orders of negative

[what]

so i was thinking earlier today about (amongst many other things) negative numbers. they're quite odd. not only are they theoretical numbers that do not exist outside abstract concepts such as finance or physics, but they also have no square root. for example, -1 * -1 =1.

in a weird train of thought that i cannot remember any more, i came to the conclusion that the square root of a negative number must belong to a different set of negatives, one more negative than the last. a different order of negative, if you like. in this way, the square root of -1 would be equal to the 2nd order negative 1. the product of two second order negatives would equal to a first order negative, the product of two third order negatives would equal to a second order negative and so on.

these numbers would be more abstract (harder to find real-life examples) than our familiar first order negatives, which makes sense because they are more negative. this would make a lot of equasions much easier because if a square root of a negative number shows up, we can get a precise value rather than being forced to leave the square root in the final answer because there is no such thing.

is this as stupid as it sounds?

 

[The Grey Man]

Just write i. Everyone else does.

Imaginary number - Wikipedia

en.wikipedia.org

There's no reason why we couldn't define a term as that which is the square root of -i, so that it is a "doubly imaginary" unit, though I can't think of a reason why one would want to.

 

[mister m 1 data]

\( \sqrt{-i} = \frac{-1 + i}{\sqrt{2}} \)

 

[Ex-User (14663)]

yes the square root of a negative does belong to a different set of numbers, except not "more" negative, but in a different dimension, rather. In particular the imaginary axis in the complex plane.

 

[patrick]

ive just finished a course in complex calculus and heres my take on the subject.
first the negative sign - implies the existence of the inverse of the sum for the object.
in this case the object is the number 1 from the set of the reals.

second the square root is not an operation in the same sense as sum or multiplication (though im pretty sure you can redefine things in a way that it could be seen as an operator), but rather a way of writing a mathematical problem, which is, which number product itself gives me -1.

now as we already know this number does not exists within the set of the reals, and what mathematicians do when the solution to a problem cannot be expressed with objects and operations previously defined on the set, is to simply put a new symbol and define a new set containing the previous set and the new object/solution to the problem. and then checking if the previous set operations hold for the new set.

so its not that it has more orders of negative than other things, its just that the problem sqrt(-1)=i implies that there exists objects outside the set with its operations from which the problem was modeled.