Division by zero in a unique way

[John]

Infinity is not the multiplicative inverse of zero. Also, I think you're slightly misunderstanding how the expressions at the end of your post are used.

 

You're obviously intelligent, and it's good that you're thinking about these concepts rather than, for instance, obsessing about some reality TV show. But you have to tread carefully when dealing with infinity, and you seem to be making a few missteps.

[Visionem Ex Illuminatio]

John. Thanks for the constructive statements.

 

"Infinity is not the multiplicative inverse of zero". Yes. You're correct; sorry for the misconception. It is, however, in the way I defined infinity (you could accept it or reject it, it's just my personal speculation outside the respect of established mathematics).

 

Your statement's true, because in the RNS, no number times zero equals one. [math]f(x)=x^{-1}[/math] shows that [math]f(0)=?[/math] undefined, as there is no limit as the function increases/decreases indefinitely to [math]\pm\infty[/math].

 

That's why I thought the inverse of zero is infinity (+- infinity actually), which is why I looked into this, and made the proof for 0 times ∞ equals the real numbers. And looking upon even weirder properties of infinity I haven't discussed here; I concluded that it all makes sense in its own right, intuitively and with the help of new definitions.

 

1335042168[/url]' post='673199']

You're obviously intelligent, and it's good that you're thinking about these concepts rather than, for instance, obsessing about some reality TV show. But you have to tread carefully when dealing with infinity, and you seem to be making a few missteps.

 

Lol. Thanks, I guess. And there's nothing wrong with reality TV shows

[AnkuR Tiwari]

I see no need to put any time into reviewing such an absurd idea. The Bhartiya New Rule for Fraction does not solve any problems, it creates problems. It would create continuity in discontinuous functions and break traditional methods of analysis used for engineering.

 

Consider for example Tan 90° = sin 90°/cos 90° = 1/0 which is both undefined and discontinuous. How would simply calling this value 0 with some flawed application of algebra be a benefit to analysis?

 

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

 

 

 

[ydoaPs]

I was under the impression that [math]\frac{1}{0}[/math] yielded complex infinity.

[Visionem Ex Illuminatio]

I was under the impression that [math]\frac{1}{0}[/math] yielded complex infinity.

 

That's what Wolfram Alpha will give you if you type in 1/0.

 

The thing is, infinity's not always considered an actual number, let alone an imaginary/complex one. Most of the time it really isn't a number, depending on which area of math you're using it and the number fields used therein. But merely a mathematical concept, meaning "limitlessness", e.g. in limits, as the graph approaches a certain point, it may be going to infinity. It can't actually go to infinity, since you can't calculate infinity in normal equations like [math]f(x)=\dfrac{x^2}{x+1}-x[/math]. Since infinity is the concept of "without bounds", the graph will go up to infinity, without bounds, not to an actual "infinite" number.

 

It's all okay as long as you don't take it to the realm of elementary algebra/arithmetic. Here, it will cause all sorts of anarchy. So numerical infinities are considered illegal there.

Edited May 3, 2012 by Visionem Ex Illuminatio

[studiot]

Hello VEI,

 

Actually infinities and their applications are already pretty well defined in mathematics.

 

Note I use the plural since there are many infinities.

 

You should look up 'cardinality'.

 

The fact that there are many infinities, some bigger than others is why we can often evalute expressions such as

 

 

 

[math]\frac{\infty }{\infty }or\frac{0}{0}[/math]

 

to yield a finite result

[imatfaal]

That's what Wolfram Alpha will give you if you type in 1/0.

/snipped

 

But if you look up Complex Infinity or Division By Zero on WolframAlpha's big-brother site you can find the following

 

There are, however, contexts in which division by zero can be considered as defined. For example, division by zero for in the extended complex plane C-* is defined to be a quantity known as complex infinity. This definition expresses the fact that, for , (i.e., complex infinity). However, even though the formal statement is permitted in C-*, note that this does not mean that . Zero does not have a multiplicative inverse under any circumstances.

 

http://mathworld.wolfram.com/DivisionbyZero.html

http://mathworld.wolfram.com/ComplexInfinity.html

[Daedalus]

I came across a formula derived by Ankur Tiwari, which he says enables division by zero.

 

The website claims

 

This formula enables us to divide in a unique way without using denominator. This formula is based on the principle that, If the value of X divided by Y (X/Y) is A than by using this formula we can find out A without dividing X by Y directly, that means without dividing X by Y we can find out its value. This is the reason why 'Bhartiya New Rule for Fraction' is capable of diving by Zero.

