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Proofs of Irrationality

Tuesday, February 14th, 2012 19:18 / 3 Comments

“Irrational numbers are those real numbers which are not rational numbers!”

A rational number is a real number which can be expressed in the form of $\frac{a}{b}$ where $a$ and $b$ are both integers relatively prime to each other and $b$ being non-zero.
Following two statements are equivalent to the definition 1.
1. $x=\frac{a}{b}$ is rational if and only if $a$ and $b$ are integers relatively prime to each other and $b$ does not equal to zero.
2. $x=\frac{a}{b} \in \mathbb{Q} \iff \mathrm{g.c.d.} (a,b) =1, \ a \in \mathbb{Z}, \ b \in \mathbb{Z} \setminus \{0\}$ .

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Gamma Function

Tuesday, February 7th, 2012 15:35 / Leave a Comment

If we consider the integral $I =\displaystyle{\int_0^{\infty}} e^{-t} t^{a-1} \mathrm dt$ , it is once seen to be an infinite and improper integral. This integral is infinite because the upper limit of integration is infinite and it is improper because $t=0$ is a point of infinite discontinuity of the integrand, if $a<1$ , where $a$ is either real number or real part of a complex number. This integral is known as Euler’s Integral. This is of a great importance in mathematical analysis and calculus. The result, i.e., integral, is defined as a new function of real number $a$ , as $\Gamma (a) =\displaystyle{\int_0^{\infty}} e^{-t} t^{a-1} \mathrm dt$ .

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The Area of a Disk

Friday, January 27th, 2012 16:22 / 1 Comment

~~[This post is under review.]~~

If you are aware of elementary facts of geometry, then you might know that the area of a disk with radius $R$ is $\pi R^2$ .

The radius is actually the measure(length) of a line joining the center of disk and any point on the circumference of the disk or any other circular lamina. Radius for a disk is always same, irrespective of the location of point at circumference to which you are joining the center of disk. The area of disk is defined as the ‘measure of surface‘ surrounded by the round edge (circumference) of the disk.

The area of a disk can be derived by breaking it into a number of identical parts of disk as units — calculating their areas and summing them up till disk is reformed. There are many ways to imagine a unit of disk. We can imagine the disk to be made up of several concentric very thin rings increasing in radius from zero to the radius of disc. In this method we can take an arbitrary ring, calculate its area and then in similar manner, induce areas of other rings -sum them till whole disk is obtained. (more…)

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Triangle Inequality

Friday, January 20th, 2012 10:57 / 5 Comments

Triangle inequality has its name on a geometrical fact that the length of one side of a triangle can never be greater than the sum of the lengths of other two sides of the triangle. If $a$ , $b$ and $c$ be the three sides of a triangle, then neither $a$ can be greater than $b+c$ , nor $b$ can be greater than $c+a$ , nor $c$ can be than $a+b$ .

: Triangle

Consider the triangle in the image, side $a$ shall be equal to the sum of other two sides $b$ and $c$ , only if the triangle behaves like a straight line. Thinking practically, one can say that one side is formed by joining the end points of two other sides.
In modulus form, $|x+y|$ represents the side $a$ if $|x|$ represents side $b$ and $|y|$ represents side $c$ . A modulus is nothing, but the distance of a point on the number line from point zero.

: Visual representation of Triangle inequality

For example, the distance of $5$ and $-5$ from $0$ on the initial line is $5$ . So we may write that $|5|=|-5|=5$ .

Triangle inequalities are not only valid for real numbers but also for complex numbers, vectors and in Euclidean spaces. In this article, I shall discuss them separately. (more…)

On Ramanujan’s Nested Radicals

Friday, December 30th, 2011 08:50 / 8 Comments

Ramanujan (1887-1920) discovered some formulas on algebraic nested radicals. This article is based on one of those formulas. The main aim of this article is to discuss and derive them intuitively. Nested radicals have many applications in Number Theory as well as in Numerical Methods.

