As a layperson who is interested in math, imaginary numbers always fascinated me. Like in the real world you taking the square of a negative makes no sense whatso ever, but in theoretical math you can just invent new imaginary numbers, make it so that *i*^2 = -1 and suddenly you have just revolutionized math. If this is useful, why can’t you break other rules and account for them with new imaginary symbols?
So let’s pretend that we call them made up numbers and use *m* to represent them. Why is *m*=1/0 impossible when something like *i*^2 = -1 is not?
In: 27
Lots of answers are giving good reasons why you can’t. But I would actually argue, you can, but people won’t use your new framework or your new mathematics until you prove that it can be useful. You can invent rules if you like, but the rules that we end up teaching everyone are the rules that end up being useful.
The reason imaginary numbers are revolutionary is because they seem totally wrong, but are very very useful.
The problem with “defining” the result of dividing by 0 is that it could be anything, depending on the context, so the result would be fairly meaningless without that context.
The way we can reason about this comes from calculus, specifically limits if you want to look it up more – consider a function such as 1/x. When x gets really small, 1/x gets really big. For example, 1/0.0001 = 10000. But, if x is negative, x gets really big and negative. This means that from one side, it’s negative infinity, and from another side, it’s positive infinity. These are two different values.
Consider a different function, x/x. Anywhere except where x = 0, this function is 1. In fact, we can also say the limit at x = 0 is 1. But we still can’t say that 0/0 actually is 1 because in other contexts it’s something else.
Outside of the context, the value is useless – within the context, we have the concept of limits to formally reason about divisions by 0.
You can always find a way to define what divide by 0 looks like.
Mathematicians don’t like being told they cannot do something; if they find something that doesn’t have a rule for it they try to come up with one, and *i^2 = -1* is a great example of this. There is no (existing) number that squares to negative one, so we will define one (or two, depending on how you look at it) and see what happens.
But for these definitions and rules to stick they really need to be consistent and, if possible, useful. The *i^2 = -1* does this. You define this new concept, but it still follows all the existing rules of algebra and number theory, and has some useful results.
Dividing by 0 doesn’t work that way. So far no one has been able to come up with a rule for dividing by 0 that is consistent with all our other rules for algebra and numbers (outside the special case of 0/0, which we tackle with limits if we are careful), never mind one that is useful.
The closest we get is the concept of infinity, but that doesn’t work as a number, it doesn’t work with algebra or geometry(ish – there are some exceptions – and this is probably where infinity exists in the most divide-by-zero way), doesn’t even work with infinitesimals and limits. Analysis uses infinity a bit, although not in the divide-by-zero sense, but it isn’t really until we get into set theory that infinity starts to become interesting and useful. But again, not really in the divide-by-zero sense.
You could always start a paper by stating “we will define 1/0 = banana”, but that won’t help you with anything.
There isn’t a way to divide by 0 in a way that is consistent with the rest of maths, or useful.
The problem with dividing by zero, or, more properly having a zero in the denominator is that it could literally be anything. We call it undefined, but I think it is better described as ambiguous. Say you have 2/0, I have handed you two things but you have no idea whether they are half of something or 533rds of something. The number i is a placeholder but it is always only one thing. That is to say, it is unexplainable but definable and predictable, something a ratio with a zero denominator can never be.
Compare it to having a 0 in the numerator, you know the line or quantity was evenly divided once thereby creating 2 equal sets or lines. If I hand you none of those equal sets, your ratio is defined as zero. If I hand you two of the resulting sets for 2/2, you have all of the even subsets and therefore your ratio is just the whole number of 1. You can do this up to infinitesimally close to infinity in the nuker provided your denominator is >0.
You **can** divide by zero in some theoretical field. It just isn’t meaningful or helpful.
Imaginary numbers are just as real as other numbers. We use them to solve problems and make accurate predictions. If you’re trying to model what happens when you turn a crank, apply electricty to something, or run water through a wheel you’ll probably want to use imaginary numbers.
Imaginary numbers are all about change. It’s easy to say “I have one orange” and “You lost one orange” because we’re used to dealing with 1 and -1. But how many oranges do you have while you hand one to your friend? Do you both have 1? No, because there’s only one orange, not two. Do you both have half an orange? No, because the orange isn’t split in half. It’s more accurate to admit that the situation is complex. You can use imaginary numbers to model the gradual change of ownership during the handoff.
Let’s say you have a big group of people handing each other oranges. You can make an equation to calculate how many oranges each person has at any point in time. Using imaginary numbers will make this equation more accurate and meaningful. If you look really closely at one person during one moment you’re equation will show that he has some real oranges and some imaginary oranges. And that’s okay. At the end of the day, you wait until all the handoffs are complete before telling everyone to stop and go home.