 

The interesting points in regard of this formula are :-

 

1.'Bhartiya New Rule for Fraction' is based on present phenomenon and rules of mathematics.

 

2. It is very simple and easy formula.

 

3.Greatest benefit of 'Bhartiya New Rule for Fraction' is that it is capable of dividing by Zero and giving its value as an integer.

 

4.'Bhartiya New Rule for Fraction' can be used to find out the value of four not defined trigonometric ratios tan90, cosec0, sec90, cot0. So that these values can be utilized in the field of astronomy and other fields related to mathematics.

 

5.If in place of simple division (X/Y), 'Bhartiya New Rule for Fraction' is used in any digital electronic device as its processing command for division in processor, it will results in permanently elimination of 'divide by Zero' error from that device.

 

How would it affect mathematics? Is there a fallacy in there?

Let us discuss...

 

As Cap'n Refsmmat pointed out:

 

The above formula approximates X / Y by taking the average of the division at points to the left and to the right of Y. It's a very informal limit.

This actually implies that the Bhartiya New Rule for Fraction is ill defined, and does not actually equal the result of the division. This mathematically flawed rule is based on the following limit:

 

[math]\lim_{n \to 0}\frac{x}{2} \left(\frac{1}{y-n}+\frac{1}{y+n}\right)=\lim_{n \to 0}\frac{x}{2} \left(\frac{y+n}{(y-n)(y+n)}+\frac{y-n}{(y+n)(y-n)}\right)=\lim_{n \to 0}\frac{x}{2} \left(\frac{2y}{y^2-n^2}\right)=\frac{2 \, x \, y}{2 \, y^2}=\frac{x}{y}[/math]

 

However, B.N.R.F. is based on the precision of a floating point processor (FPU). Therefore, [math]n[/math] does not approach 0, but some other value [math]\epsilon[/math]. This affects the result of the limit:

 

[math]\lim_{n \to \epsilon}\frac{x \, y}{y^2-n^2}=\frac{x \, y}{y^2-\epsilon^2}=\frac{x}{y}\left(\frac{1}{1-\frac{\epsilon^2}{y^2}}\right) \ne \frac{x}{y}[/math]

 

The reason why this method produces incorrect results when dealing with division by zero is as follows:

 

[math]\lim_{y \to 0}\frac{x \, y}{y^2-\epsilon^2}=\frac{0}{0-\epsilon^2}=0, \ \left \{\epsilon \in \mathbb{R} \, | \, \epsilon \ne 0 \right \}[/math]

 

Where the actual limits for [math]\frac{x}{y}[/math] as [math]y[/math] approaches zero are:

 

[math]\lim_{y \to 0^-}\frac{x}{y}=-\infty[/math]

 

and

 

[math]\lim_{y \to 0^+}\frac{x}{y}=\infty[/math]

Edited May 13, 2012 by Daedalus

[AnkuR Tiwari]

 

As Cap'n Refsmmat pointed out:

 

 

This actually implies that the Bhartiya New Rule for Fraction is ill defined, and does not actually equal the result of the division. This mathematically flawed rule is based on the following limit:

 

[math]\lim_{n \to 0}\frac{x}{2} \left(\frac{1}{y-n}+\frac{1}{y+n}\right)=\lim_{n \to 0}\frac{x}{2} \left(\frac{y+n}{(y-n)(y+n)}+\frac{y-n}{(y+n)(y-n)}\right)=\lim_{n \to 0}\frac{x}{2} \left(\frac{2y}{y^2-n^2}\right)=\frac{2 \, x \, y}{2 \, y^2}=\frac{x}{y}[/math]

 

However, B.N.R.F. is based on the precision of a floating point processor (FPU). Therefore, [math]n[/math] does not approach 0, but some other value [math]\epsilon[/math]. This affects the result of the limit:

 

[math]\lim_{n \to \epsilon}\frac{x \, y}{y^2-n^2}=\frac{x \, y}{y^2-\epsilon^2}=\frac{x}{y}\left(\frac{1}{1-\frac{\epsilon^2}{y^2}}\right) \ne \frac{x}{y}[/math]

 

The reason why this method produces incorrect results when dealing with division by zero is as follows:

 

[math]\lim_{y \to 0}\frac{x \, y}{y^2-\epsilon^2}=\frac{0}{0-\epsilon^2}=0, \ \left \{\epsilon \in \mathbb{R} \, | \, \epsilon \ne 0 \right \}[/math]

 

Where the actual limits for [math]\frac{x}{y}[/math] as [math]y[/math] approaches zero are:

 

[math]\lim_{y \to 0^-}\frac{x}{y}=-\infty[/math]

 

and

 

[math]\lim_{y \to 0^+}\frac{x}{y}=\infty[/math]

 

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

[Daedalus]

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). ...

