Limit

What is the limit? Question can strike in mind for those who study mathematics in senior secondary level. Answer to the question is quite simple but not clear. When a function f(x) is undefined at a point say x = a, then we try to find its value very very close to that point. If this value exist, we say it limiting value of f(x) at a.
                                               

Geometric Progression

A sequence obtained by successive multiplication of a fixed term to the first term is called in geometric progression. In other words " If ratio of any two consecutive terms of a sequence is constant called in G.P."
Example:  1, 2, 4, 8, .......
                  1, 3, 9, 27, 81, .........

Symmetry

                                           

Symmetry in nature always attracted people.  It develops beauty in nature. There are plenty of things available in nature which has linear symmetry. Symmetry in English alphabets, Symmetry in birds, symmetry in flowers and symmetry in human being. 

           Due to symmetry, flowers are so beautiful


Beauty in human being is Due to symmetry 

           











                   
        Mathematical figures also look nice Due to symmetry

Symmetry in Alphabets

सुखार्थिनः कुतो विद्या …… 

सच है सुख की कामना रखने वालों विद्या की प्राप्ति कैसे हो सकती है ? सुख पूर्वक समय बिताने वाला सफलता के ऊँचे पायदान पर अपना कदम कैसे रख सकता है ? माना ज़माना बदल गया परन्तु कठिन परिश्रम और लगन से अध्ययन  करने वाला हीं सफल हो कर समाज में अपनी पहचान बना सकता है ।  Facebook , internet , mobile , T.V. , ए. सी. , कूलर,  फैन्सी ड्रेस  यहां तक  कि चटपटे भोजन की कामना रखने वालों को असफलता अविलम्ब अपने साथ लगा  लेती है ।  

पाना चाहो जीवन में कुछ , ना करना विश्राम । 

जैसी करनी करोगे प्यारे , वैसे पाओ परिणाम ॥

Check for accuracy of a product

Subject:  How to check for the accuracy of product of two numbers.
                   Example           123476
                                              x 654879
                                            7645983254
      Now to check for the accuracy no need to go to the actual product. Follow the following simple steps.
1.    add all the digits of the first number i.e. 1+2+3+4+7+6= 23
2.  divide it by 9 and record the remainder i.e. 23 /9 , remainder = 5
3. add all the digits of second number 6+5+4+8+7+9 = 39
4.  divide it by 9 and record the remainder i.e. 39 /9 , remainder = 3
5.  multiply the two remainders i.e. 5x3 = 15
6.  divide the number obtained in step 5 by 9 and record the remainder. i.e. 15/9, remainder = 6
7.  add all the digits of the product of two numbers i.e. 7645983254 you get
      7+6+4+5+9+8+3+2+5+4= 53
8.  divide it by 9 and record the remainder i.e. 53 /9 , remainder = 8
Conclusion: If remainders obtained in step 6 and 8 are equal then your product is correct otherwise wrong.
Clearly above product is wrong. As remainder in step 6 is and that in step 8 is 8 , unequal. 

Vedic Mathematics

How to multiply two numbers which are near 100.

Example:  1.   95x96

                       95       -5

                   X  96      -4               either 95 – 4 = 91

                     9120                       Or       96 – 5 = 91

                                                    And   -5x-4 = 20

Example:  2.   102x105

                       102       +2

                   X  105      +5            either 102 + 5 = 107

                     10510                     Or       105 + 2 = 107

                                                    And   5x2 = 10

Now multiply 96 and 99

          Multiply 89 and 97

                  Vedic Mathematics

Subject:         How to square a number ending with 5

Formula:       Ekadiken purven

Meaning:      One more than the previous one.

Method:       To make square of  a number which has 5 at its unit place, take the number leaving out 5 , add one in it. Now multiply them and write 25 after the product so obtained. You will get the square of that number.

Example:           (25)² = (2x3)25 = 625

                        (35)² = (3x4)25 = 1225

                        (45)² = (4x5)25 = 2025  e.t.c

                        Now make practice for other numbers yourself i.e.

