Solving Division Of Zero By Zero

 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,



so on.

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

Author: Narinder

I am graduate in Electronics and Electrical Communication Engineering from Punjab Engineering College, Chandigarh. I worked as Scientist in Solid State Physics Lab DELHI then National Fertilizers Ltd , Bhatinda as Instrumentation Engineer, then Ministry of Labour, Employment, Training as Deputy Director. Thereafter, I joined Civil Services in the year 1986, worked in different capacities on administrative posts and retired on September 30, 2013 as Secretary to Government Punjab from Indian Administratve Services. My interests are Astronomy, Physics, History, Music, Law, Spirituality, Administration and writings. I believe in hard work, determination and consistency in efforts. I love to write on topics related to Astronomy Daily life experience and human sufferings. My favourite writers are Leo Tolstoy, Rabindranath Tagore, Mulk Raj Anand and Munshi Prem Chand.

21 thoughts on “Solving Division Of Zero By Zero”

    1. You seem to be a busy person. If you find time, kindly advise me the name of sub atomic particles which can give solution of 0/0 as informed by you earlier.


      1. Hello Santino, sorry to have kept you waiting apparently I missed my email yesterday. As far as fundamental particles and the question of zero mass and the interaction with the higgs boson, Narinder explained that better than I. I argued that to answer the question of 0/0 is completely dependent on how one defines 0. In mathematics, 0 is a place holder for values between -1 and 1. One can, theoretically, infinitely approach zero but in reality one will reach zero and cross over to the negative intergers. All that demonstrates its’ value as a place holder.
        Now when we define zero as the absence of content, it is no longer a place holder. It is nothing and nothing is not divisible because there is simply nothing to be divided. Thus 0/0= empty set not 0
        This difference of definitions is the reason your calculator cannot divide by zero. In mathematics zero is a bit of a paradox, it has value yet in computations its’ value is none. As an example of defining zero as the absence of content, the problem of 5/0= 5 becomes possible because there is an absence of division of 5 therefore you get left with the 5 you started with but when zero is a place holder it assumes an infinitesimally small real value that is not specified so the answer for calculations such as 0/0 become unknowable.


      2. As far as the names of the sub atomic particles involved in the 0/0 question, I reason that the equation of E=mc2 can be altered to show E/c2=m which indicates that light is not massless. Obviously Einsteins equation is a specific form of the force equation (F=ma). To me this indicates that each frequency of light is one of the fundamental particles and that due to light speed being the fastest speed attainable for matter in the universe all the light particles are all the fundamental particles. Hopefully I have given an appropriate answer to the question. Of course the logic here is only one of 3 necessary parts to say that something is a fact, so this being true remains to be proven.


      3. As far as the names of the sub atomic particles involved in the 0/0 question, I reason that the equation of E=mc2 can be altered to show E/c2=m which indicates that light is not massless. Obviously Einsteins equation is a specific form of the force equation (F=ma). To me this indicates that each frequency of light is one of the fundamental particles and that due to light speed being the fastest speed attainable for matter in the universe all the light particles are all the fundamental particles. I also need to add that the differing frequencies is due to slightly different speeds of light which is an indicator of differing sizes of photons. Hopefully I have given an appropriate answer to the question. Of course the logic here is only one of 3 necessary parts to say that something is a fact, so this being true remains to be proven.


  1. U r touching my nerves as I was at bottom in math but my grand daughter has understood up to math portion whereas u proved by physics route. Great


    1. I am pleased to know your grand daughter has potential for higher learning in mathematics. She would be too young to go to college but keep building her interest in mathematics. If she would be good in mathematics, physics knowledge would be complimentary. Enjoy Sunday and take care against cold.


