If you are familiar with complex numbers, the "imaginary" number i has the property that the square of i is -1. It is a rather curious fact that i raised to the i-th power is actually a real number!
In fact, its value is approximately 0.20788.
Presentation Suggestions:
This makes a great exercise after learning the basics about complex numbers.
The Math Behind the Fact:
From Euler's formula, we know that exp(i*x) = cos(x) + i*sin(x), where "exp(z)" is the exponential function e power z. Then exp(i*Pi/2) = cos(Pi/2) + i*sin(Pi/2) = i.
Raising both sides to i-th power, we see that the right side is the desired quantity i power i, while the left side becomes exp(i*i*Pi/2), or exp(-Pi/2), which is approximately .20788.
(Actually, this is one of many possible values for i to the i, because, for instance, exp(5i*Pi/2)=i. In complex analysis, one learns that exponentiation with respect to i is a multi-valued function.)
9/23/10
Does the Orbit of Earth Stable?
We have some better understanding about the rotation of earth. This article made some clear idea of rotation of Sun, Earth and moon. The motion of Sun and Earth understand by us by using two dimension pictures. Here the video given below makes our thoughts about the motion of Sun and Earth very clear. Also, it try to describe the orbit of Earth is not Stable.
9/22/10
WHAT IS ZERO?
We already discuss that there is nothing to explain to understand the whole thing about something. This article built the base in your mind to understand the emptiness (zero). No one can explain zero completely. But any one can understand something about zero. Here I try to explain how to know the zero completely.
- A born baby felt that the emptiness of his stomach. And start crying to get food.
- 5 years old kid could understand his/her water bottle or Tiffin box is empty.
Consider the example,
I give you 1 unit of product (1kg rice) in your hand. And allow you to give exactly half to others. For the first person you can give half unit of the product (half kg rice). You can easily understand that at some finite stage there is only-one rice in your hand. After some another stages an atom (10-10) size of product is in your hand.
But no one is willing to receive the half of the atom. Because they felt that there is nothing. It is real, because you couldn’t see that but it is not true, because you had half of the atom. So your hand is not empty. It contains a half atom. The truth is your hand will not be empty at any finite stage.
It will be empty in infinite stages. You could not explain the zero without infinite and you could not explain the infinite without zero. To understand the zero and infinite, the following model is used,
Consider the real line as a closed envelope, the top arrow that is the starting as well as ending of the envelope that means it is infinite. The bottom arrow is meant for zero. Now it is clear that this envelop represents whole real number system. Consider any real point in the circumstance of this envelope; it is static in its position.
If you travel at any speed from any real point towards bottom arrow (zero) or top arrow (infinite) in this envelope, you will be in static position.
9/19/10
WHAT IS BUFFON'S NEEDLE?
BUFFON'S NEEDLE PROBLEM:
what is Buffon's needle problem?
It is interesting, it is the problem of calculating the approximate value of pi.
what's the problem is?
Throw a needle of length L at random on a floor covered by equidistant parallel lines d units apart. What is the probability that the needle will cross at least one of the lines?
SOLUTION:
Let the length of the needle is one unit and the distance between the lines is also one unit. There are two variables, the angle at which the needle falls (theta) and the distance from the center of the needle to the closest line (D). Theta can vary from 0 to 180 degrees and is measured against a line parallel to the lines on the paper. The distance from the center to the closest line can never be more that half the distance between the lines.
The needle will hit the line if the closest distance to a line (D) is less than or equal to 1/2 times the sine of theta. That is, D <= (1/2)sin(theta).
we now plot D along the ordinate and (1/2)sine(theta) along the abscissa. The values on or below the curve represent a hit (D <= (1/2)sin(theta)). Thus, the probability of a success it the ratio shaded area to the entire rectangle.
The shaded portion is found with using the definite integral of (1/2)sin(theta) evaluated from zero to pi. The result is that the shaded portion has a value of 1. The value of the entire rectangle is (1/2)(pi) or pi/2. So, the probability of a hit is 1/(pi/2) or 2/pi. That's approximately .6366197.
