User:Purushottam Kabra
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INFINITY DISSOLVED
[edit] Division by Zero: A different approach
by: Purushottam R. Kabra
Mathematicians of ancient India have mentioned that any process done on a digit with zero results into zero. Here ‘process’ means division and /or multiplication. On the contrary today’s mathematicians believe that i) multiplication by zero results into zero & ii) division by zero is (a) not possible, (b) it results in infinity, (c) it gives infinite numbers of results (answers) & (d) illegal or forbidden, which is illogical. Unfortunately, the explanations or arguments of Indian Mathematicians are not available. The whole world believes in the wisdom of Indian Mathematicians but is not ready to accept the answer of “division by zero” which is most contradictory.
It is strongly believed that answer for division by zero must be taken as zero. Let us try to explain it as under with some examples and logic behind it. In first place let me mention that ‘Division’ and ‘Ratio’ are not synonymous terms. We are messing the things by taking them at par. Ratio is explained in Part two.
We shall discuss this subject in three parts: I) division by zero relating to numbers only, II) division by zero in relation to ratios, proportions and percentages, III) Circular progression of number system vs. linear progression.
PART ONE: DIVISION BY ZERO RELATING TO NUMBERS ONLY:
1) Let us take a commonly taken example of an apple. When we divide one apple by 2 we get 2 equal pieces of the apple. In mathematical form we put it as ‘1/2’. We call ‘one’ as numerator (dividend), ‘two’ as denominator (divisor). After doing the operation what we get is ‘result (quotient)’ & it is ‘0.5’. In this case we are not taking any remainder.
1 / 2 = 0.5
Currently, in our example, we are taking ‘1’ as a ‘thing, ‘2’ as a ‘knife’ and getting ‘0.5’ as result of action of cutting with knife. It is what we learn in schools and our mind/memory has photographed the picture.
But I humbly desire to differ the above and present it in other form.
Numerator = a pot/plate/container containing the ‘thing’, here it is one apple (on our left side).
Denominator = number of pots/plates/containers/groups in which we desire to put the thing in equal parts (pots to be kept on our right side). In our example it is ‘2’ (but shown under 1 on left side). Or say available number of pots.
Quotient = Quantity/amount/part of the thing in ‘one’ container out of the given number of containers (on our right side and there are other container/s which contain the equal amount). In our example there is one containing ‘half’ an apple and the other containing the equal amount. In current system we are taking thing in one plate as the final answer. But it is not point of difference.
Remainder = (If we are not following decimal system) what remains out of numerator which is not divisible. In decimal system also sometimes a part remains as remainder. For example: 10/3 = 3.33 and 0.01 remains. Unless we add this remainder to the quotient test we do not get the exact 10 as answer. Undistributed or undistributable and available for distribution.
If there are three apples and we want to divide it in 2 parts we will get ‘1.5’ apple in each of the plates. This can be done for any number of things taken as numerator and any number as denominator. Currently we are doing it and getting the proper result. But when we take zero as denominator we answer it as ‘infinity’ which is an unmeasurable answer.
If above presentation is considered in proper perspective, we can put any number as ‘numerator’ and take ‘0’(zero) as denominator and the result will be ‘0’. This is because we have no container on our right side. As there are zero containers on right side, we are not going to put part of the numerator thing in any of the plate. It may be noted that zero is not a digit/number and it is sign for absence of anything. Further what remains on left side is ‘remainder’ which is numerator itself & it is still available for distribution.
It is possible that somebody may ask then why we are not taking the numerator itself as the answer? This is an irrelevant question, as we are not doing any process of division and, as such, taking/putting nothing on the right side. Here again take note that what remains on left is a remainder & it is still available for distribution
2) Surely the above preposition will satisfy the ‘quotient test’ as ‘0 x 0 + remainder’ (quotient multiplied by divisor plus remainder) will give us the original numerator. Accordingly this test of multiplication will not give us so called innumerable answers though innumerable numbers (taken for quotient) till infinity may satisfy the test. But that may also not happen as we have numerator as remainder. It is astounding fact how so far we forgot to consider the ‘remainder’. This is the reason we are getting the confusing answer. Further, we are taking ratio and division at par, that is one more reason that we get such confusing answer. Again consider the example of division of ten by three given in definitions of remainder above.
Computers and calculators are not programmed to give us the remainder, which further added to our confusion.
