AnkuR Tiwari

Hi, my friend, I am sure that you have a vast knowledge in mathematical field and you are the relative expert of the same. However you are somewhat misunderstanding B.N.R.F. due to these reasons. I personally request you to please read this carefully and completely. 1. You are taking B.N.R.F. in the aspects of differential and integral calculus, the domain of mathematics where non-uniform velocity (displacement/time) can also be treated as uniform velocity by using the concepts of differentiation and integration. Which in real life doesn't exists at all but in mathematical and scientific aspects it is believed to be happening. However B.N.R.F. deals with the real life aspects of division. The divisional aspects which takes place in our daily life or/and in our calculating device. Because the calculating device is nothing but the subset of real number line. For example: Two sets associated with Bhartiya New Rule for Fraction when we are using ten (10) digits calculating device are: A. Set of real (Actual) number line. Set A = { X ; X ∈ R } B. Set of calculating device number line. Set B = { X ; -L ≤ X ≤ +L and X ∈ 10 digits number } From the two sets above it is clear that set B is subset of set A. Therefore if we apply any function on that Set B or derive any formula for them than it will be automatically applicable in all those elements of real number line which are in that subset B. At this point once again I like to clear the fact that we are only studying the property of Set B (Set of calculating device number line) to generalize and obtain some concrete results on the aspects of division by Zero. 2. Now, Second thing, when we are using any calculating device which is of ten (10) digits than its largest positive number is (+L) +9999999999 and smallest negative number is (-L) -9999999999. Now if there is one fraction X divided by Y (X/Y) and Y ≠ ±L than for that fraction two fractions whose values will be nearest to it will be X divided by largest number preceding Y (X/Y') and X divided by smallest number succeeding Y ( X/Y''). In this situation X/( Y')> X/Y>X/( Y'') and all three will be approximately equidistant from each other. Because the value of X divided by Y' (X/Y') will be (very little bit) more than X divided by Y (X/Y) and the value of X divided by Y'' (X/Y'') will be equally less than X divided by Y (X/Y). So that when mean of both of them will be taken it will be equal to X divided by Y (X/Y). Once again have a look over my point of view that, we are only studying the property of Set B (Set of calculating device number line) to generalize and obtain some concrete results on the aspects of division by Zero. And the above result is only for the particular calculating device. So, in this case there will be no any equation in three variables as mentioned in your last post and it will simply be equal to X/Y for that particular subset of real number line. 3. While using Bhartiya New Rule for Fraction we are calculating only up to limited number of digits because of problem of non-termination of division. According to modern Mathematics If denominator of any fraction not having any common factor other than one (1) has its prime factors as only 2 and 5 than it have a terminating value which reveals that the value of it is a perfect number, whereas if the same has its prime factor as other numbers than that of only 2 and 5 than it have a non-terminating value. In such situations we use their approximate values and calculating device too shows their values with appropriate approximation. (Article from Mystery of Zero - Shoonya Ka Rahasya, Chapter 3, Page no.31) So, we have to use the appropriate approximation in case of B.N.R.F. also because to implement B.N.R.F. first requirement is to fix a limit and if a limit is fixed up to any digits than only the numbers up to that digits can be considered and we have to use the appropriate approximation up to that digits. Additionally, I like to clear one more point that of course the value of X/Y in case where Y≠ 0 obtained by B.N.R.F. is using limited digits approximation but in case of X/0 it will be a true value of it because if we observe the denominators of all the fractions that are used in B.N.R.F. in case of X/0 than they all will have its prime factors as only 2 and 5. Therefore it will have a terminating value which reveals that the value of it is a perfect number