To calculate Pi from the needle drops, simply take the number of tries and multiply it by two, then divide by the number of hits, or
2(total tries) / (number of hits) = pi (approximately).
what is Buffon's needle problem?
It is interesting, it is the problem of calculating the approximate value of pi.
what's the problem is?
Throw a needle of length L at random on a floor covered by equidistant parallel lines d units apart. What is the probability that the needle will cross at least one of the lines?
SOLUTION:
Let the length of the needle is one unit and the distance between the lines is also one unit. There are two variables, the angle at which the needle falls (theta) and the distance from the center of the needle to the closest line (D). Theta can vary from 0 to 180 degrees and is measured against a line parallel to the lines on the paper. The distance from the center to the closest line can never be more that half the distance between the lines.
The needle will hit the line if the closest distance to a line (D) is less than or equal to 1/2 times the sine of theta. That is, D <= (1/2)sin(theta).
we now plot D along the ordinate and (1/2)sine(theta) along the abscissa. The values on or below the curve represent a hit (D <= (1/2)sin(theta)). Thus, the probability of a success it the ratio shaded area to the entire rectangle.
The shaded portion is found with using the definite integral of (1/2)sin(theta) evaluated from zero to pi. The result is that the shaded portion has a value of 1. The value of the entire rectangle is (1/2)(pi) or pi/2. So, the probability of a hit is 1/(pi/2) or 2/pi. That's approximately .6366197.
To calculate Pi from the needle drops, simply take the number of tries and multiply it by two, then divide by the number of hits, or
2(total tries) / (number of hits) = pi (approximately).
9/16/10
INFINITE- DIMENSIONAL SPACE
Here are two common but unhelpful was to think about infinity.
1. Infinity makes things harder.
2. Infinity is a useless academic abstraction.
Neither of these is necessarily true. Problems are often formulated in terms of infinity to make things easier and to solve realistic problems. Infinity is usually a simplification. Think of infinity as “so big I don’t have to worry about how big it is.”
Infinite dimensional spaces guide our thinking about large finite dimensional spaces. If you want to solve a practical problem in high dimensions, the infinite dimensional case may be a better guide ordinary three dimensional space. Continuity in infinite dimensional spaces requires structure that may not be apparent in low dimensions. Thinking about the infinite case may prepare you to exploit that structure in a large finite dimensional problem.
In mathematics and physics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify each point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it. A surface such as a plane or the surface of a cylinder or sphere has a dimension of two because two coordinates are needed to specify a point on it (for example, to locate a point on the surface of a sphere you need both its latitude and its longitude). The inside of a cube, a cylinder or a sphere is three-dimensional because three co-ordinates are needed to locate a point within these spaces.
The concept of infinite-dimensional spaces can be formulated in a meaningful way in Hilbert spaces, which are an essential concept in quantum mechanics.The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions.Infinite dimensional spaces guide our thinking about large finite dimensional spaces.
1. Infinity makes things harder.
2. Infinity is a useless academic abstraction.
Neither of these is necessarily true. Problems are often formulated in terms of infinity to make things easier and to solve realistic problems. Infinity is usually a simplification. Think of infinity as “so big I don’t have to worry about how big it is.”
Infinite dimensional spaces guide our thinking about large finite dimensional spaces. If you want to solve a practical problem in high dimensions, the infinite dimensional case may be a better guide ordinary three dimensional space. Continuity in infinite dimensional spaces requires structure that may not be apparent in low dimensions. Thinking about the infinite case may prepare you to exploit that structure in a large finite dimensional problem.
In mathematics and physics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify each point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it. A surface such as a plane or the surface of a cylinder or sphere has a dimension of two because two coordinates are needed to specify a point on it (for example, to locate a point on the surface of a sphere you need both its latitude and its longitude). The inside of a cube, a cylinder or a sphere is three-dimensional because three co-ordinates are needed to locate a point within these spaces.
The concept of infinite-dimensional spaces can be formulated in a meaningful way in Hilbert spaces, which are an essential concept in quantum mechanics.The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions.Infinite dimensional spaces guide our thinking about large finite dimensional spaces.