3) One more logical point: As division by zero does not satisfy the ‘quotient test’, we say that it has innumerable answers or infinity is the answer or indeterminate answer. Out of these innumerable answers the last answer is the infinity. Instead of taking the last in row ‘infinity’ as answer why we do not accept the first answer i.e.’0’ as the logically correct answer is not understandable. Logically we should take the first natural answer as the correct answer instead of taking last imaginary answer. (Here I again submit that the ‘0’ as answer satisfies the quotient test).
4) Zero is neutral: actually zero is a neutral thing. Further it represents absence of anything.
I) If we add to or subtract ‘0’ from anything, it does not affect the number/set of digits.
II) Multiplication is a process of adding the original ‘set of digits/number’ (multiplicand) for number of times (given as multiplier). When we add a ‘set of digits/number’ ‘0’ time, it will give only ‘0’ as the answer. For example ‘1 x 3’ is ‘1+1+1’ giving the product as 3. If we want to do ‘1 x 0’ we are taking ‘1’ for ‘0’ times and naturally gives ‘0’ as the product.
III) As stated in “Mathematical Handbook (elementary mathematics)” by “M. Vyogodsky” published by “Mir Publishers, Moscow” English translation.1979 edition on page 66:
a) Division is the process of finding one of two factors from the product and the other factor.
b) It is the process of determining how many times one number is contained in another.
I simply take it as division is a process of subtracting the ‘quotient’ from the ‘numerator’ for the number of times stated as ‘denominator’ to give ‘0’ balance (minus ‘remainder’ if we are following the remainder system). Or in other words, ‘quotient’ multiplied by ‘denominator’ plus the ‘remainder’ equals the ‘numerator’. Accordingly, if we take division by zero resulting in zero and adding the remainder to it gives us the numerator. It satisfies all the tests needed to verify correctness of the division. It may be put in other words that ‘0’ is deducted from the ‘numerator’ for ‘0’ times and the ‘remainder’ (which is the numerator itself) is added to the result (which is ‘0’) gives us the ‘numerator’ hence all the tests for verifying correctness of division are satisfied and accordingly answer ‘0’ after division of any number by ‘0’ is correct.
IV) Dividing ‘0’ by any other number is also an absurd preposition as it is attempt to divide ‘nothing’ into given parts which is not possible.
V) It is humble presentation that zero should not be taken as a digit and should be taken as absence of any digit/number.
5) If we attempt to divide the smallest whole number ‘1’ by zillion zillions (or say googol googols) we will get some answer which has zillion zillions (or say googol googols) zeroes after decimal point followed by ‘1’ but it will never be ‘zero’ & has some value. It tends to zero but will never touch the zero position. If hypothetically we take that it gives zero as the answer, then it will also be possible that it may enter in minus territory giving ‘negative (-) answer if we put more zillion zeroes on to the denominator, and it is not possible. (If it is possible, then it may also be possible that a time will come when division of a positive integer by a number followed by multi-zillion zeroes after decimal point give us a negative answer. Which is not at all possible. Further if we stretch our logic, it can said that if division by zero gives infinity as the answer then division by negative number shall give us multi-infinity as the answer, but that is not the thing.). Its limit may be zero but not zero itself. Conversely, division by ‘0’ cannot give you the inexhaustible (greater than any quantity that can be assigned) or say infinite answer or any INFINITESIMAL (an infinitely small) or even an indeterminable answer. In another way nobody can give a number dividing by which the quotient answer will be zero. Therefore Only zero is the answer if division by zero is performed on any number. The zero is not infinitesimal part of ‘one’; it is an independent entity, it has no speed of itself, it is static, it is starting point only & it is ‘equidistant’ backward from ‘one’ as is ‘two’ in forward direction.
6) The question of dividing ‘0’ by ‘zero’ does not arise, because a ‘nothing’ thing cannot be divided in to zero (nothing) parts, though it may be considered for LIMITS calculations. Further as I am of the opinion that independent zero is not a number and is ‘an expression’ for nothing, as such; I call it ‘death of the limit/number’. Please consider the following examples:
5/5 =1, 5/4=1.25, 5/3=1.6666…, 5/2=2.5, 5/1=5, 5/0.5=10, 5/0.1=50, 5/0.01=500.
