Divi­sion by zero; or, guide to kick­ing a hornet’s nest

Why do we not divide by zero, and what hap­pens if we dare to? Also, some thoughts on infini­ties.

It is incred­i­bly funny that the reason we say divi­sion by zero is unde­fined is because we lit­er­ally have no idea how to define it. Per­haps con­flict­ing opin­ions exist’ would be a better state­ment. Bah’ would undoubt­edly be fun­nier since when­ever we encounter a $0/0$ in physics we des­per­ately rush to find ways of get­ting rid of it.

Gen­er­ally, divi­sion is treated as repeated sub­trac­tion, e.g. $18/6 = 3$ can be writ­ten as $18 – 3 – 3 – 3 – 3 – 3 – 3 – 3=0$. In other words, if $x/y=z$ it means we can sub­tract $z$ from $x$ and repeat this process $y$ times to arrive at zero. Of course the process is not so straight­for­ward if $z$ is a frac­tion, but we have no reason to rack our brains about such spe­cific cases right now.

The fact that zero is an awk­ward number to divide some­thing by has made it some­what of a recre­ation. With choco­lates, this is easy: divid­ing a Toblerone among zero people means you get to keep it all. So if divid­ing Toblerone by zero equals Toblerone, why does divid­ing $n$ by zero not equal $n$ too? This sounds good until you realise you are not really divid­ing Toblerone among zero people; you exist, which means the divi­sor is one, not zero. And $n$ over one is $n$ just as we expect it.

Not all exam­ples are so triv­ial, though. There are some that prove’ absur­di­ties like $2=1$. Con­sider the fact that $0 \cdot 1=0$ and $0 \cdot 2=0$, which means $0\cdot 1=0\cdot 2$ as a result of which $1=2$. But this is really a proof against divi­sion by zero. That is to say, if $0 \cdot 1=0\cdot 2$ it is because we divide by zero, as $(0/0) \cdot 1=(0/0)\cdot 2$, that $1=2$, which is clearly a fal­lacy, so $0/0\ne1$.

It appears that we decided only around the penul­ti­mate decade of the 1800s that divi­sion by zero is an impos­si­bil­ity (see Cajori, Florian’s 1929 paper, Absur­di­ties due to divi­sion by zero’) as far as ped­a­gog­i­cal alge­bra was con­cerned. This was in spite of the fact that the defin­i­tive result was stated half-a-cen­tury ear­lier by Martin Ohm. The form we know today was stated by Bernard Bolzano in the 1850s. If we solve the simul­ta­ne­ous equa­tions $a – b=a – b$ and $b – a=b – a$ we arrive at $a – a=b – b$ which can be re-writ­ten as $a(1 – 1)=b(1 – 1)$, which divid­ing by zero (since $1 – 1=0$) gives us $a=b$ even for two obvi­ously unequal quan­ti­ties.

We have nar­rowed things down now: the reason we do not divide or mul­ti­ply by zero is because, although doing so seems just fine, it implies incor­rect results else­where. So divi­sion by zero is essen­tially like taking a poker and stab­bing a hornet’s nest before diving into it head first.

Look­ing at it another way, the more dreaded result of $0/0$ actu­ally has a per­fectly mean­ing­ful mul­ti­plica­tive form: if $0/0=v$ then $v\cdot 0=0$ is a valid state­ment. This is what gives me some hope for $0/0$, but take another step and it becomes bleak: if $0/0=v$ then so can $0/0=a$ or $b$ or any of the other twenty-three alpha­bets in eng­lish or twenty-four in Greek and so on. Or, for that matter, any of the infi­nite pos­si­ble num­bers. This is why we safely state that $0/0$ cannot be defined’.

How­ever, could there be an as yet undis­cov­ered, spe­cial number that does sat­isfy $0/0$? One way to look at this is through limits. If any $a/b=c$ we quickly realise that, as $a ⟶ 0$ and $b ⟶ 0$, we must also end up with $c ⟶ v$. Except, if you keep making $a$ and $b$ smaller, you can get any arbi­trary answer you want that depends com­pletely on the rate of change of $a$ and $b$. In other words, if you keep halv­ing them, $c$ turns out to be dif­fer­ent from what you would get if you keep taking their square-root.

