Couldn’t the result of division by zero be “defined”, just like the square root of -1?
Edit: Wow, thanks for all the great answers! This thread was really interesting and I learned a lot from you all. While there were many excellent answers, the ones that mentioned Riemann Sphere were exactly what I was looking for:
https://en.wikipedia.org/wiki/Riemann\_sphere
TIL: There are many excellent mathematicians on Reddit!
In: 1691
>Couldn’t the result of division by zero be “defined”, just like the square root of -1?
No. The very cornerstone of the numbers we use is that a number cannot equal its own successor: that 0 does not equal 1, 1 does not equal 2, 2 does not equal 3, etc forever. Two very important properties of number spring up from this idea: the Zero Property, and the Multiplicative Identity.
The Zero Property says zero times anything is zero. The Multiplicative Identity says for all numbers that are not zero, there is a number you can multiply with it to get one. These properties are mutually exclusive. We can show these properties are true in proofs *using only the above concept of successors* (numbers not equaling the number that comes after them).
The Multiplicative Identity is how we define division. That for any number that is multiplied with its prime version, so that their product is one, that prime version is a divisor. Since this exists for every number except zero, division is defined for every number except zero. We could say, for example, 2 times the multiplicative inverse of 3, is 2 divided by 3.
If we were to define a number that multiplies with zero to get one, to create a situation where zero is a valid divisor, then we would have to say that number times 0, ***equals both zero and one***, so zero equals one. That’s a violation of the premise of successors that our number system is based on.
That’s why division by zero cannot just be defined.
For comparison, the square root of one is an operation defined on all inputs. You can put any number into it and get an output with only the catch of the even root of -1 being not yet defined, but being consistent in value. Meaning, we can have an input of negative one without undermining any other rules of numbers, and as such we can create and define an output for it.
Yes! You absolutely can. This is done in several mathematical objects. The terms to google are the prjectively extended real line and riemann sphere.
In all such objects you do lose properties, as others have shown. But this isn’t any different to imaginary numbers. Including i means you lose some key properties of the real numbers, namely the ordering.
It is consistent, if you make it so. Sit down, say `x / 0 = q` or whatever, and see what happens. What is q multiplied by a number? What about adding it? And so on.
What you’re really asking is, why don’t I hear about this `q`, and why don’t mathematicians seem to use it?
Imaginary numbers were invented to solve a problem: mathematicians were trying to solve cubic polynomials and were constantly getting square roots of negative numbers. They decided, well what if that did work, and the imaginary numbers ended up canceling out so you got “normal” solutions. That’s why we know about imaginary numbers, they ended up being helpful for solving problems. (They also ended up having lots of applications in engineering and computer science, so lucky them I guess).
If you sit down and try to solve problems with your `q` you don’t really get anything useful. Assuming `x / 0 = q` causes too many contradictions to be helpful. For example, `q * 0` is equal to every number, which makes every number equal to each other. You can probably design some way out of this, but you can see how this is hard to use as a tool to solve problems.
The fun thing about math is that you can assume anything to be anything, the hard part is assuming something useful. Imaginary numbers are. Dividing by zero isn’t really.
This is basically a slightly involved question. My mathematical answer would be that including a square root of -1 simply creates a larger ring. Not only that, but starting with something nice (like the integers, the rationals, or the real numbers), and including i, gives you something equally nice or possibly nicer.
The integers (a Euclidean domain) become the Gaussian integers (also a Euclidean domain).
The rationals (a field, namely the field of fractions of the integers) become another field, the field of fractions of the Gaussian integers.
The reals (another field, this one being “Cauchy complete”) become the complex numbers (also a Cauchy complete field, but it’s also algebraically closed – even better).
To put this in ELI5 terms, you don’t really “lose” that much by including a square root of -1. And in the case of real to complex, you actually gain something extremely important (algebraic closure).
On the other hand, including something like 1/0 (a multiplicative inverse of 0) breaks a lot of things. Your structure can no longer be a ring, unless it’s the trivial ring (in other words, if you want “normal” arithmetic, you are forced to have 0 = 1, which it turns out is super boring).
Here’s a quick proof that should be ELI5 compatible. Suppose x = 1/0, a number we want to include. By definition, we are saying that 0x = 1 (it “cancels out” the 0). The problem is that 0 = 0+0, so we can substitute that in: (0+0)x = 1. Now, an important thing we want multiplication to do is distribute (expanding the brackets, in school math terms). So we should be able to say 0x+0x = 1. Now we have a “problem”, because 0x = 1 at the start, so this equation says 1+1 = 1, in other words 2=1, and 1=0.
So, again in ELI5 terms, being able to divide by 0 causes you to “lose” a whole lot. Your nice number system can no longer have normal arithmetic, unless you want 0 to equal 1, which it turns out means this is the only number (every number is equal to this “0 & 1” number). So you have to start making weird arithmetic rules, but in doing this you lose other nice things, and so on. You can do this, but these number systems are not as nice as the things I talked about with the imaginary number i.
Defined by what, though? Take any random fraction, let’s say, 6/2=?. Let’s replace the question mark and do some simple algebra. Now you have 6=x*2. Now you should clearly see a/b=c is the same as a=c*b. Now there’s a problem when b=0 *because* no matter what number you multiply with the result is always 0 and cannot be a. Let’s see that in action using 6/0
6/0=x
6=0x
These 2 equations are equivalent due to the formal rules of algebra. But let’s throw out the rules and try it anyway. So let’s define the answer to zero division as @ and try those 2 equations again
6/0=@ Seems legit, let’s continue
6=0*@
Uh oh. Now there’s a problem because no matter what @ “equals”, when multiplied by 0 the result is always 0. And 6=0 is false
Let’s even look at this another way. As children we were taught basic division by repeated subtraction. So 6/2, if you have 6…idk…apples. Yes 6 apples, and you remove 2, you now have 4 apples. Remove 2 again and you have 2 apples. Remove the last 2 and you have no more apples. So 6/2 equals 3. But what happens when you remove *no* apples? There is no change. In other words 6-0=6. No matter how many times you repeat this it will **always** be 6. Sometimes people try to say the result of zero division is infinity, but you should see very clearly that isn’t true either. And even if it was true, infinity is also undefined lol
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