  2. Hello Narinder, I am certainly no expert but I do spend a lot of time contemplating the universe. As mathematics would say 0/0= 0, but this assumes that zero is holding a place for another potentially measurable value (the first zero). But if we define zero as true nothingness then there is nothing to be divided and no matter what value you are trying to divide nothing by there will be nothing but the second zero states we aren’t actually creating any divisions. Therefore our issue is with defining the first zero. Is it a place holder or does zero truly represent pure absence of content. As a place holder, I would have to agree that 0/0=0, but if zero represents the pure absence of content then 0/0= empty set. One cannot divide what is not. I know my logic isn’t perfect but I’m certain the concept I’m trying to convey is worth a moment of contemplation. I know that this is somewhat off topic but I have also hypothesized that photons are actual particles and each frequency category is actually a slightly different sized photon (example: x-rays vs. radio waves) the radio waves are larger particles than x-ray particles and therefore travel slightly slower than the higher energy particles thus the speed of light used in equations today is an average. I could go on and would love to elaborate if you are interested in that conversation. Very intriguing post 🙂


    1. Your philosophical views on zero are correct. You rightly said, it is nothing or absence. When there is something and if one is asked to divide it in infinitesimal small parts, then the number of such part is infinite or infinity. But here 0 is to be divided by 0, it can be divided amongst 1 or 2 or 3 or 4 or 5 or any number. Because zero multiplied by any number is zero. Therefore, 0/0 is called indeterminate.
      In this article, an attempt has been made to find its value on the basis of relativistic mass of photon because on the basis of Theory of Relativity, relativistic mass takes the form of 0/0.
      On the basis of De Broglie hypothesis, energy of photon depends upon its frequency and Energy is equal to Planck constant multiplied by frequency. But according to Theory of Relativity, Energy is equal to mass multiplied by velocity of light squared. Equating these two, relativistic mass takes the value of 0/0 which been attempted by me for its solution.
      Regarding photons,
      1 These move with constant velocity of 30000000000 centimetres per second. We can not change its velocity in a medium. But we can decrease its velocity by passing these into denser medium like glass or water.
      2 Frequency of photon is a measure of its energy. More frequency, more energy. Frequency is number of times a photon vibrates in one cycle. Radio Frequency —> Very High Frequency VHF —–> Ultra High Frequency UHF —-> Infra Red Frequency IRF —–> Ultra Violet Frequency UHF ——–> X Rays Frequency———> Gamma Rays.
      These are the electromagnetic frequency in increasing order.
      3 Wavelength is proportional to inverse of frequency and is equal to velocity divided by frequency. More the frequency shorter the wavelength. Higher wavelength means less energy. Wavelength is distance travelled by wave in one vibration or one cycle.
      4 Velocity of light or photon or wave is multiplication of frequency and wavelength. V = frequency multiplied by wavelength.
      If you are interested in wave and energy, you may save this reply for future reference. This will refrain you from mixing up Energy, velocity, frequency, electromagnetic wavelengths.
      N B Photons and light are synonymous here.


      1. So if we accept that different photons have different levels of energy by the equation E/c2=m we can then determine that photons do have mass and that mass should be applied in the equations involving photons. Yet we consider photons to be massless as a particle. I reason that this is due to only being able to measure its’ quantum field at this point in our knowledge collection. My overall hypothesis is that photons are the fundamental smallest particles that make up all larger matter and that energy transfer/conversion always has a bit of matter to facilitate the change or production of energy. Obviously, I am a determinist to deduce such ideas. For instance, my hypothesis would state for quantum entanglement, the entangled particles are tethered together by a thread of gravity not some “spooky” mysterious and unfindable magic.
        I think one of the leading problems with today’s experimentation is that we misinterpret our use of probabilities. Probabilities are like crowd statistics being applied to individuals and we know it’s helpful but can lead us to some very inaccurate conclusions. For instance, string theory and the multiverse interpretation trying to say that matter can disappear and reappear because we’re interacting with undetectable parallel universes. This interpretation is just a symptom of not understanding the behaviour of the smallest individual particles because we use the information of a crowd of particles to determine the possible activity of an individual particle. Like the photon, the smallest particles in the universe are single units but have a variety of sizes and amounts of energy. I know that with only a concept and no mathematics and data to back me up, I’m really reaching to assert these ideas. Although, I’m not the first to have such a thought. I believe it was Paul Dirac who had a similar idea. OK, I have gotten way off the topic of your original post but I don’t know anyone personally that is capable of having this conversation. I’m definitely curious to read your feedback on my thoughts. Thank you for teaching me a few things I was not specifically aware of.


