0

I'm having trouble formulating my question.

I understand how complex numbers work on a basic level (I had to work with Quaternions for 3d rotations, etc), but I'm curious about why i2=-1 in particular allows for a whole number system to be based around it, while other types of "rule breaks" don't seem to do so. Something like i/0=1, or i0=-1, i0=1, and so on. I understand why the root of -1 is important, I'm just wondering why other "imaginary numbers" like that don't seem to be around.

I'm sure I sound naive saying this, but I would imagine there are endless combinations on how to create "imaginary" number systems like that. How do mathematicians know they don't have an importance of their own, or do other systems like that exist, that are based on numbers that shouldn't "exist"?

In essence, I'm not confused about imaginary numbers - I'm confused why there aren't many more imaginary number systems.

(I'm using the word "imaginary" as placeholder for numbers that break rules / shouldn't exist on paper, not necessarily to refer to imaginary/complex numbers as they exist now.)

Katai
  • 353
  • 1
    The definition of $i^2 =-1 $ was not arbitrary, it is the solution to the equation $x^2+1=0$, which you need in your field in order to reach a field such that any polynomial has a root. – FreeZe Feb 28 '22 at 12:27
  • $(1)$ Division by $0$ is also impossible within the complex numbers $(2)$ Complex numbers have no order. This can be easily proven by considering the cases $i>0$ and $i<0$. In both cases we get $i^2>0$ contradicting $i^2=-1$ $(3)$ "imaginary" (or better "non-real") just means that the number is not on the real line. It is however in the plane of the complex numbers. – Peter Feb 28 '22 at 12:28
  • You can extend the complex numbers to the quaternions and you can continue. But in every step you lose an important property. Quaternions for example do not satisfy the commutativity of multiplication. – Peter Feb 28 '22 at 12:30
  • You don't need HTML; you need MathJax. $i^2=-1$ is simply $i^2=-1$. – Randall Feb 28 '22 at 14:32

1 Answers1

1

To be fair, there are others, but they don't break the rules in the same way. Hyperreal numbers allow for infinitesimals, and to some extent match some of your recommendations. For instance, $\epsilon\cdot\omega = 1$, where epsilon is really close to zero and omega is an infinity.

Another entire set of systems, which is similar to imaginary numbers, are those made by domain extensions. $i$ allowed the extension of the square root function into negative numbers. There are similar extensions of other functions. The complex log function does a similar thing with the logarithm. The $\Gamma$ function extends the factorial to the reals. We now know how to extend exponentiation to the complex numbers.

In a similar semi-rule-breaking fashion, hyperreal numbers can be used to extend valid summations to include divergent sums.

The problem, in general, is finding a system in which the meanings of basically "everything" translate over, or there is a minimal of breakage to existing mathematics. It's probably possible to devise a system where multiplying by zero gives a valid result, or raising to zero power gives a different number than 1, but it is probably going to break a lot of things.

johnnyb
  • 3,509