Solving division of zero by zero has remained a puzzle in mathematics and physics. Mathematics books say that zero divided by zero is indeterminate and can not be found out. Solving it has become all the more important particularly when we come across a tensor component which is equal to zero and is also required to be divided by another tensor which is also equal to zero.
Zero is naught or nothing or no presence or absence. Interestingly, zero was not considered significant prior to Aristotle probably due to the fact that when something is nonexistent, there is no use of pondering over it. It was Brahmagupta (597 AD – 667AD) an Indian mathematician and astronomer who gave detailed account of zero in his Brahmasphutasiddhanta which lays down that positive multiplied by positive is positive, positive multiplied by negative is negative, negative multiplied by negative is positive, negative multiplied by positive is negative amongst other formulae. However, division by zero was considered naught.
Thereafter, Bhaskar, Indian Mathematician and astronomer, explained the division by zero as infinity. The work of Hindu mathematician travelled to Arabian countries in the west and China in the east. From there it traversed to Europe.Thereafter, English mathematician John Wallis gave it the symbol ∞. Thence on division by zero was recognised as a quantity which is larger than the largest.
Zero divided by zero can be written as (0+0+0+0)/0, since 0+0+0+0+0 is 0. It can also be written as 0+0+0+0+0 as 5 multiplied by 0 as anything multiplied by 0 is 0. Therefore, (0+0+0+0+0)/0 = 5 x 0/0. Considering division of zero by zero is permissible, then 5 x 0/0 becomes equal to 5 on cancelling 0 of numerator with 0 of demoniator. Also since 4 multiplied by zero is zero, therefore, I can write 0/0 = 4 x 0/0 or 4 after cancelling zero in the denominator with zero in the numerator.That means 0/0 is equal to 5 and also 4. That means 5 is equal to 4 but it is absurd since both have different values.
Where is the problem then? While solving 0/0, it was assumed by me that that 0 can be divided by zero as I cancelled 0 in the numerator with zero in the denominator. Thus my assumption that zero can be divided by zero is wrong, I can not cancel zero in the denominator with zero in the numerator. If I do, it leads to absurdity.
Now let us look at zero as a quantity which is extremely small and is tending to zero. Sin x tends to zero when angle x tends to zero. I know calculate Sin x/x when x tends to zero.
That is limit x——>0, Sin x/ x = very very small quantity tending to zero divided by very very small quantity tending to zero. Let this very very small quantity tending to zero is’a’
then Limit x——>0, Sin x/ x = a/a = 1.
But if we were to find the value of Sin x/ x at x = 0, then it becomes 0/0. For the information of my learned readers, here x is not tending to zero but it is zero. There is a huge difference between zero and tending to zero.
I submit that it should not always be considered division of a quantity tending to zero by another quantity tending to zero is always one. In this regard, I give an example of Sin(x/3)/x , limit x——->0 is not one but it is equal to 1/3. To know how it is done, one will have to study chapter of limits in Calculus. However, it is clear from above that a quantity which tends to zero divided by another quantity that also tends zero is divisible but 0/0 is not divisible. We can say that we can obtain limiting value of (—–>0)/(—>0) but not 0/0.
How can then we solve zero divided by zero? For that I refer to Albert Einstein who propounded the theory of relativity which states that quantities mass, time, length are not absolute or constant, their value changes, according to the frame of reference in which these are. For example, time elapsed in completion of an event on earth will be different from that passed at some other planet for the same event depending upon their relative velocity. Time stops for a person who moves with velocity of light with reference to inertial frame of reference. Similarly, mass of an object also varies depending upon its velocity, higher the velocity, more will be the mass.
All particles except some type of Nutrinos, Bosons and Photons which are massless, have mass. Photons which have rest mass zero, move with velocity of light and thus have energy and mass on account of their motion. Einstein deduced the relation between relativistic mass and rest mass as
M = Mo/Gamma ……………………………….(1)
where Gamma = Square root of quantity (1- v squared/c squared),
v is velocity of mass, c velocity of light, Mo rest mass, M relativistic mass.