      2. Mass of photon is due to its motion but when it comes to rest, it transfers its mass by way of energy to the object with which it interacts and it itself becomes massless.
        Regarding constitution of matter, some theorists hypothesise that photons constitute matter as you have explained. Your hypothesis “ the entangled particles are tethered together by a thread of gravity not some “spooky” mysterious and unfindable magic,” will require convincing theory and explanation for its acceptance.
        One of the differences between classical and quantum theory is that in classical theory, we can ascertain the outcome but in quantum theory, we can have probability of outcome. That is the outcome may occur and may not occur but occurrence and non occurrence has probability.
        Paul Dirac was a theoretical physicist who is known for Fermi Dirac Equation and other works in quantum physics.
        It is good to know you are interested in research work in the field of Physics. Keep it up. Regards.


      3. Thank you for the conversation and your polite and unbiased responses. I like your past topics so I eagerly await your next post.


      4. Thanks. Intuitively, I found you a gentle and sober person. It delighted me to know you have a keen interest in Physics.
        Keep it up.
        Again thanks for going through the article.
        Regards. Have a nice day!


  3. After looking over my posts again I realized that without my premise, it all looks like a fun time of pulling ideas out of nowhere. As a premise, for how the differing frequencies point to differing velocities to establish differing masses of each photon frequency, I have to go back to the singularity. All the energy that all the unverses’ matter has was from the single big bang so that the force applied to all matter was the same so that differing speeds of matter points to differing sizes of matter.
    As far as photons losing mass when they hit something, I cannot entirely agree because the object hit by the photon is energized and another flies out of the object that was hit. The first photons appears to have stopped when we look at that particular photon but again it has given more movement to the object that was hit and is therefore still moving as it is now one with the object. If all photons cease to move when they hit some object then by now 13.8 billion years later there would be some serious issues with a lack of entropy in the universe. Large matter would not continue to move if the smallest matter stopped by joining the larger matter. There would be truly dead matter all over the universe because it would be made of massless motionless sub atomic particles. So that is the perspective my assertions are coming from. Right or wrong these are really awesome concepts for the mind to play with.
    Well Narinder I hope I have contributed something worth the trouble of posing the original question.


      1. Good afternoon Narinder,
        I accept that new photons and other larger particles are being generated by stars. So do we agree that the photon becomes one with the matter it collides with or does it remain separate? From my perspective, the photon becomes one with the matter it collides with and therefore shares its’ energy with the matter rather imparting all of the energy to become massless. This would mean that the overall mass of the matter would be shared in an appropriate proportion to all the particles that constitute the matter.
        Is this a controversial issue and I’m proposing something maybe even new or has it been already decided with certainty that what I’m saying is pure rubbish?


  4. Photons are characterised by their energy. No energy photons means nothing. I take the case of energetic photons also known as gamma rays, X rays, when these interact with matter, these form electron and positron ( with same mass of electron but positive charge). Electron and positron then annihilate and reemit photons.
    2. These may knock out an electron from atom thus making it an ion. Compton Effect.
    3. These may increase quanta state of energy bringing the atom in excited state and when it returns to ground state, it will again emit energy.
    4. If photons are in optical range and interact with matter, these may be fully absorbed as in black body absorption or partly absorbed, partly refracted and partly reflected or fully reflected. In reflection, photons are absorbed and remitted. In refraction, photons pass into the matter, a part of it is absorbed, a part passes into matter with slower velocity if the medium is optically denser.
    5. Low energy photons when fall on conductor, these are absorbed exponentially with depth of the conductor and phenomenon is called skin effect. When absorbed by matter, these are re radiated as in case of antenna, this is due to the fact that conductor is a receiver and transmitter.
    Photons in fact gives its energy to the matter and loses its existence.


    1. Thank you for your instruction and patience with my ignorance Narinder, it appears I have some homework before I can even ask another question let alone contribute to this topic.
      Until our conversation continues, I wish you well.


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