 

Just because you are taking a limit does not mean that you are dividing by zero. You are applying Calculus to analyze the following equation:

 

[math]\frac{x \, y}{y^2 - z^2}[/math]

 

which is not equal to:

 

[math]\frac{x}{y}[/math]

 

I would recommend that you learn some Calculus before making such assertions. You are incorrectly comparing two different indeterminate forms.

[doG]

...Do we really want to teach kids that these equal 0 because that's exactly what they'll think if their calculator says so. What next? Do we start teaching that the limits of these equations approach 0 as the denominator approaches 0? What effect would this have on engineering?

You still haven't answered these questions Ankur! Division by zero is undefined by definition for good reason. Why should we throw good reason out with the bath water for an unneeded, irrational solution?

Edited May 14, 2012 by doG

[Daedalus]

You still haven't answered these questions Ankur! Division by zero is undefined by definition for good reason. Why should we throw good reason out with the bath water for an unneeded, irrational solution?

doG, I wouldn't pay much attention to what he is selling. I have already shown that he is confused about dealing with indeterminate forms. He is comparing a function of three variables with a function of two, and trying to say that both are equal to each other.

[AnkuR Tiwari]

Just because you are taking a limit does not mean that you are dividing by zero. You are applying Calculus to analyze the following equation:

 

[math]\frac{x \, y}{y^2 - z^2}[/math]

 

which is not equal to:

 

[math]\frac{x}{y}[/math]

 

I would recommend that you learn some Calculus before making such assertions. You are incorrectly comparing two different indeterminate forms.

 

 

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.

Edited May 16, 2012 by AnkuR Tiwari

[Daedalus]

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

The above assumption is incorrect. The largest / smallest possible number for any electronic calculating device is not [math]\pm 9999999999[/math]. It is actually based on the the precision of the FPU as described here:

 

Double-precision floating-point format

 

Furthermore, you have neglected to consider calculators that are based on arbitrary precision arithmetic.

 

In computer science, arbitrary-precision arithmetic, also called bignum arithmetic, multiple precision arithmetic, or sometimes infinite-precision arithmetic, indicates that calculations are performed on numbers which digits of precision are limited only by the available memory of the host system.

You have erroneously related the base of the number system to the number of digits for your largest / smallest number. My TI-89 can handle base 10 numbers above and below your specified range.

 

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.

Here you are trying to apply the squeeze theorem. However, there is a huge problem with your method. You do not apply the limit correctly. Let me demonstrate the issue. You are claiming that if:

 

[math]z = \frac{x}{y}[/math] then [math]z = \frac{x}{2}\left(\frac{1}{y'}+\frac{1}{y''}\right)[/math] where

 

[math]y' = \lim_{n \to 0^-} (y-n)[/math] and [math]y'' = \lim_{n \to 0^+} (y+n)[/math] such that

 

[math]\frac{x}{y'} > \frac{x}{y} > \frac{x}{y''}[/math].

 

However, as stated in your method, you are not taking the limit as [math]n[/math] approaches zero. Instead, you are claiming that calculating devices cannot cover the entire set of reals, which makes it possible to choose some value [math]\epsilon > 0[/math] such that:

 

[math]z = \frac{x}{2}\left(\frac{1}{y'}+\frac{1}{y''}\right)=\frac{x}{y}[/math]

 

The above statement is wrong because we are dealing with the set of real numbers!!! What you are actually doing is as follows:

 

[math]z = \lim_{n \to \epsilon} \frac{x}{2}\left(\frac{1}{y-n}+\frac{1}{y+n}\right)=\frac{x\, y}{y^2-\epsilon^2} \ne \frac{x}{y}[/math]

 

Example 1 - you claim that:

 

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 [to] approximate [the] value of both we get]

 

= 50 [ 0.2 + 0.2 ]

= 50 [ 0.4 ]

= 20

 

...

 

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 problem with the above example is that:

 

[math]\left(\frac{1}{5-0.000000001}+\frac{1}{5+0.000000001}\right) \ne \frac{2}{5}[/math]

 

Your method only approximates the result of the division:

 

[math]\left(\frac{1}{5-0.000000001}+\frac{1}{5+0.000000001}\right) \approx \frac{2}{5}[/math]

 

You cannot claim that this allows you to divide by zero over the set of reals. It is a logical fallacy:

 

[math]\left(\frac{1}{0-0.000000001}+\frac{1}{0+0.000000001}\right) \ne \frac{2}{0}[/math]

 

Now I will address your latest post:

 

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.