9/15/10
Fuzzy Set
A fuzzy set is a pair (A,m) where A is a set and m : A \rightarrow [0,1].
For each x\in A, m(x) is called the grade of membership of x in (A,m). For a finite set A = {x1,...,xn}, the fuzzy set (A,m) is often denoted by {m(x1) / x1,...,m(xn) / xn}.
Let x \in A. Then x is called not included in the fuzzy set (A,m) if m(x) = 0, x is called fully included if m(x) = 1, and x is called fuzzy member if 0 < m(x) < 1. The set \{x\in A\mid m(x)>0\} is called the support of (A,m) and the set \{x\in A\mid m(x)=1\} is called its kernel.
Sometimes, more general variants of the notion of fuzzy set are used, with membership functions taking values in a (fixed or variable) algebra or structure L of a given kind; usually it is required that L be at least a poset or lattice. The usual membership functions with values in [0, 1] are then called [0, 1]-valued membership functions. This kind of generalizations was first considered in 1967 by Joseph Goguen, who was a student of Zadeh.
For each x\in A, m(x) is called the grade of membership of x in (A,m). For a finite set A = {x1,...,xn}, the fuzzy set (A,m) is often denoted by {m(x1) / x1,...,m(xn) / xn}.
Let x \in A. Then x is called not included in the fuzzy set (A,m) if m(x) = 0, x is called fully included if m(x) = 1, and x is called fuzzy member if 0 < m(x) < 1. The set \{x\in A\mid m(x)>0\} is called the support of (A,m) and the set \{x\in A\mid m(x)=1\} is called its kernel.
Sometimes, more general variants of the notion of fuzzy set are used, with membership functions taking values in a (fixed or variable) algebra or structure L of a given kind; usually it is required that L be at least a poset or lattice. The usual membership functions with values in [0, 1] are then called [0, 1]-valued membership functions. This kind of generalizations was first considered in 1967 by Joseph Goguen, who was a student of Zadeh.
Irrational number 'e'
Euler's number, "e", is one of the most important (if not the most important) fundamental constants in mathematics. It forms the base of the "natural" logarithm. It is an irrational number (meaning that it has no exact value, much like pi and can continue to be written forever), and while no one is exactly sure why, it is incredibly useful in describing a great many real-world phenomena, such as the continuous growth of compound interest and other exponential functions.
The mathematical constant e is the unique real number such that the value of the derivative (slope of the tangent line) of the function f(x) = e^x at the point x = 0 is equal to 1. The function e^x so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base e.
The number e is sometimes called Euler's number after the Swiss mathematician Leonhard Euler. (e is not to be confused with the Euler–Mascheroni constant, sometimes called simply Euler's constant.)The exponential function is one of the most important functions in mathematics. For a variable x, this function is written as exp(x) or e^x, where e is a mathematical constant, the base of the natural logarithm, which equals approximately 2.718281828, and is also known as Euler's number.
Also,The number e is irrational; it is not a ratio of integers. Furthermore, it is transcendental; it is not a root of any non-zero polynomial with rational coefficients. The numerical value of e truncated to 20 decimal places is e=2.718281828459045235360287471352662497757...
Some applications of the exponential function include modeling growth in populations, economic changes, fatigue of materials, and radioactive decay.
One of the original defining attributes of e, which is an interesting tidbit today, is the fact any bank account having a 100% APR interest rate which is compounded continuously, will grow at the exponential rate e^t, where t is time, discovered by Jacob Bernoulli. Intuitively, compounding an initial account of $1 would yield $e after one year.
The mathematical constant e is the unique real number such that the value of the derivative (slope of the tangent line) of the function f(x) = e^x at the point x = 0 is equal to 1. The function e^x so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base e.
The number e is sometimes called Euler's number after the Swiss mathematician Leonhard Euler. (e is not to be confused with the Euler–Mascheroni constant, sometimes called simply Euler's constant.)The exponential function is one of the most important functions in mathematics. For a variable x, this function is written as exp(x) or e^x, where e is a mathematical constant, the base of the natural logarithm, which equals approximately 2.718281828, and is also known as Euler's number.
