Zero divided by Zero

Zero divided by Zero

This might look very absurd but is interesting as a concept.  I believe that was sometime while I was in my 4th or 5th class, I asked my father about One divided by Zero (1/0).  He said that it cannot be determined.  He then asked a counter question of what was Zero divided by Zero (0/0).  I blinked as I didn't know nor could I deduce. He then said that it could be Zero, relating to the philosophy of the Upanishads that the Whole remains a Whole how many ever wholes you take out or don't. 

The Isaa-vaasya Upanishad says:

ऊँ पूर्णमदः पूर्णमिदम् पूर्णात् पूर्णमुदच्यते ।

पूर्णस्य पूर्णमादाय पूर्णमेवावशिष्यते ।।

OM poornamadah poornamidam

poornaat poornamudachyate

poornasya poornamaadaaya

poornamevaavashishyate

That (God) is whole; this (world) is whole;

The whole manifested from the Whole.

From the Whole, even if whole is taken away,

what remains again is the Whole.

Thus Zero being considered a whole number in the Number System, just as its philosophical interpretation the above verse is, in terms of God/That as Infinity, the same applies to Zero too.  However, the young boy couldn't digest much of what he said then and it was left there.

When later on, I pursued my graduation in the Mathematics (1985-88), with the study of Real Numbers, Complex Numbers etc., the same problem returned for a better analysis again. 

During this time, I had few discussions with my classmates and with one of my Mathematics professors at St. Joseph's College, Calicut on my understanding of the concept of Zero divided by Zero. While one of my classmates thought there was merit, the professor rubbished though!. 

About that time, I had written an article in a magazine called 'Aarsha-naadam' combining the concept of Vedic Mathematics in arriving at the result of dividing Zero by Zero. (Though I had preserved the copy of the book for a long time, am unable to trace it now after many relocations in the many years that followed. My father used to subscribe to  copies of that bi-monthly magazine those days).

However, I will briefly explain the concept that I had tried to impress on the readers then.  

In the methodology of Vedic Mathematics, (Nikhilam sutra), in case of a division, instead of subtraction, addition is performed by 'Nikhilam' of the divisor, so as to equalize the multiplier to the quotient part of resultant (i.e. excluding the remainder).  (Nikhilam is defined as "nikhilam navatas-charamam dasatah" ie. difference all from 9, last from 10. This in effect becomes the tens complement).  The resultant will have two parts namely the quotient part and the remainder part;  the remainder part will have as many digits as there are in the divisor.  Initially, the dividend itself is considered as the resultant and the multiplier is assumed to be equal (or about) the quotient part of the resultant.  

To be more detail, let us say we want to divide 9 by 7.  For this, the dividend is 9, the divisor is 7.  Considering, the dividend itself as the initial resultant, reserving as many digits as there are in the divisor for the remainder, which is 1 (as the divisor 7 is single digit), we get the initial multiplier as 0 and initial remainder as 9 (assuming that the dividend is 09).  Taking Nikhilam of 7, we get 3 which is (10 - 7). Multiply this 3 by a suitable multiplier in this case 1 giving 1 x 3 = 3 and Add this product 3 to the dividend 9 to get 12.  Since the base of the nikhilam was 10, and the power of the base being 1, we partition the resultant into two parts; the last 1 digit forms the remainder part and the remaining forms the quotient part.  Partitioning the resultant 12 into two by removing the last 1 digit we have 1 in the quotient part and 2 in the remainder part.  Well that is exactly the answer provided that the remainder is not bigger than the value of the quotient in which case, the next higher multiplier is to be considered. This condition hasn't arisen here, since the quotient part of the resultant is 1 which happens to be equal to the multiplier, we can conclude that our quotient is 1 and remainder is 2. 

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Eg: Divide 9 by 7:

		Resultant
		(quotient-part)	(remainder-part)
	divisor	dividend		multiplier
	7	0	9	0
Nikhilam  of divisor =	3		3	1
		12
(Note:- multiplier to be maintained equal to quotient always)		1	2
		quotient	remainder

In the above example, if we were to consider 0 as the quotient, then the remainder would have been 9 which is not allowed since the remainder cannot be greater than the divisor.  Hence the next number to be taken ie. 1.  Multiplying the multiplier 1 by the Nikhilam of 7 which is 3, we get 1 x 3 = 3 which is put below the dividend.  The sum gives 12 with 1 in the quotient part and 2 in the remainder part.  The number of digits in the remainder part is understood from the number of digits in the divisor (because the remainder has to be less than the divisor).  Thus the end result is that 9 divided by 7 gives 1 as quotient and 2 as remainder.