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A Trip to Mathematics: Part V Equations-I

Saturday, December 3rd, 2011 14:25 / Leave a Comment

Applied mathematics is one which is used in day-to-day life, in solving tensions (problems) or in business purposes. Let me write an example:

George had some money. He gave 14 Dollars to Matthew. Now he has 27 dollars. How much money had he?

If you are familiar with day to day calculations —you must say that George had 41 dollars, and since he had 41, gave 14 to Matthew saving 27 dollars. That’s right? Off course! This is a general(layman) approach. ‘How will we achieve it mathematically?’ —We shall restate the above problem as another statement (meaning the same):

George had ~~some money~~ $x$ dollars. He gave 14 dollars to Matthew. Now he has 27 dollars. ~~How much money he had?~~ Find the value of $x$ .

This is equivalent to the problem asked above. I have just replaced ‘some money’ by ‘x dollars’. As ‘some’ senses as unknown quantity— $x$ does the same. Now all we need to get the value of x.
When solving for $x$ , we should have a plan like this:

George had	$x$ dollars.
He gave to Matthew	14 dollars
Now he must have	$x-14$ dollars

But problem says that he has 27 dollars left. This implies that $x-14$ dollars are equal to 27 dollars.
i.e., $x-14=27$

$x-14=27$ contains an alphabet x which we assumed to be unknown—can have any certain value. Statements (like $x-14=27$ ) containing unknown quantities and an equality are called Equations. The unknown quantities used in equations are called variables, usually represented by bottom letters in english alphabet (e.g., $x,y,z$ ). Top letters of alphabet ( $a,b,c,d$ ..) are usually used to represent constants (one whose value is known, but not shown).

Now let we concentrate on the problem again. We have the equation x-14=27.
Now adding 14 to both sides of the equal sign:
$x-14 +14 =27 +14$
or, $x-0 = 41$ (-14+14=0)
or, $x= 41$ .
So, $x$ is 41. This means George had 41 dollars. And this answer is equal to the answer we found practically. Solving problems practically are not always possible, specially when complicated problems encountered —we use theory of equations. To solve equations, you need to know only four basic operations viz., Addition, Subtraction, Multiplication and Division; and also about the properties of equality sign.
We could also deal above problem as this way:
$x-14= 27$
or, $x= 27+14 =41$
-14 transfers to another side, which makes the change in sign of the value, i.e., +14.

When we transport a number from left side to right of the equal sign, the sign of the number changes and vice-versa. As here -14 converts into +14; +18 converts into -18 in example below:
$x+18 =32$
or, $x=32 -18 =14$ .
Please note, any number not having a sign before its value is deemed to be positive—e.g., 179 and +179 are the same, in theory of equations.
Before we proceed, why not take another example?

Marry had seven sheep. Marry’s uncle gifted her some more sheep. She has eighteen sheep now. How many sheep did her uncle gift?

First of all, how would you state it as an equation?
$7 + x = 18$
or, $+7 +x =18$ (just to illustrate that 7=+7)
or, $x= 18-7 =9$ .
So, Marry’s uncle gifted her 9 sheep. ///
Now tackle this problem,

Monty had some cricket balls. Graham had double number of balls as compared to Monty. Adam had also 6 cricket balls. They all collected their balls and found that total number of cricket balls was 27. How many balls had Monty and Graham?

As usual our first step to solve this problem must be to restate it as an equation. We do it like this:
Monty had (let) x balls.
Then Graham must had $x \times 2=2x$ balls.
Adam had 6 balls.
The total sum= $x+2x+6=3x+6$
But that is 27 according to our question.
Hence, $3x+6=27$
or, $3x=27-6 =21$
or, $x=21 /3 =7$ .
Here multiplication sign converts into division sign, when transferred.
Since $x=7$ , we can say that Monty had 7 balls (instead of x balls) and Graham had 14 (instead of $2x$ ).
///

Types of Equations

They are many types of algebraic equations (we suffix ‘Algebraic’ because it includes variables which are part of algebra) depending on their properties. In common we classify them into two main parts:

1. Equations with one variable (univariable algebraic equations, or just Univariables)

2. Equations with more than one variables (multivariable algebraic equations, or just Multivariables)

Univariable Equations

Equations consisting of only one variable are called univariable equations.

