So in grade 7 math we learned that you can’t take the square root of negative numbers because any number squared is always going to be positive. A few years later we learn that you can actually calculate with the square root of negative numbers. You replace the square root from -1 with i. So why aren’t we replacing something divided by zero also with a letter?
In: Mathematics
There are certain rules that we want certain kinds of number systems (what a mathematician would call “fields”) to satisfy. Namely, on a set *F* we want two operations + and * such that
* Addition and multiplication are commutative and associative.
* There is an additive and a multiplicative identity, usually called *0* and *1* respectively. We require that *0* and *1* not be the same.
* Every element *a* has an additive inverse, usually called *-a.*
* Every element *a* except *0* has a multiplicative inverse, usually called *a****^(-1)****.*
* Multiplication distributes over addition.
We want these properties because we consider them “essential” to our usual field of rational numbers or real numbers. In particular, we want *0* and *1* to be distinct because then every number in our field would be equal to *0* (a fact which I prove at the end). There are lots of different fields, some of which are very weird. The complex numbers, in which negative numbers have square roots, and finite fields, in which I can add *1* to itself enough times and eventually get *0*, are some examples.
Now, it follows from the bullets above that anything times *0* is *0*. Indeed, if I have any *x*, then *x*0=x*(0+0)=x*0+x*0*, and subtracting *x*0* from both sides gives me *0=x*0.*
Now let’s see what happens if we allow *0* to have a multiplicative inverse in one of these fields. Say *x* is the inverse of *0*, so that *x*0=0*x=1*. Well, then I could multiply this equation by zero and get *0=1*0=(x*0)*0=x*(0*0)=x*0=1*. So in this case, *0=1*! Since we explicitly required that *0* not equal *1*, this shows that the inverse for *0* which we called *x* cannot exist if we want all of the bulleted properties.
Well now the question is, what if we don’t want all of the above properties? Well, mathematicians study a bunch of other objects which satisfy only some of the above properties (try googling “rings”, “groups”, “semigroups”, etc.). It turns out that none of these can have inverses for *0* either (assuming they even have a *0*). I have heard of one object, called a “wheel,” which tries to formalize divison by zero. I played around with the concept in undergrad once but I quickly became uninterested. I’m not sure how useful this concept is. Google “wheel theory” to learn more (although it’s not very easy to understand).
The point here is that if you want to invent a new number system where you allow an operation that is usually not allowed, then you need to prove (in the precise mathematical sense) that your number system doesn’t have any contradictions like the one we showed above. Introducing a number *i* such that *i**^(2)**=-1* indeed introduces no such contradictions, so mathematicians consider it to be perfectly valid, hence the field of complex numbers.
EDIT: For those curious, if *0=1*, then if we have a number *x*, then *x=x*1=x*0=0*. Hence every number is equal to zero.
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