                      Find square of 55

                      Find square of 75

                      Find square of 85

PARABOLA

What is parabola? Where this appear? What is the use of  it? Why we study conic sections and the parabola? These are the general question arise in students mind. And the answer is just question and answer of various label i.e. easy to very difficult. After doing a lot of problem students get nothing. As a result students lose their interest and forget in time span.
Study of parabola , ellipse and hyperbola must be based on reality. Teacher must visualize these conics in their surrounding. For example, there is a hanging bridge on the river Ganga in Rishikesh. The bridge has no pillars. A long iron rope is tied with two poles situated on two banks of the river and the whole bridge is hanging with some vertical supporting ropes. The long iron rope which is tied with two pillars and the vertical ropes takes the shape of a parabola.  

What is Russell's paradox?

Aug 17, 1998

John T. Baldwin and Olivier Lessmann of the Department of Mathematics, Statistics and Computer Science at the University of Illinois at Chicago offer the following explanation.

Russell's paradox is based on examples like this: Consider a group of barbers who shave only those men who do not shave themselves. Suppose there is a barber in this collection who does not shave himself; then by the definition of the collection, he must shave himself. But no barber in the collection can shave himself. (If so, he would be a man who does shave men who shave themselves.)

BERTRAND RUSSELL confounded mathematicians when he published his famous paradox in 1903.

Bertrand Russell's discovery of this paradox in 1901 dealt a blow to one of his fellow mathematicians. In the late 1800s, Gottlob Frege tried to develop a foundation for all of mathematics using symbolic logic. He established a correspondence between formal expressions (such as x=2) and mathematical properties (such as even numbers). In Frege's development, one could freely use any property to define further properties.

Russell's paradox, which he published in Principles of Mathematics in 1903, demonstrated a fundamental limitation of such a system. In modern terms, this sort of system is best described in terms of sets, using so-called set-builder notation. For example, we can describe the collection of numbers 4, 5 and 6 by saying that x is the collection of integers, represented by n, that are greater than 3 and less than 7. We write this description of the set formally as x = { n: n is an integer and 3 < n < 7} . The objects in the set don't have to be numbers. We might let y ={x: x is a male resident of the United States }.

Seemingly, any description of x could fill the space after the colon. But Russell (and independently, Ernst Zermelo) noticed that x = {a: a is not in a} leads to a contradiction in the same way as the description of the collection of barbers. Is x itself in the set x? Either answer leads to a contradiction.

When Russell discovered this paradox, Frege immediately saw that it had a devastating effect on his system. Even so, he was unable to resolve it, and there have been many attempts in the last century to avoid it.

Russell's own answer to the puzzle came in the form of a "theory of types." The problem in the paradox, he reasoned, is that we are confusing a description of sets of numbers with a description of sets of sets of numbers. So Russell introduced a hierarchy of objects: numbers, sets of numbers, sets of sets of numbers, etc. This system served as vehicle for the first formalizations of the foundations of mathematics; it is still used in some philosophical investigations and in branches of computer science.

Zermelo's solution to Russell's paradox was to replace the axiom "for every formulaA(x) there is a set y = {x: A(x)}" by the axiom "for every formula A(x) and every set bthere is a set y = {x: x is in b and A(x)}."

What became of the effort to develop a logical foundation for all of mathematics? Mathematicians now recognize that the field can be formalized using so-called Zermelo-Fraenkel set theory. The formal language contains symbols such as e to express "is a member of," = for equality and ¿ to denote the set with no elements. So one can write formulas such as B(x): if y e x then y is empty. In set-builder notation we could write this as y = {x : x = ¿} or more simply as y = {¿}. Russell's paradox becomes: let y = {x: x is not in x}, is y in y?

Russell's and Frege's correspondence on Russell's discovery of the paradox can be found in From Frege to Godel, a Source Book in Mathematical Logic, 1879-1931,edited by Jean van Heijenoort, Harvard University Press, 1967.