All of the equations we solved above are univariables since they contain only one variable (x). Other examples are:
$3x+2=5x-3$ ;
$x^2+5x +3=0$ ;
$e^x =x^e$ (e is a constant).

Univariables are further divided into many categories depending upon the degree of the variable. Some most common are:

Linear Univariables: Equations having the maximum power (degree) of the variable 1.
$ax+b=c$ is a general example of linear equations in one variable, where a, b and c are arbitrary constants.
Quadratic Equations: Also known as Square Equations, are ones in which the maximum power of the variable is 2.
$ax^2+bx+c=0$ is a general example of quadratic equations, where a,b,c are constants.
Cubic Equations: Equations of third degree (maximum power=3) are called Cubic.
A cubic equation is of type $ax^3+bx^2+cx+d=0$ ; where a,b,c,d are constants.
Quartic Equations: Equations of fourth degree are Quartic.
A quartic equation is of type $ax^4+bx^3+cx^2+dx+e=0$ .

Similarly, equation of an n-th degree can be defined if the variable of the equation has maximum power n.

Multivariable Equations:

Some equations have more than one variables, as $ax^2+2hxy+by^2=0$ etc. Such equations are termed as Multivariable Equations. Depending on the number of variables present in the equations, multivariable equations can be classified as:

1. Bi-variable Equations - Equations having exactly two variables are called bi-variables.
$x+y=5$ ; $x^2+y^2=4$ ; $r^2+\theta^2=k^2$ , where k is constant; etc are equations with two variables.
Bivariable equations can also be divided into many categories, as same as univariables were.

A.Linear Bivariable Equations: Power of a variable or sum of powers of product of two variables does not exceed 1.
For example: $ax+by=c$ is a linear but $axy=b$ is not.
B. Second Order Bivariable Equations: Power of a variable or sum of powers of product of two variables does not exceed 2.
For example: $axy=b$ , $ax^2+by^2+cxy+dx+ey+f=0$ are of second order.
Similarly you can easily define n-th order Bivariable equations.

2. Tri-variable Equations: Equations having exactly three variables are called tri-variable equations.
$x+y+z=5$ ; $x^2+y^2-z^2=4$ ; $r^3+\theta^3+\phi^3=k^3$ , where k is constant; etc are trivariables. (Further classification of Trivariables are not necessary, but I hope that you can divide them into more categories as we did above.)
Similary, you can easily define any n-variable equation as an equation in which the number of variables is n.

Out of these equations, we shall discuss only Linear Univariable Equations here (actually we are discussing them). ////

We have already discussed them above, for particular example. Here we’ll discuss them for general cases.
As told earlier, a general example of linear univariable equation is $ax+b=c$ .
We can adjust it by transfering constants to one side and keeping variable to other.
$ax+b = c$
or, $ax = c-b$
or, $x = \frac{c-b}{a}$
this is the required solution.
Example: Solve $3x+5=0$ .
We have, $3x+5=0$
or, $3x = 0-5 =-5$
or, $x = \frac{-5}{3}$ ////

A Trip to Mathematics: Part IV Numbers

Monday, November 21st, 2011 16:07 / Leave a Comment

If logic is the language of mathematics, Numbers are the alphabet. There are many kinds of number we use in mathematics, but at a broader aspect we may categorize them in two categories:
1. Countable Numbers
2. Uncountable Numbers
The names are enough to explain the properties of above numbers. The numbers which can be counted in nature are called Countable Numbers and the numbers which can not be counted are called Uncountable Numbers.

Well, this is not the correct way to classify the bunch of types of numbers. We have some formal names for special types of numbers, like Real numbers, Complex Numbers, Rational Numbers, Irrational Numbers etc.. We shall discuss these non-interesting numbers (let me say them non-interesting) at first and then some interesting numbers(those numbers are really interesting to learn). Although in this post I have concisely described the classification, I will rigorously discuss them later.
Let me start this discussion with the memorable quote by Leopold Kronecker:

“God created the natural numbers, and all the rest is the work of man.”