Even when you’re working with imaginary numbers you usually do the math so that those imaginary numbers are elminated from the final result. You don’t want to leave things partway complete; you want things to settle down into something more tangible.
We can, and it is called [projectively extended reals](https://en.wikipedia.org/wiki/Projectively_extended_real_line). As others have said, we have to consider the consequences. Every time we add something to a number system, we lose some nice properties.
Going from real numbers to complex numbers isn’t “free” either. The biggest one (I’m aware of) is we lose total order. Namely, for any real numbers a and b, either a > b, a < b or a = b. This does not hold for all complex numbers.
Each of the following holds for all real numbers a, but does not hold for all projectively extended real numbers a.
– a × 0 = 0 × a = 0
– a – a = 0
Also, you know how when doing algebra in real numbers, you have to be careful with division by zero? For example, when given x^(2)-5x+4 = x-4, you can’t just divide both sides by (x-4). If you want to divide you have to break it into two cases where x=4 and x≠4 to arrive at (x^(2)-5x+4)/(x-4) = 1 OR x = 4
When dealing with projectively extended reals, this spreads to addition, subtraction and multiplication. In reals, only division can produce undefined results. In projectively extended reals, all four basic arithmetic operations can. For example, given x+a = x+b, we **cannot** subtract x from both sides and conclude a = b. Like the example above, we have to consider the two cases where x=∞ and x≠∞ to arrive at a=b OR x=∞.
Is defining 1/0 = ∞ useful? Maybe. Is it worth the cost of losing some nice arithmetic properties as discussed above? That is for you to decide.
You can define 1/0 in [wheel theory](https://en.m.wikipedia.org/wiki/Wheel_theory) it equals infinity. The problem is that many of the other basic rules of algebra have to be redefined subtly, and the result doesn’t have as much of a connection to the real world
Full disclaimer: I don’t think there’s a way to ELY5 on this one.
So, imaginary numbers have been given a bad rap; they’re not made up or invented or anything, they exist. They’re a different “dimension” of value. It’s not gonna really make sense in a day-to-day type of explanation because they don’t describe a value we interact with in our everyday lives, the same way numbers in the real number kingdom do. But they do describe a value, a particular piece of information. That’s what numbers do; they’re descriptors of value. Think about two: two is two ones, it’s four halves, it’s three less than five, it’s eight more than negative six, it’s the square root of four. These are all functions that describe the value of two and no other value. The problem with attempting to “divide by zero” is that the function doesn’t define any particular value. It’s actually called “undefined” for that very reason. If you were to attempt to use mathematics to approximate an answer to 1/0, you might find that your ‘answer’ could equal +/- infinity (which doesn’t really work in mathematics, infinity is a concept, not a value). And then you find that you get the same ‘answer’ when you try 2/0, or 3/0, or any other value.
Basically, dividing by zero doesn’t describe a value; it’s just kinda mathematical nonsense, but the square root of negative one _is_ an actual value, just not one that exists in the same dimension as values we’re traditionally familiar with.
Others have given you a lot of examples of why **m** cannot be a real number – if we assume that division is the inverse of multiplication, this results in an inconsistency.
We can append **m** to the real numbers, similar to how we appended **i**. The rules will be different, but this is what is done to form the [Riemann sphere](https://en.wikipedia.org/wiki/Riemann_sphere) or the less useful [real/complex wheel](https://en.wikipedia.org/wiki/Wheel_theory).
We can also give up on division being the inverse of multiplication. This was done by a programming language called Pony, and [this post by Hillel Wayne](https://www.hillelwayne.com/post/divide-by-zero/) is a soft defense of allowing 1/0 = 0. It’s no longer generally true that x * (1 / x) = 1, but only true if x != 0. However, that might just be ok sometimes.
There’s a lot of helpful stuff here about why 1/0 causes problems in “regular” math — but you’re exactly right that we can define it anyway! That’s what happened with negative numbers (from positives), rational numbers (from integers), irrational numbers (from rationals), and complex numbers (from reals) — none of them fit with all the axioms of previous, simpler systems of numbers, but mathematicians found them all useful enough that they (eventually) figured out new axioms that could correctly describe the behavior of these new numbers. Take a look here to see how adding 1/0 to the club changes things: [https://en.wikipedia.org/wiki/Projectively_extended_real_line#Dividing_by_zero](https://en.wikipedia.org/wiki/Projectively_extended_real_line#Dividing_by_zero)
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