Also,The number e is irrational; it is not a ratio of integers. Furthermore, it is transcendental; it is not a root of any non-zero polynomial with rational coefficients. The numerical value of e truncated to 20 decimal places is e=2.718281828459045235360287471352662497757...
Some applications of the exponential function include modeling growth in populations, economic changes, fatigue of materials, and radioactive decay.
One of the original defining attributes of e, which is an interesting tidbit today, is the fact any bank account having a 100% APR interest rate which is compounded continuously, will grow at the exponential rate e^t, where t is time, discovered by Jacob Bernoulli. Intuitively, compounding an initial account of $1 would yield $e after one year.
9/13/10
Golden Number
There are a lot of interesting recurrence sequences, but the most popular one is the Fibonacci sequence. Every number or term of this sequence is the sum of the two direct preceding terms:
Fn + Fn+1 = Fn+2
and lim.n to infinite: Fn / Fn-1 = 1.618... or Phi.
...-1, 1, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377...
which is a golden number.
THE GOLDEN RECTANGLE
One of the most interesting properties of the golden rectangle is that if you cut off a square section whose side is equal to the shortest side, the piece that remains is also a golden rectangle.
The golden rectangle R, constructed by the Greeks, has the property that when a square is removed a smaller rectangle of the same shape remains. Thus a smaller square can be removed, and so on, with a spiral pattern resulting. The sides are in the "golden proportion" (1 : 1.618034 which is the same as 0.618034 : 1) has been known since it occurs naturally in some of the proportions of the Five Platonic Solids .
Also,The golden rectangle was considered by the Greeks to be of the most pleasing proportions, and it was used in ancient architecture.
Fn + Fn+1 = Fn+2
and lim.n to infinite: Fn / Fn-1 = 1.618... or Phi.
...-1, 1, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377...
which is a golden number.
THE GOLDEN RECTANGLE
One of the most interesting properties of the golden rectangle is that if you cut off a square section whose side is equal to the shortest side, the piece that remains is also a golden rectangle.
The golden rectangle R, constructed by the Greeks, has the property that when a square is removed a smaller rectangle of the same shape remains. Thus a smaller square can be removed, and so on, with a spiral pattern resulting. The sides are in the "golden proportion" (1 : 1.618034 which is the same as 0.618034 : 1) has been known since it occurs naturally in some of the proportions of the Five Platonic Solids .
Also,The golden rectangle was considered by the Greeks to be of the most pleasing proportions, and it was used in ancient architecture.
9/5/10
MATHEMATICS IS DIVINE
MAHEMATICS IS DIVINE
Mathematics is tool to understand, experience, and feel the nature of god. This statement does not mean that there is a god. Also, I didn’t try to explain that there is no god. Actually the statement means that you can feel status of your holiness. You can get a power of god very close to you, with in you, and every where do you want. Mathematics has a bundle of philosophical ideas that is not real but that is true. There are many concepts, terms that are not easily understandable.
What is the meaning of understandable?
Consider the three situations, some one enlighten
· Something about something.
· Whole thing about nothing.
· Nothing about whole thing .
Which one is understandable?
The answer for the question “What you understand from the above sentence?” differs by every individual. Some one felt that what ever he understands by reading this sentence is the actual meaning of it. It is real but it is not true.
For each three situations the truth is
· You could not say I understand whole thing.
· You could not say I understand nothing
· You could say I understand something.
I felt there is nothing to explain to understand the whole thing about something.
Before understand the term understandable. The great invention of emptiness or nothing (that is zero) is made by Indians. I feel that it happens only because of the nature of spirituality. Some one understands that the concept of whole thing about nothing and nothing about whole thing is not understandable.
Zero
Infinite
We will discuss and understand something about the emptiness and wholeness (that is zero and infinite) in next article. If you understand that you will get a tool. If you experience the whole thing by using that tool, you will become a God.
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