On squaring and rearranging, we get
Mo squared/M squared = 1 – v squared/c squared…………,,,.(2)
At rest, photon mass Mo, its relivistic mass M are both equal to zero. And also velocity v is zero since photons are being considered at rest.
On substituting these value in equation (2), we get
0/0 = 1 – 0/c squared,
or 0/0 = 1.
This value 1 has been considered by some followers of mathematics to be correct but applicable in case of relativistic photons. In this regard I submit that this value may not be universally applicable and other conditions would also have to be considered in cases other than that of photons. To prove my point, I examine it to ascertain whether the value of 0/0 as 1, passes the tests of scrutiny.
I consider equation (1) and to calculate relativistic mass M, I substitute Mo as 0, velocity of photons v as c.
Then M becomes equal to 0 divided by square root of (1-1/1),
or M = 0/0 but we have calculated 0/0 as 1,
therefore, M = 1. That means, photons should always have same constant mass and therefore should have same constant energy, since E = M x c squared. But practically photons have different energy depending upon their frequencies. Photons constituting ultraviolet light are more energetic than photons constituting infra red light. Therefore, the fact that photons have constant mass as 1, does not match with practical results. It, is therefore, incorrect to conclude that relativistic mass of photon is always one. It is also incorrect to arrive at the value of 0/0 as 1.
It is submitted that light has dual nature, it travels as a wave and also as a particle of quantum of energy photon. Physicist De Broglie gave the hypothesis that matter when it moves, it moves in a wave whose wavelength is equal to Planck constant divided by momentum. Mathematically, this equation is generally written as Lambda = h/p.
Further, he related the energy of the matter in motion with its frequency as Planck constant multiplied by frequency. That is E = h.f ……………………………(3),
where E is energy and f frequency.
Therefore, if the photon has the frequency f, its energy would be h.f and momentum as h/wavelength. That also means if we increase the frequency of photon, its energy will increase. That is why ultraviolet light has more energy than red light.
On equating this energy h.f with M.c squared, we get
M.c squared = h.f or M = h.f/c squared.
This shows that photon acquires mass due to its motion whereas its rest mass is zero. Since velocity of light in a medium is constant, therefore, relativistic mass of photon varies with its frequency.
We have earlier found that relativistic mass of photon is 0/0.
Therefore 0/0 becomes equal to h multiplied by f divided by c squared. But quantity h multiplied by f divided by c squared varies according to the frequency of wave. That means 0/0 will have different values depending upon the frequency of photon.
Can a numerical fraction have different values? No, it should not, will be the natural answer. Let us consider whether 0/0 can have different values simultaneously. We have determined the value of 0/0 as h multiplied by f divided by c squared. Putting different values of frequency as f1, f2, f3 so on, we well get value of 0/0 as h.f1/c squared, h.f2/c squared, h.f3/c squared. Let us say, these are equal to l, m, n ……. so on.
Then 0/0 =l,
On crossmultiplication, we get
0 = l X 0 = 0,
0 = m X 0 = 0,
0 = n X 0 = 0
These equalities 0=0 are universal identities, therefore, 0/0 unlike any other fraction, can assume different values defying mathematical principles. That indicates value of 0/0 is not constant but it depends upon the parameters and type of equation where it occurs, in the case of photon, it is h.f/c squared. 0/0 may assume a different value if it happens to occur in some other physical formula involving different parameters.
Before parting with this article, I request my learned readers to check what our electronic gadgets say about zero divided by zero. Interestingly, on enquiring value of zero divided by zero, ‘Siri’ on iPhone, ends his explanation by saying, “You are sad that you have no friends.” I wonder how loneliness is connected with the solution of zero divided by zero. Anyway, let us call it a day.
Have a great day!
NB: It is an attempt to solve zero divided by zero and your comments are welcome.
Writer is an Electronics and Electrical Communication Engineering graduate and was earlier Scientist, then Instrument Maintenance Engineer, then Civil Servant in Indian Administrative Service (IAS). After retirement, he writes short stories and also on subjects, Astronomy, Mathematics, Yoga, Humanity etc