Once again, $c$ (and $v$ in turn) can be arbi­trary, making $0/0$ unde­fined. Or, if you want to look at the glass half-full, it can mean $z$ does exist but is a func­tion of that $a$ and $b$ which approached zero and we can define $0/0$ if we knew the nature of $a$ and $b$, and, there­fore, $0/0$ in gen­eral remains unde­fin­able. (Stu­dents of cal­cu­lus may notice this whole ‘$a/​b$ tends to some $a/​c$ as $b$ tends to $c$ approach’ that I took as some­thing like the rule of l’Hôpital.)

Getting such var­i­ous answers is noth­ing new in math­e­mat­ics or physics. For instance, the right-hand limit as $x ⟶ 0$ of $1/​x$ gives $+\infty$ and the left-hand limit of the same func­tion gives $ – \infty$. In other words, it can be tempt­ing to say that two infini­ties exist but this is not what a limit means; it simply means that we approach arbi­trar­ily large pos­i­tive or neg­a­tive num­bers. But look a little more and you will realise that many infini­ties do exist (just not in terms of limits) and that infin­ity’ itself is really just an umbrella term for an extremely large number.

There are math­e­mat­i­cal ways of look­ing at this (for instance, Cantor’s the­o­rem) but for the pur­poses of this essay, I will state an intu­itive approach: we all agree that there can be infi­nite num­bers simply because we cannot think of any jus­ti­fi­ca­tion to end count­ing at some point, which means the set of even num­bers has an infin­ity as does the set of odd num­bers and the two infini­ties cannot be the same. Another such inter­est­ing, non-math­e­mat­i­cal exam­ple would be that of Hilbert’s hotel.

If you really crave math­e­mat­ics though, here is another look: one can start count­ing from zero to one in suf­fi­ciently small frac­tions as to never reach one’ in their life­time. More inter­est­ingly, there can exist an infi­nite number of divi­sions between 0 and 1. The same is true of the 1 to 2 domain, and, clearly, the infin­ity that exists between 0 and 1 cannot be the same as that between 1 and 2. How­ever, you can make these two infini­ties equal by simply defin­ing a one-to-one cor­re­spon­dence between the set 0 to 1 and the set 1 to 2. That way, there are as many num­bers between 0 and 1 as there are between 1 and 2 and the infin­ity you reach while count­ing in one set (what­ever that means) is then the same as the infin­ity you reach in the other.

That is so far as only count­ing the number of ele­ments between the extremums of the sets goes. If you counted one set as zero plus the number of ele­ment, i.e. start count­ing the first ele­ment from zero instead of simply call­ing it the first ele­ment, and, like­wise, if you started count­ing the second set as one plus the number of ele­ment instead of simply as the first or second or nth ele­ment, you would still end up with two dif­fer­ent infini­ties with two dif­fer­ent weights and the weights would then differ by one, since the points from which you started count­ing, zero and one, differ by one. And remem­ber that this whole mess is char­ac­ter­is­tic of real or com­plex num­bers since the set of inte­gers and whole num­bers is well-defined: there are $9 – 4=5$ num­bers between 4 and 9, not infin­ity.

How­ever, none of these equal­i­ties exists for the even and odd infini­ties we came across ear­lier, or, say, a ratio­nal and an irra­tional infin­ity that we can sim­i­larly describe. That is to say, two infini­ties of the same kind can be shown as equal, but not two dif­fer­ently clas­si­fied infini­ties.

It does get mind-bog­gling. As physi­cists we then know­ingly throw some more tantrums: what, we ask, is the phys­i­cal impli­ca­tion of so many infini­ties; and what might be the hope­fully exis­tent mag­i­cal number, v, that can solve all our trou­bles? For the time being, that is a can of worms I do not want to open.


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