Your method is based on limits, which forces us to evaluate it with the tools of calculus. The above statement is pure speculation because you are trying to relate your method to a physical application of dividing by zero. We are not talking about physical processes (although it would not matter if we were because the mathematics of indeterminate forms - [math]1/0, 0/0[/math], etc... - can have limits that are zero, infinity, etc...). We are discussing the result of a division by zero as defined by your method:

 

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.

 

This is where you reveal that you are not truly dividing by zero. Defining a subset of real numbers does not allow for division by zero:

 

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'').

 

...

 

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.

The problem with this statement is that it does not matter which set you choose, A or B. We are not studying the properties of set B. We are analyzing the operations that you are applying to the set. Once you realize this, you will see that your method is not dividing by zero:

 

[math]\frac{x \times 0}{0-\epsilon^2} \ne \frac{x}{0}[/math]

 

It does not matter if we write your method as an equation of two variables or three. The results are the same:

 

[math]\frac{x \, y}{(y-0.000000001)(y+0.000000001)} \ne \frac{x}{y}[/math]

 

or

 

[math]\frac{x \, y}{(y-\epsilon)(y+\epsilon)} = \frac{x \, y}{y^2-\epsilon^2} \ne \frac{x}{y}[/math]

 

The only reason why your method produces a zero when [math]y = 0[/math] is because:

 

[math]\frac{x \times 0}{0 - \epsilon^2} = \frac{0}{-\epsilon^2} = 0[/math]

 

 

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)

The key point that I will make here is that you are calculating up to a limited number of digits. This means that your method has two discontinuities where [math]y^2 = \epsilon^2[/math]. However, because you state that you are only considering set B, your method cannot approximate values of [math]|y| > L[/math] because you incorrectly suggest that a base 10 calculating device cannot handle numbers larger or smaller than [math]\pm L[/math].

 

There is one more thing I would like to add in response to your claims:

 

The interesting points (or claims being made) in regard of this formula are:

 

1. 'Bhartiya New Rule for Fraction' is based on present phenomenon and rules of mathematics.

 

2. It is very simple and easy formula.

 

3. Greatest benefit of 'Bhartiya New Rule for Fraction' is that it is capable of dividing by Zero and giving its value as an integer.

 

4. 'Bhartiya New Rull for Fraction' can be used to find out the value of four not defined trigonometric ratios tan 90, cosec 0, sec 90, cot 0. So that these values can be utilized in the field of astronomy and other fields related to mathematics.

 

5. If in place of simple division (x/y), 'Bhartiya New Rule for Fraction' is used in any digital electronic device as its processing command for division in processor, it will results in permanently elimination of 'divide by Zero' error from that device.

1.) Your method is not based on the rules of mathematics for division. It is based on the function [math]\text{B}(x,y)=\frac{x\, y}{y^2-\epsilon^2}[/math] that you have defined where [math]x[/math] and [math]y[/math] are variables, and [math]\epsilon[/math] is a constant such that [math]\epsilon > 0[/math].

 

2.) Stating that your method is simple and easy to use is an opinion and not a fact.

 

3.) Your method when [math]y=0[/math] is equivalent to [math]\frac{x \times 0}{0-\epsilon^2} = -\frac{0}{\epsilon^2} \ne \frac{x}{0}[/math] and does not divide by zero. However, [math]f(x, 0)=0, \ \ \{x \in \mathbb{R} \ | \ -\infty < x < \infty\}[/math].

 

4.) Your method produces incorrect values for the ratios tan 90, csc 0, sec 90, and cot 0.

 

Proof for cot 0:

 

 

Trigonometric functions:

 

[math]\text{tan} \ \angle A = \frac{a}{b}, \ \ \ \ \text{cot} \ \angle A = \frac{b}{a}[/math]

 

Bhartiya New Rule for Fraction:

 

[math]\text{B}(x, y) = \frac{x \, y}{y^2 - \epsilon^2}[/math] where [math]\epsilon[/math] is a constant such that [math]\epsilon > 0[/math]

 

The slope of a line is equal to the change in y divided by the change in x, which corresponds to the definition:

 

[math]m=\frac{\Delta y}{\Delta x}=\text{tan} \ \angle A[/math] where [math]\angle A[/math] is the angle of the line with respect to the [math]x[/math] axis.