What does it mean? What did Kronecker think when he made this quote? Why is this quote true? —First part of this article is based on this discussion.
Actually, he meant to say that all numbers, like Real Numbers, Complex Numbers, Fractions, Integers, Non-integers etc. are made up of the numbers given by God to the human. These God Gifted numbers are actually called Natural Numbers. Natural Numbers are the numbers which are used to count things in nature.

Eight pens, Eighteen trees, Three Thousands people etc. are measure of natural things and thus ‘Eight’, ‘Eighteen’, ‘Three Thousands’ etc. are called natural numbers and we represent them numerically as ’8′, ’18′, ’3000′ respectively. So, if 8, 18, 3000 are used in counting natural things, are natural numbers. Similarly, 1, 2, 3, 4, and other numbers are also used in counting things —thus these are also Natural Numbers.

Let we try to form a set of Natural Numbers. What will we include in this set?

1?    (yes!).
2?    (yes).
3? (yes).
….
1785?    (yes)
…and so on.

This way, after including all elements we get a set of natural numbers {1, 2, 3, 4, 5, …1785, …, 2011,….}. This set includes infinite number of elements. We represent this set by Borbouki’s capital letter N, which looks like $\mathbb{N}$ or bold capital letter N ( $\mathbf{N}$ where N stands for NATURAL. We will define the set of all natural numbers as:

$\mathbb{N} := \{ 1, 2, 3, 4, \ldots, n \ldots \}$ .

It is clear from above set-theoretic notation that $n$ -th element of the set of natural numbers is $n$ .
In general, if a number $n$ is a natural number, we right that $n \in \mathbb{N}$ .
Please note that some mathematicians (and Wolfram Research) treat ’0′ as a natural number and state the set as $\mathbb{N} :=\{0, 1, 2, \ldots, n-1, \ldots \}$ , where $n-1$ is the nth element of the set of natural numbers; but we will use first notion since it is broadly accepted.

Now we shall try to define Integers in form of natural numbers, as Kronecker’s quote demands. Integers (or Whole numbers) are the numbers which may be either positives or negatives of natural numbers including 0.
Few examples are 1, -1, 8, 0, -37, 5943 etc.
The set of integers is denoted by $\mathbb{Z}$ or $\mathbf{Z}$ (here Z stands for ‘Zahlen‘, the German alternative of integers). It is defined by
$\mathbb{Z} := \{ \pm n: n \in \mathbb{N} \} \cup \{0\}$
i.e., $\mathbb{Z} := \{\ldots -3, -2, -1, 0, 1, 2, 3 \ldots \}$ .

Now, if we again consider the statement of Kronecker, we might ask that how could we prepare the integer set $\mathbb{Z}$ by the set $\mathbb{N}$ of natural numbers? The construction of $\mathbb{Z}$ from $\mathbb{N}$ is motivated from the requirement that every integer can be expressed as difference of two positive integers (i.e., Natural Numbers). Let $a,b,c,d \in \mathbb{N}$ and a relation ρ is defined on $\mathbb{N} \times \mathbb{N}$ by $(a,b) \rho (c,d)$ if and only if $a+d = b+c$ . The relation ρ is an equivalence relation and the equivalence classes under ρ are called integers and defined as $\mathbb{Z} := \mathbb{N} \times \mathbb{N} /\rho$ . Now we can define set of integers by an easier way, as $\mathbb{Z}:= \{a-b; \ a,b \in \mathbb{N}\}$ . Thus an integer is a number which can be produced by difference of two or more natural numbers. And similarly as converse defintion, positive integers are called Natural Numbers.
After Integers, we head to rational numbers. Say it again– ‘ratio-nal numbers‘ –numbers of ratio.

