Why can’t you invent an imaginary number for division by zero like you can for a square root of a negative?
In: Mathematics
I think the truth is we got lucky with imaginary numbers. I don’t know the history, but it’s not really correct to say that i is the square root of minus 1, in the way that we normally understand square roots. We sort of stumbled onto a whole system of arithmetic, which contains the real numbers within it, and in which there’s a “number” that when “multiplied” by itself gives – 1. Which is neat. But it wasn’t invented exactly.
Someone might stumble on something analogous to that for 1/0, or might have already (possibly in multiple ways). But it wouldn’t really be division as we know it.
You can divide by zero if you are on the extended complex plane, which is the complex number (x i +y, so the combination of real and imaginary numbers ) and ∞( infinity). It is often represented as the [https://en.wikipedia.org/wiki/Riemann_sphere](https://en.wikipedia.org/wiki/Riemann_sphere)
There is only one ∞, not a +∞ or -∞ that you can get to in limes valuation for a real number.
A number on the complex plane is usually represented with z.
z/0 = ∞ and z/∞ =0 that is if z is not 0 or ∞
∞/0 = ∞ and 0/∞ =0
What is still undefined is ∞/∞ and 0/0
But if you use the extended complex plane you need to know what it lacks some stuff you are used to in for example real numbers.
One example is that numbers do not have a well-defined order. 4> 3 is well known but is i >1, i<1 or i=1? The answer is complex numbers do not have a single defined order, you can only compare the absolute value that is the distance from 0 and is represented by |z|
The result is that the distance to |i| and |1| both are 1. This is also why there is only one ∞ and not +∞ and -∞
This shows that if you add stuff like the ability to take the square root of -1 the resulting number will loo some other stuff like an absolute order.
This is true for complex numbers not just if you extend the plane and include ∞. But is a relatively simple example that if you gain stuff you loos stuff too. So you can divide by zero in some situations if you know what to do and what other consequences that have.
It has lots of practical applications and for example, https://en.wikipedia.org/wiki/Control_theory where https://en.wikipedia.org/wiki/Zeros_and_poles is a common tool to know how to control systems.
Mathematicians like when things work nicely and in almost every case, allowing division by 0 will make things not work nicely.
For example, one of the nice things about numbers is that we can multiply numbers in whatever order we want and still get the same result. For example (2 * 5) * 4 = 2 * (5 * 4) and 2 * 3 = 3 * 2. (Mathematicians call these “associativity” and “commutativity” of multiplication.)
Another nice property is that a(x + y) = ax + ay. For example, 2*3 + 2*4 is the same as 2(3 + 4). (Mathematicians call this “distributivity” of multiplication over addition.)
Let’s see what happens if we allow division by 0. Let’s just make a new thing called X and we will define that 0 * X = X * 0 = 1. So 1 / 0 = X.
Then (3 * 0) * X = 0 * X = 1. Except that 3 * (0 * X) = 3 * 1 = 3. But 3 is not equal to 1!
So this breaks one of the rules we like about multiplication. Maybe this is no biggie, maybe we can just let it slide. Let’s keep exploring.
Let’s check if distributivity still holds. How about X * (0 + 0)?
X * (0 + 0) = X * 0 = 1, but X * 0 + X * 0 = 1 + 1 = 2. Another problem… we lost distributivity.
If you keep exploring, you’ll likely find even more problems with assuming that X = 1/0 exists.
We now have two options here: First, we can allow division by 0, in which case we would have to abandon a bunch of things that work nicely with arithmetic. Or, second, we could just say that X does not exist. The second option is almost always the best option.
You can. People have; one example is the projectively extended real line, which is more or less the real numbers and a point at infinity. In this number system, 1/0 = ∞. The real question is “why don’t we use a number system where we can divide by 0 in everyday life?”
Part of it is cultural; “we just don’t.” But there are good reasons for it. Number systems that allow division by 0 inevitably lose some useful properties of the real numbers (side note: the real numbers aren’t any more or less “real” than any other kind of number; mathematicians just suck at naming things). For instance, ∞ introduces all sorts of weirdness. ∞-∞, in the projectively extended real line, is undefined, just like 1/0 is in the reals. So, now you can say 1/x=∞, but you can’t say x-x=0! Generally, trying to allow division by 0 is more trouble than it’s worth for everyday purposes.
You can. All the people saying it’s impossible are wrong. For example, you could add a value called ∞ to the real numbers, and say that x/0 = ∞ for any nonzero value of x, [turning the number line into more of a number circle](https://en.wikipedia.org/wiki/Projectively_extended_real_line).
However, this sort of thing is not as useful as the idea of imaginary/complex numbers, for two reasons. First, the resulting system isn’t that interesting: it’s basically just the real numbers with a single extra point, and that point has rather boring properties like ∞ + x = ∞ and ∞ – x = ∞ for any real number x. The complex numbers have far more interesting behavior. Second, and more importantly, the complex numbers provide additional insight into the real numbers, especially in calculus and related fields. The mathematician Jacques Hadamard once said that “the shortest and best way between two truths of the real domain often passes through the imaginary one”. Adding the point ∞ to the real numbers, on the other hand, doesn’t tell us much at all.
There are two ways to tackle this question: symbolically and physically. Let’s go with the physical first.
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Consider what division represents, using the expression `6 / 2 = 3` as a guide.
>You have six apples. Dividing them into two groups leaves you with three apples per group.
Similarly, you can extend this to division by fractions: `6 / .5 = 12`.
>You have six apples. Dividing them into half a group means that one whole group would have twelve apples.
So far, so good, right?
But: `6 / 0 = ?`.
Let’s divide six apples into zero groups. How many apples per group? …Well, there are zero groups, so…you can’t answer the question.
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Now, symbolically. Let’s do what you suggest, and invent a new number to represent the multiplicative inverse of zero — the number such that z = 0^-1 .
This means that 0z = 1.
But we know that 0z = 0.
By defining a number to be the multiplicative inverse of 0, we end up attempting to assert that 1 = 0, which we know to be false. Therefore, there can be no number for division by zero.
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