5th March, 2017

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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