Image via Wikipedia

A rational number $\frac{p}{q}$ is defined as a ratio of an integer p and a non-zero integer q. (Well that is not a perfect definition, but as an introduction it is great for understanding.) The set of rational numbers is defined by $\mathbb{Q}$ .
Once integers are formed, we can form Rational (and Irrational numbers: numbers which are not rational ) using integers.
We consider an ordered pair $(p,q):=\mathbb{Z} \times (\mathbb{Z} \setminus \{0 \})$ and another ordered pair $(r,s):=\mathbb{Z} \times (\mathbb{Z} \setminus \{0\})$ and define a relation ρ by $(p,q) \rho (r,s) \iff ps=qr$ for $p,q,r,s \in \mathbb{Z}, \ q, r \ne 0$ . Then ρ is an equivalence relation of rationality, class (p,q). The set $\mathbb{Z} \times (\mathbb{Z} \setminus \{0\})/\rho$ is denoted by $\mathbb{Q}$ (and the elements of this set are called rational numbers).
In practical understandings, the ratio of integers is a phrase which will always help you to define the rational numbers. Examples are $\frac{6}{19}, \ \frac{-1}{2}=\frac{-7}{14}, \ 3\frac{2}{3}, \ 5=\frac{5}{1} \ldots$ . Set of rational numbers includes Natural Numbers and Integers as subsets.
Consequently, irrational numbers are those numbers which can not be represented as the ratio of two integers. For example $\pi, \sqrt{3}, e, \sqrt{11}$ are irrationals.
The set of Real Numbers is a relatively larger set, including the sets of Rational and Irrational Numbers as subsets. Numbers which exist in real and thus can be represented on a number line are called real numbers. As we formed Integers from Natural Numbers; Rational Numbers from Integers, we’ll form the Real numbers by Rational numbers.
The construction of set $\mathbb{R}$ of real numbers from $\mathbb{Q}$ is motivated by the requirement that every real number is uniquely determined by the set of rational numbers less than it. A subset $L$ of $\mathbb{Q}$ is a real number if L is non-empty, bounded above, has no maximum element and has the property that for all $x, y \in \mathbb{Q}, x < y$ and $y \in L$ implies that $x \in L$ . Real numbers are the base of Real Analysis and detail study about them is case of study of Real Anlaysis.
Examples of real numbers include both Rational (which also contains integers) and Irrational Numbers.

The square root of a negative number is undefined in one dimensional number line (which includes real numbers only) and is treated to be imaginary. The numbers containing or not containing an imaginary number are called complex numbers.
Some very familiar examples are $3+\sqrt{-1}, \sqrt{-1} =i, \ i^i$ etc. We should assume that every number (in lay approach) is an element of a complex number. The set of complex numbers is denoted by $\mathbb{C}$ . In constructive approach, a complex number is defined as an ordered pair of real numbers, i.e., an element of $\mathbb{R} \times \mathbb{R}$ [i.e., $\mathbb{R}^2$ ] and the set as $\mathbb{C} :=\{a+ib; \ a,b \in \mathbb{R}$ . Complex numbers will be discussed in Complex Analysis more debately.
We verified Kronecker’s quote and shew that every number is sub-product of postive integers (natural numbers) as we formed Complex Numbers from Real Numbers; Real Numbers from Rational Numbers; Rational Numbers from Integers and Integers from Natural Numbers. //
Now we reach to explore some interesting kind of numbers. There are millions in name but few are the follow:
Even Numbers: Even numbers are those integers which are integral multiple of 2. $0, \pm 2, \pm 4, \pm 6 \ldots \pm 2n \ldots$ are even numbers.

Odd Numbers: Odd numbers are those integers which are not integrally divisible by 2. $\pm 1, \pm 3, \pm 5 \ldots \pm (2n+1) \ldots$ are all odd numbers.

Prime Numbers: Any number $p$ greater than 1 is called a prime number if and only if its positive factors are 1 and the number $p$ itself.
In other words, numbers which are completely divisible by either 1 or themselves only are called prime numbers. $2, 3, 5, 7, 11, 13, 17, 19, 23, 29 \ldots$ etc. are prime numbers or Primes. The numbers greater than 1, which are not prime are called Composite numbers.
Twin Primes: Consecutive prime numbers differing by 2 are called twin primes. For example 5,7; 11,13; 17,19; 29,31; … are twin primes.