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Eg: Divide 4536132 by 8996:-

		Resultant
		(quotient-part)	(remainder-part)
	divisor	dividend		multiplier
	8996	0453	6132	0453
Nikhilam  of divisor =	1004	000	0	0
		40	16	4
		5	020	5
			3012	3
multiplier not tallying:-		0499	0944	0045 / 0498
		000	0	0
		00	00	0
		4	016	4
			5020	5
multiplier not tallying:-		0503	6124	0004 / 0502
		000	0	0
		00	00	0
		0	000	0
			4016	4
multiplier not tallying:-		0504	0140	0002 / 0504
		000	0	0
		00	00	0
		0	000	0
0504 - 0502 = 2			2008	2
(Note:- multiplier to be maintained equal to quotient always)		0504	2148
		quotient	remainder

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A non-detailed way of writing the above would be as follows:-

		Resultant
		(quotient-part)	(remainder-part)
	divisor	dividend		multiplier
	8996	0453	6132	0453
Nikhilam of divisor =	1004	40	16	4
		5	020	5
			3012	3
		4	016	4
			5020	5
			4016	4
multiplier not tallying:-		0504	0140	0002 / 0504
0504 - 0502 = 2			2008	2
(Note:- multiplier to be maintained equal to quotient always)		0504	2148
		quotient	remainder

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Thus 4536132 divided by 8996 gives 504 as quotient and remainder 2148. 

Having understood the above division technique let us summarize the following rules:-
1. The number of digits of the remainder should be equal to the number of digits of the Nikhilam of the divisor. 
2. If the remainder is 0, then the divisor is fully divisible.  Otherwise, the remainder should be less than the divisor so that it ends up as a proper fraction.

3. The multiplier of the Nikhilam of the divisor should be equal to the quotient. 
   

Let us now try to divide 0 by 0 using this Vedic Mathematics method. 
The Nikhilam of 0 is expressed as a single digit below as 'T' indicating the value Ten.  Having taken T to represent 10, the arithmetic of T could be defined as:- 

1  x  T   =  T

 2  x  T  =  1T

  3  x  T  =  2T

  4  x  T  =  3T

.... 

....

 T  x  T  =  9T

		Resultant
		(quotient-part)	(remainder-part)
	divisor	dividend		multiplier
	0	0	0	0
Nikhilam  of divisor =	T		0
multiplier tallying:-		0	0	0 / 0
(Note:- multiplier to be maintained equal to quotient always)		0	0
		quotient	remainder

While the rules have been applied properly and the right answer has been obtained as quotient = 0 and remainder = 0,  for academic interest let us see, what would happen if the multiplier was taken as 1 instead of 0?

		Resultant
		(quotient-part)	(remainder-part)
	divisor	dividend		multiplier
	0	0	0	1
Nikhilam  of divisor =	T		T
multiplier not tallying:-		0	T	1 / 0
(Note:- multiplier to be maintained equal to quotient always)		0	T
		quotient (Erroneous)	remainder  (Erroneous)

Here we can see that it is not possible to have the multiplier equal to the quotient part as the quotient part is 0 while the multiplier part is 1.    Hence, it has to be concluded that the quotient for 0 divided by 0 is 0 and not any other number. 

Caution:-  However, there is a word of caution in this answer.  This is simple Arithmetic and not Mathematics.  In Arithmetic, I would hold that 0 divided by 0 = 0.  However in Mathematics, I would like to believe that it could be something else too.  

The reason for this is because, Arithmetic deals with absolute numbers.  Whereas Mathematics deals will approximate numbers as well.  Just as in physics, mathematics has for eg. in case of differentiation, say, 

                                 Limit    [Sin(A) /  A]   =   1

A --> 0

If A were to be substituted by 0, then Sin(A) would have been 0 and A itself is 0 which could have given [ Sin(A) / A ] =  [0 / 0]   = 0.   Whereas since we deal here with approximate 0 and not absolute 0,  we get the result 1. 

Therefore, what I would like to mention is that even though arithmetically,  0 / 0 = 0 in its absolute terms, mathematically, it cannot be considered so due to the fact that mathematics and physics consider numbers as approximates.  Another example to the same is the number  2.99999999999......  which in mathematics is equal to 3.  The number 3.00000000........000001  is also equal to 3 in mathematics.

x -  0 -  x -  0  -  x