As we reduce the denominator, the quotient increases in certain proportion/progression. We may get infinitely great quotient if we go on increasing zeros in the denominator before the figure ‘1’. But we can not eliminate the figure ‘1’ from the denominator after so many zeroes. If we take out figure ‘1’ which we put after so many zeroes we are doing the silly thing, absurd thing. We may progress towards zero by putting any number of zeroes between decimal point and the figure ‘1’ but can’t touch the ‘zero’ position. Accordingly we may get the infinitely great answer by dividing a number by any smallest number near to ‘zero’ position but can’t divide by the independent zero which gives us conclusion that division by zero results in giving zero.
Take another example,
5/4=1.25, 4/3=1.3333.., 3/2=1.5, 2/1=2 the quotient is increasing in certain proportion, then how come 1/0=infinity? Or even anything like ‘0.0001/0 = infinity?
This leads us to the conclusion that the ‘independent zero’ is a neutral position, unaffected position, unaffectable position. In can’t be negative or positive.
PART TWO: DIVISION BY ZERO IN RELATION TO RATIOS ETC.
A)RATIOs: We are nowadays misusing the term ‘ratio’. As described in the book referred supra “ratio once had a different meaning from ‘division’.” The term ratio was once applied only to cases when it was required to express one quantity as a ‘fraction’ of another homogeneous quantity. It means that the denominator must be greater than the numerator, which will give the ratio or relation between them. In the case of ratios the dividend (numerator) is called antecedent and the divisor (denominator) the consequent of the ratio. Presently we use the term ‘ratio’ for non-homogeneous quantities. Examples: One area as a fraction of another area is case of ratio of homogeneous quantity
Weight of a solid object to its volume is case of ratio of non-homogeneous quantities.
As such the term ‘ratio’ may not be treated as ‘division’. The division by ‘0’ is not at all involved in case of ratios.
B) Proportions: The relevant terms in case of proportions are ‘means and extremes’. Two equal types of ratios form a proportion and hence as explained in case of ratios the division by ‘0’ is not at all involved.
C) Percentage: The percent means hundredth part. It is implied that the division by zero is not involved.
We love the infinity too much; hence we can coin a new term called ‘zero infinity’.
LINEAR PROGRESSION VERSUS CIRCULAR PROGRESSION:
Instead of taking the number system in linear progression, if we learn the number system in circular progression we will be able to consider the ‘0’ at the tangent position. Infinity will be very close to the zero but will never merge in zero. Multiplication can be taken as speed of ‘floating point’ in forward direction till the multiplicand and division can be taken as speed in backward direction till we reach the denominator. This full circle will be of positive integers. The tangent at point ‘0’ can be assumed as an imaginary powerful absolute mirror which can reflect the numbers on both sides. What will be seen in the mirror will be ‘negative’ circle and the tangent will be common. The mathematical operations involving ‘+’ will be answered in positive circle and operations (except addition and subtraction) involving single/odd ‘-ve’s will be answered in negative circle. But operations involving even ‘-ve’s (except addition and subtraction) will be answered in positive circle. (One negative will give answer in its own negative circle but then its mirror image due to second negative operation will be in positive circle.) All the confusions will die. No rule has been flouted.
It is my inability to refer ‘Modern Algebra’ (groups, rings & fields). It is believed that what is true for Arithmetic, must be true for any branch or division like calculus, limits, sets, rings, groups, fields, vectors etc
It will be an interesting project to find out how ‘remainder’ remained to be considered while doing division by zero and how ratio became synonym for division.
The infinity is dissolved. SALUTE THE INDIAN RISHI-MUNIES AND GREAT MATHEMATICIANS.
Let me mention here a Sanskrit verse in praise of ‘The ALMIGHTY INFINITY’. Find its relevance in the present context.
“AUM. POORNAMADAH POORNAMIDAM POORNAT POORNAMUDACHYATE, POORNASYA POORNAMADAYA POORNAMEWAWASHISHYATE
.
Simple meaning: AUM = The key to ULTIMATE KNOWLEDGE. THAT is TOTAL, absorbs the TOTAL, and from TOTALITY attains TOTALITY, THAT gives COMPLETENESS to the COMPLETE and still remains COMPLETE.
I am looking forward your feedback. Email ID: pooraak@yahoo.co.in or my blog. 59.95.12.52 (talk) 16:47, 7 February 2008 (UTC)
