a point which is placed between the two or more digit value. Students who want to understand the concept of decimal, they first required to understand the concept of Decimal Place value. In mathematics, decimal position can be defined as a place of any number set through which the value of any number can be determined by their corresponding position. The number value which contains a decimal value within it then value of each number is very important.

In the normal form, each value of any number has their specific position or place value. Suppose we have a number 1234. Here the number 1234 has a specific number place value. The value 4 can be consider as unit value, value 3 can be consider in the position of tens, in the same aspect 1 and 2 can be consider as a place value of thousand and hundred position. (know more about Decimal Place value, here)

But when we put a decimal point between the digits of any number that has more then two digit into their value then place value of digits gets change into two categories. It means to say that right side digit of number has different place value and left side has different place value. Suppose we have number 123.456. Here we can see that there is a decimal point between the number values and both sides have different place value. Here left side of decimal value remains as like normal number value but right side has different place value.

After decimal first digit can be consider as a tenths place value means 4 tenths. In the same aspect the value 5 and 6 after decimal point can be consider as a fifth hundredths and sixth thousandths. Surface Area of Cone can be calculated by using the following formula that are given below:

Surface Area of Cone = pi * r * s + pi * r * r

icse board stands for Indian Certificate of Secondary education that conducts national level board examination for class 10th and 12th. In the next session we will discuss about Comparing Fractions. 

Comparing Fractions

In mathematics, fraction can be defined as a ration representation of two whole numbers. Fraction is a part of number system that helps the students for resolving various kinds of mathematical problem. Fraction is most important mathematical concept that performs the task of expressing any number into ‘n’ number of equal parts. Suppose there is pizza which is required to be distributed between three friends. At that time the concept of fraction resolve the problem by distributing the pizza equally through which each friend get one- third part of pizza to eat.

In the concept of fraction, we perform the representation of two whole numbers in role of dividend and divisor. Like 2 / 3 and 4 / 5. Here the value 2 and 4 can be consider as dividend known as numerator for fraction and 3 and 5 can be consider as divisor known as denominator for fraction number. (know more about Comparing Fractions, here)

Here we are going to discuss about one of the fractional concepts which is popularly known as comparing fraction. As a name specified as comparing fraction is task of performing a comparison between the two fractional values.

It means to say that Comparing Fractions is to calculate which number is greater and which is lower. The comparing fraction can be performed by using some symbol like <, > and =. To perform the comparing fraction successfully between fractional values can be done by following some step that is given below:

Step a) First we need to calculate LCD.

Step b) When we perform the step A then we need to calculate the product of LCD value with top and bottom value of fraction number for equalizing the denominator value of both fractional number.

Step c) In the last step perform the comparison of numerator value and put the respective sign between them.

Surface Area of a Prism in rectangular form can be calculated by using following formula

2 * length * width + 2 ( length + width ) height

For board examination icse guess papers 2013 guide the students to make their exam preparation better. In the next session we will discuss about Decimal Place value. 

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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