Pseudoprimes: Chinese mathematicians claimed thousands years ago that a number $n$ is prime if and only if it divides $2^n -2$ . In fact this conjecture is true for $n \le 340$ and false for upper numbers because first successor to 340, 341 is not a prime ( $31 \times 11$ ) but it divides $2^{341}-2$ . This kind of numbers are now called Pseudoprimes. Thus, if n is not a prime (composite) then it is pseudoprime $\iff n | 2^n-2$ (read as ‘n divides 2 powered n minus 2‘). There are infinitely many pseudoprimes including 341, 561, 645, 1105.

Carmichael Numbers or Absolute Pseudoprimes: There exists some pseudoprimes that are pseudoprime to every base $a$ , i.e., $n | a^n -a$ for all integers $a$ . The first Carmichael number is 561. Others are 1105, 2821, 15841, 16046641 etc.

e-Primes: An even positive integer is called an e-prime if it is not the product of two other even integers. Thus 2, 6, 10, 14 …etc. are e-primes.

Germain Primes: An odd prime p such that 2p+1 is also a prime is called a Germain Prime. For example, 3 is a Germain Prime since $2\times 3 +1=7$ is also a prime.
Relatively Prime: Two numbers are called relatively prime if and only their greatest common divisor is 1. In other words, if two numbers are such that no integer, except 1, is common between them when factorizing. For example: 7 and 9 are relatively primes and same are 15, 49.

Perfect Numbers: A positive integer n is said to be perfect if n equals to the sum of all its positive divisors, excluding n itself. For example 6 is a perfect number because its divisors are 1, 2, 3 and 6 and it is obvious that 1+2+3=6. Similarly 28 is a perfect number having 1, 2, 4, 7, 14 (and 28) as its divisors such that 1+2+4+7+14=28. Consecutive perfect numbers are 6, 28, 496, 8128, 33550336, 8589869056 etc.

Mersenne Numbers and Mersenne Primes: Numbers of type $M_n=2^n-1; \ n \ge 1$ are called Mersenne Numbers and those Mersenne Numbers which happen to be Prime are called Mersenne Primes. Consecutive Mersenne numbers are 1, 3 (prime), 7(prime), 15, 31(prime), 63, 127.. etc.

Catalan Numbers: The Catalan mumbers, defined by $C_n = \dfrac{1}{n+1} \binom{2n}{n} = \dfrac{(2n)!}{n! (n+1)!} \ n =0, 1, 2, 3 \ldots$ form the sequence of numbers 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, …

Triangular Number: A number of form $\dfrac{n(n+1)}{2} \ n \in \mathbb{N}$ represents a number which is the sum of n consecutive integers, beginning with 1. This kind of number is called a Triangular number. Examples of triangular numbers are 1 (1), 3 (1+2), 6 (1+2+3), 10(1+2+3+4), 15(1+2+3+4+5) …etc.

Square Number: A number of form $n^2 \ n \in \mathbb{N}$ is called a sqaure number.
For example 1 ( $1^2$ ), 4 ( $2^2$ ), 9( $3^2$ ), 16 ( $4^2$ )..etc are Square Numbers.

Palindrome: A palindrome or palindromic number is a number that reads the same backwards as forwards. For example, 121 is read same when read from left to right or right to left. Thus 121 is a palindrome. Other examples of palindromes are 343, 521125, 999999 etc.

A Trip To Mathematics: Part II Set(Basics) (wpgaurav.wordpress.com)
Do complex numbers really exist? (math.stackexchange.com)
Definitions (gowers.wordpress.com)

MY DIGITAL NOTEBOOK

Category Archives: Study Notes

Proofs of Irrationality

Gamma Function

The Area of a Disk

Triangle Inequality

On Ramanujan’s Nested Radicals

A Trip to Mathematics: Part V Equations-I

Types of Equations

A Trip to Mathematics: Part IV Numbers

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