 

It is known that two lines, [math]L_1[/math] and [math]L_2[/math], are perpendicular if their slopes [math]m_1 \times m_2 = -1[/math]. If [math]L_1[/math] has a slope of [math]m_1=\text{tan} \ \angle A[/math], then a line [math]L_2[/math] perpendicular to [math]L_1[/math] will have a slope [math]m_2=-\text{cot} \ \angle A[/math]:

 

[math]m_1 \times m_2=(\text{tan} \ \angle A) \times (-\text{cot} \ \angle A) = -1[/math]

Now consider a horizontal line that has a slope of zero [math]m_1=0[/math]. The line perpendicular to it has an undefined slope [math]m_2=\infty[/math]. However, if we replace division with [math]\text{B}(x,y)[/math], the affected tan and cot functions produce:

 

[math]m_1 = \text{tan} \ 0 = 0[/math]

[math]m_2 = -\text{cot} \ 0 = 0[/math]

 

This results in two horizontal lines that are parallel to each other:

 

[math]y_1=0 \times x + b_1[/math]

[math]y_2 = 0 \times x +b_2[/math]

 

You have you successfully broken trigonometry and linear algebra.

As for claim 5, you are more than welcome to modify your calculating devices to use your method instead of actually doing division. Just leave my devices alone : ) In closing, you should consider the following:

 

[math]\lim_{y \to 0} \frac{0}{y} = 0[/math]

 

which is similar to:

 

[math]\lim_{y \to 0} \frac{x \, y}{y^2 - \epsilon^2} = 0[/math]

 

However, these limits are not equal:

 

[math]\lim_{y \to 0^-} \frac{1}{y} = -\infty \ \ \ \ \lim_{y \to 0^+} \frac{1}{y} = \infty[/math]

This is why we call these things indeterminate forms.

Edited May 16, 2012 by Daedalus

[doG]

...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...

 

AnkuR Tiwari.

You are still missing and/or avoiding the point that this alleged proposal is unneeded and irrational. We have the technology to make calculators give whatever answer we want but we don't because we want them to be mathematically correct and your proposal purposely dismisses an error they are designed to display on purpose. Even in our daily life we do not need to ignore the FACT that division by zero is undefined. Why would anyone want to teach anything but the truth?

[juanrga]

I was messing with 1/0 .. what do you think ?

 

[math] \frac{1}{0} = ? [/math] if and only if [math] 0 \cdot ? = 1 [/math]

 

Now any real number ? multiplied by zero gives zero and no number solves the equation [math] 0 \cdot ? = 1 [/math]. Conclusion: the division of one by zero is not defined.

Edited May 19, 2012 by juanrga

[bnrf]

Hi,

All of you, the reason of posting here in this thread after a period of more than two years is that, now AnkuR Tiwari has reinitiated his battle against modern mathematicians for the rights of our great ancient Indian mathematicians Brahmagupta, Mahavira and Bhaskar II.
Many of you here, stated him, to go and get some knowledge of higher calculus and now he has done so, after making all the things well known, he has presented all his works in yet a level of mathematics that is considered as of PhD or M.Sc. level.
You all are invited to view the all the associated articles here in this website:
link removed
At least go through all the four articles over 'Division by Zero' in this website and share your views on the Portal of BNRF, the link is link removed
Since, AnkuR Tiwari will not be able to reply through this forum; do necessarily post your views/queries on the BNRF Portal, if after going through this website also, you consider modern mathematics as right and our ancient Indian mathematicians as wrong in the aspect of division by Zero.

[rktpro]

When you say a battle is initiated against modern mathematics for the rights of Indian giants, you are dis-respecting them. I am very much sure that Bhaskar and others would not mind it when they were wrong. Now, there is no Indian or Russion mathematics, it is universal. Also, if you can, please post the updated result/proof here. Your idea of claiming royality(as is pointed out in the site) will not be hampered.

[swansont]

Hi,

All of you, the reason of posting here in this thread after a period of more than two years is that, now AnkuR Tiwari has reinitiated his battle against modern mathematicians for the rights of our great ancient Indian mathematicians Brahmagupta, Mahavira and Bhaskar II.

Many of you here, stated him, to go and get some knowledge of higher calculus and now he has done so, after making all the things well known, he has presented all his works in yet a level of mathematics that is considered as of PhD or M.Sc. level.

You all are invited to view the all the associated articles here in this website:

link removed

At least go through all the four articles over 'Division by Zero' in this website and share your views on the Portal of BNRF, the link is link removed

Since, AnkuR Tiwari will not be able to reply through this forum; do necessarily post your views/queries on the BNRF Portal, if after going through this website also, you consider modern mathematics as right and our ancient Indian mathematicians as wrong in the aspect of division by Zero.

 

 

!

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