and it is a true value of it. Once again I like to clear the fact that we are only studying the property of Set B (Set of calculating device number line) to generalize and obtain some concrete results on the aspects of division by Zero. 4. Three Indian Pioneer Mathematicians who had work in the field of division by Zero are :- A. Brahmagupta (598 - 670) B. Mahavira (about 800 - about 870) C. Bhaskara II (1114 - 1185) All these three ancient Indian Mathematicians had work in the field of division by Zero by stating out the rules in their books. A. Brahmagupta (598 - 670) In 628 A.D. an Indian mathematician Brahmagupta had given out rules of using Zero in his book Brahmasphutasiddhanta. These also include rules related to division by Zero. According to Brahmagupta; • Zero divided by Zero is Zero. • A positive or negative number when divided by zero is a fraction with the zero as denominator. As stated in book "Mystery of Zero – Shoonya KA Rahasya" ISBN 978-81-8465-678-7 we may conclude from this that Brahmagupta may want to say that any number divided by Zero is Zero because when we put the value of Zero as Zero divided by Zero in Fraction of any positive or negative number divided by Zero than we will get answer as Zero B. Mahavira (about 800 - about 870) In 830 A.D. Mahavira the successor mathematician of Brahmagupta related to division by Zero in his book Ganita Sara Samgraha. According to Mahavira; • The number remains unchanged when divided by Zero. C. Bhaskara II (1114 - 1185) Later on an ancient Indian Mathematician Bhaskara had stated that the value of any positive or negative number divided by Zero tends to infinity. As mentioned in my last post you can look that B.N.R.F. resembles with all of them as according to B.N.R.F. the value of any positive or negative number X divided by any positive or negative number Y tends infinity (positive or negative) as the value of Y tends to Zero. However, when any number N is ultimately divided by Zero it gives Zero as the quotient and that number itself as the remainder. Once again I like to clear the fact that we are only studying the property of Set B (Set of calculating device number line) to generalize and obtain some concrete results on the aspects of division by Zero and these results is obtained by defining a continuous function on Set B. 5. B.N.R.F. is mainly derived for a calculating devices and IT sector to remove out the error of division by Zero it is mainly designed for computer algorithms to permanently remove division by Zero error from the computer era without effecting our other calculations and aspects. Regards, AnkuR Tiwari.

Derivation of Bhartiya New Rule for Fraction To understand derivation of Bhartiya New Rule for Fraction it is important to know some facts like, between any two numbers there are infinite numbers therefore, it is impossible to say which number (Y’) is the largest number which precedes any number (Y). Similarly, it is also impossible to say which number (Y’’) is the smallest number which succeeds any number (Y). But both these numbers are used in Bhartiya New Rule for Fraction because when we are talking about any particular calculating device than we are talking about limited number of numbers and we have a complete control over all numbers because of which answer of these types of questions were primarily not impossible for us. Second thing, when we are using any calculating device which is of ten (10) digits than its largest positive number is (+L) +9999999999 and smallest negative number is (-L) -9999999999. Now if there is one fraction X divided by Y (X/Y) and Y ≠ ±L than for that fraction two fractions whose values will be nearest to it will be X divided by largest number preceding Y (X/Y’) and X divided by smallest number succeeding Y ( X/Y’’). In this situation X/( Y')> X/Y>X/( Y'') and all three will be approximately equidistant from each other. Because the value of X divided by Y’ (X/Y’) will be (very little bit) more than X divided by Y (X/Y) and the value of X divided by Y’’ (X/Y’’) will be equally less than X divided by Y (X/Y). So that when mean of both of them will be taken it will be equal to X divided by Y (X/Y). Mathematical representation of above is given below – Since, X/( Y')> X/Y>X/( Y'') and all three are approximately equidistant from each other. Therefore, X/Y = X/( Y') - U [ 1 ] X/Y = X/( Y'') + U [ 2 ] ‘U’ is taken here as a