How come there isn’t an imaginary system for dividing a number by 0, while there is one for taking the square root of a negative number?



How come there isn’t an imaginary system for dividing a number by 0, while there is one for taking the square root of a negative number?

In: Mathematics

First of all, you can’t take the square root of a negative number. If you put that in any calculator you’ll get something like undefined or domain error because it’s more or less impossible. You can however multiply the square root of a positive number by a negative number to make the root negative. Like if you do the square root of -4 you’ll get an error but if you do -1 x square root of 4 you’ll get -2. Now onto your main question. Dividing by 0 doesn’t really boil down to the mathematics of it all it boils down to logic. For an equation 10 divided by 2, say you have 10 apples and you need to divide them into 2 crates evenly, you would get 5 in each crate. Pretty simple. Well what happens if you have no crates. You have nowhere to put the apples so there’s no way to define what happens to them. If mathematicians wanted to they could just make the answer be 0 or give the number the properties of 1(like the imaginary system you’re talking about), but then down the line you’ll get other errors that won’t make sense. Does that help?

Taking the square root of an imaginary number works within regular math. The imaginary number i is just equal to the sqrt(-1), which algebraically is a-ok so long as it doesn’t need to be evaluated. You can do useful math using i and it’s not a whole new system.

Dividing by zero just makes no sense. How many groups of 0 can you make out of x number of items? (A way of thinking about division). The answer is infinite- or maybe it’s 0? Maybe there’s 1 group, but no matter what it’s up for debate (to some extent) and dividing by zero, algebraically, is not useful. Also, if you could divide by zero, you can prove that any number x is equal to any number y, and x and y could be -39104 and 2 respectively

tl;dr: imaginary numbers follow the rules of algebra. Dividing by zero is against the rules because the result just can’t be determined, and it wouldn’t be a useful system

So, it boils down to one thing:

Allowing division by zero completely and utterly breaks mathematics.

Literally 2=4 now and 21334124234 = 9.45412231. Every number is equal to every other number and you just can’t get anything useful done.

But imaginary numbers (which by the way is a terrible name it’s not more “imaginary” than any other number) don’t do that at all. They are formalized and it doesn’t break math at all.

There can be – the extended real lines add an infinity, and extend division by zero to equal infinity. This still leaves 0/0 undefined, but even that can be defined in a consistent way if we want.

The issue is simply how useful it is. In the complex numbers it is no longer true that sqrt(ab) = sqrt(a) * sqrt(b), which is an annoying thing to lose (along with some other exponent identities), but not that big of a deal. What we gain is enormously useful, since all sorts of cyclic phenomena are well modeled by complex numbers.

By extending into the lines with infinities, we lose core division identities – it’s no longer true that a * (1/a) = 1, because allowing 0 * (1/0) = 1 creates contradictions. The division operator makes more intuitive sense (since 1/0 certainly feels like Infinity), but it means division is not always the opposite of multiplication, so actually doing algebra becomes more difficult. This is still sometimes useful, but mostly only in analysis (higher level calculus) and a few very abstract physical models.

i is defined. It is the square root of -1. It is always that value. Because it has a defined value you can assign a value to it and do stuff with it.

Dividing by zero is undefined. The closest thing to what you’re talking about is Limits, where you investigate what happens as parts of a function tend towards either zero or infinity.

Eg: what does a (x^2 -4) / (2x-4) equal when x = 2? At exactly x = 2 the function is undefined, but you can take limits and see if it it tends towards a value, tends towards zero or tends towards infinity.

One of the critical issues of assigning a value to zero is that there are different natures of zeros. X^2 at x=0 is a different zero than x^3 at x=0. They both have the value of zero, but if you take x^2 / x^3 and compare it to x^3 / x^2 you are going to see very different behavior as you approach zero, even though the top and bottom of the equation are both zero – in one you can cancel to 1/0 that tends to infinity, in the other you can cancel to 0/1 that tends to zero.

Using i to represent the root of -1 has lead us to a lot of cool stuff. We have used it to find new formulas and we apply them to things that aren’t imaginary.

However in the case of dividing by zero, we haven’t found a way to make it consistent or useful.

Why are imaginary roots of negative numbers consistent while imaginary divisions by 0 are not? Let’s take a closer look:

If we have sqrt(-9) = 3i

Then 3i * 3i must be -9.

Multiplication is associative, meaning it works any order you it in.

3 * 3 = 9. Then 9 * i = 9i. Finally, 9i * i= -9

Or 3i * i = -3. Then -3 * 3 = -9

We can get the same answer in two ways because of how multiplication works.

Now let’s try that trick with dividing by 0.

Let’s say that 1 / 0 = q

And q * 0 gets us back to 1.

Then, 5 / 0 = 5q

So 5q * 0 = 5

If we do the q*0 part first it works fine and we get 5, but if we use the associative property of multiplication, things go wrong.

5 * 0 = 0. Then 0*q = 1.

We got 1 when we should have gotten 5. So the system is already broken.

Projective Geometry uses Homogeneous Cartesian Coordinates (HCC), which can be mapped with conventional coordinates by an additional “scaling” number at the end. So (3,5) would be (3,5,1) or (6,10,2) or (4.5, 7.5, 1.5), for example. But if you have (3,5,0) it refers to the point at infinity along the line from (0,0,1) and (3,5,1). The General Projective Transformation (GPT) is used to handle these coordinates.

If you “draw” a line between any two points at infinity, they define the line at infinity, which goes through all the points at infinity. BTW, (3,5,0) refers to the single point at infinity in both “directions.” Using an appropriate GPT, you can transform the point at infinity to become local, but another point, previously local, becomes inaccessible, i.e., at infinity.

Because of the convenience of using HCC via the GPT, this has taken over computer graphics internally. I learned this at MIT in 1960 when our math was mostly done by hand.

If you have 3D coordinates, (3,4,5) becomes (3,4,5,1) and so on.

There actually are systems for this, sort of. They have names like hyperreal numbers and surreal numbers. I don’t know much about them but they kinda sorta allow division by zero. (It’s actually division by a number than is less than any real number, so zero in a way but not zero in a way).

The thing with these number systems is that they turn out not to he that much use beyond pure mathematics, and they are quite difficult. Complex numbers, on the other hand, are very useful in lots of physics and engineering, and are relatively easy to understand.

Division is just a fancy way to subtract. If you take 100 and divide by 2 what you are doing is seeing how many times you can subract 2 from 100 until you reach 0. So the answer is 50. But if you take 100 and divide by 0 it just doesn’t work. No matter how many times you subtract 0 from 100 you’ll never reach 0.

Extending numbers into the complex world is useful – it allows us to solve cubic equations. It appears to be as natural and useful as extending numbers into the negatives or irrationals.

Extending numbers into worlds where division by zero is well-defined ends up leading to contradictions, and, while interesting, is less than useful.