constant. Now, 2 × X/Y = X/Y + X/Y Using equation [1] and [2] – 2 × X/Y = X/( Y') - U + X/( Y'') + U X/Y = (X/Y' + X/Y'')/( 2 ) X/Y = 1/2 × [ X/( Y') + X/( Y'') ] X/Y = X/2 × [ 1/( Y') + 1/( Y'') ] Now if the value of X divided by Y (X/Y) is A than, A = X/2 [ 1/( Y') + 1/( Y'') ] Example of Implementation of B.N.R.F. If without dividing 100 by 5 we have to find value of 100 divided by 5 i.e. 100/5, than we have to use Bhartiya New Rule for Fraction. If we are using any calculating device which is of ten (10) digits than according to Bhartiya New Rule for Fraction ±L= ± 9999999999. It is clear that Y ≠ ±L Therefore we can use Bhartiya New Rule for Fraction. According to Bhartiya New Rule for Fraction, A = X/2 [ 1/Y' + 1/Y'' ] Here, X= 100, Y’= 4.999999999 and Y’’= 5.000000001 Therefore, A = 100/2 [ 1/4.999999999 + 1/5.000000001 ] [since we are using ten digits calculating device therefore we can use only ten digits. Therefore using ten digits approximate value of both we get,] = 50 [ 0.2 + 0.2 ] = 50 [ 0.4 ] = 20 (Article from Mystery of Zero - Shoonya Ka Rahasya, Chapter 3, Page no.33) Alternate Method of solving the same: If without dividing 100 by 5 we have to find value of 100 divided by 5 i.e. 100/5, than we have to use Bhartiya New Rule for Fraction. If we are using any calculating device which is of ten (10) digits than according to Bhartiya New Rule for Fraction ±L= ± 9999999999. It is clear that Y ≠ ±L Therefore we can use Bhartiya New Rule for Fraction. According to Bhartiya New Rule for Fraction, A = X/2 [ 1/Y' + 1/Y'' ] Here, X= 100, Y’= 4.999999999 and Y’’= 5.000000001 Therefore, A = 100/2 [ 1/4.999999999 + 1/5.000000001 ] Since we can see that denominators Y’ and Y’’ are not having their prime factors as only two (2) and five (5) therefore using L.C.M. technique we get, A = 100/2 [ (5.000000001+ 4.999999999)/((4.999999999)× (5.000000001) ) ] = 100/2 [ 10/((5 - 0.000000001)× (5 + 0.000000001) ) ] Using identity of (a – b) (a + b) = a^2 - b^2 we get, A = 100/2 [ 10/(5^2 –〖 (0.000000001)〗^2 ) ] = 100/2 [ 10/(25 – 0.000000000000000001) ] = 100/2 [ 10/24.999999999999999999 ] Since we are using ten digits calculating device therefore we can use only ten digits. Therefore using ten digits approximate value we get, A = 100/2 [ 10/25 ] = 100/5 = 20 The same way can be applied on the case of X/0 also (where X ∊ R). In Number line Zero is in center and on its one side there are negative and on another side there are positive numbers. So the values of Y' and Y'' can be obtained. If we will proceed with Bhartiya New Rule for Fraction (B.N.R.F.) to calculate the value of any number divided by Zero than we will be glad to know that we will get the same answer which ancient Indian Mathematician Brahmagupta had mentioned in his book Brahmasphutasiddhanta on 628 A.D. But the difference between him and we is that we have a Mathematical proof of it but unfortunately Brahmagupta didn’t have it because of which modern mathematics had denied out his research. Brahmagupta has calculated the value of Zero divided by Zero as Zero, Bhartiya New Rule for Fraction will also calculate its value as Zero. Like this, In 0/0 both X and Y are 0 . If without dividing 0 by 0 we have to find value of 0 divided by 0 i.e. 0/0 , than we have to use Bhartiya New Rule for Fraction. If we are using any calculating device which is of ten (10) digits than according to Bhartiya New Rule for Fraction ±L= ± 9999999999. It is clear that Y ≠ ±L Therefore we can use Bhartiya New Rule for Fraction. According to Bhartiya New Rule for Fraction, A = X/2 [ 1/Y' + 1/( Y'') ] Here, X= 0, Y’= (-0.000000001) and Y’’ = (+0.000000001) Therefore, A = 0/2 [ 1/(-0.000000001) + 1/(+0.000000001) ] = 0 [0] = 0 (Article from Mystery of Zero - Shoonya Ka Rahasya, Chapter 3, Page no.35) Now we will calculate the value of 1 divided by 0 i.e. 1/0 . In 1/0 , X = 1 and Y = 0 . If we are using any calculating device which is of ten (10) digits than according to Bhartiya New Rule for Fraction ±L= ± 9999999999. Now without dividing 1 by 0 we have to find out the value of 1 divided by 0 i.e. 1/0 . It is clear that Y ≠ ±L Therefore we can use Bhartiya New Rule for Fraction. According to Bhartiya New Rule for Fraction, A = X/2 [ 1/Y' + 1/( Y'') ] Here, X = 1, Y’= (-0.000000001) and Y’’= (+0.000000001) Therefore, A = 1/2 [ 1/(-0.000000001) + 1/(+0.000000001) ] = 1/2 [0] = 0 (Article from Mystery of Zero - Shoonya Ka Rahasya, Chapter 3, Page no.37) Results of Bhartiya New Rule for Fraction Therefore, according to Bhartiya New Rule for Fraction when we divide any number by Zero we get quotient as a Zero and by using Euclid’s Division Lemma we can Find out the remainder in case of division By zero and when we work on it we get, According to Euclid’s Division Lemma; Dividend = Divisor x Quotient + Remainder ∴ Remainder = Dividend – Divisor x Quotient ( By Rearranging) According to Bhartiya New Rule for Fraction, in case of X divided by 0; Dividend = X, Divisor = 0, Quotient = 0 ∴ Remainder = X – 0 x 0 = X This means that the number remain unchanged when divided by Zero Conclusion We can see that both these results obtained by Bhartiya New Rule for Fraction completely resembles with the results stated by Brahmagupta on 628 A.D. and Mahavira on 830 A.D. and this only enables us to establishment of Universal Results for Division By Zero. 1. According to Brahmagupta Zero divided by Zero is Zero and furthermore any positive or negative number divided by Zero is a fraction with that number on numerator and Zero as denominator from which we can conclude that he had stated that any number divided by Zero is Zero. 2. According to Mahavira the number remains unchanged when divided by Zero i.e. when any number is divided by Zero than it gives us the same number as remainder. 3. According to Bhaskara II any number divided by Zero tends to infinity (∞). 4. According to Bhartiya New Rule for Fraction any number divided by Zero give us quotient as Zero and that number as remainder. On combining all these four above statements we get ; According to Bhartiya New Rule for Fraction; The value of fraction any positive or negative number say X divided by Y tends infinity as the value of Y tends to Zero. However when any number say X is ultimately divided by Zero it gives Zero as a quotient and that number as a remainder. From, AnkuR Tiwari, Inventor of ‘Bhartiya New Rule for Fraction’ Writer of ‘Mystery of Zero – Shoonya Ka Rahasya’ ISBN 978-81-8465-678-7 Website:- http://www.bnrf.co.cc

Jai Mata Kii... Let us see the statements of three great ancient Indian Mathematicians regarding division by Zero, which are as follow: 1. According to Brahmagupta Zero divided by Zero is Zero and furthermore any positive or negative number divided by Zero is a fraction with the Zero as denominator from which we can conclude that he may had stated that any number divided by Zero is Zero. (Brahmagupta, See page 21Mystery of Zero - Shoonya Ka Rahasya, ISBN 978-81-8465-678-7) 2. According to Mahavira the number remains unchanged when divided by Zero i.e. when we divided any number by Zero than we get that same number as remainder. (Mahavira, See page 39Mystery of Zero - Shoonya Ka Rahasya, ISBN 978-81-8465-678-7) 3. According to Bhaskara II the value of (any) number divided by Zero is (tends to) infinity (∞). (Bhaskara II, See page 19, Mystery of Zero - Shoonya Ka Rahasya, ISBN 978-81-8465-678-7) Additionally we had seen that the result obtained by Bhartiya New Rule for Fraction resembles with all these. According to Bhartiya New Rule for Fraction conclusion of above mentioned points are as follow: The value of any positive or negative number X divided by any positive or negative number Y tends infinity (positive or negative) as the value of Y tends to Zero. However, when any number N is ultimately divided by Zero it gives Zero as the quotient and that number itself as the remainder. Anyone can use this conclusion with the fraction of any number divided by Zero by using the statement of Euclid's Division Lemma and you all will be surprised to now that any mathematical problem related with X/0 will be solved within two-three steps which needs a huge struggle time for solving by other traditional methods. You can use this results with problems related to limits, derivatives and trigonometric functions. For More and complete details please see the detailed summary of BN.R.F. present in my website, use the link given below: http://bnrf.co.cc/index_files/Page467.htm

Hello Friend, I am very glad to see that you all are discussing on the topics of My research. Thank You Very much. Kindly have a proper look over the complete detailed presentation of Bhartiya New Rule for Fraction. Use the link give below: https://docs.google....=1#slide=id.p13 Regards, AnkuR Tiwari, Personal info removed