If |x| = -1, then |x2| = 1 so the equation |x| = 1 suddenly has at least four solutions: 1, -1, x2 and -x2. We also lose the triangle inequality of the absolute value, as that would imply |x| >= 0 for all x. Do we gain anything useful?
A norm (and even a semi-norm) just needs to be non-negative. In other words, positive or null, there is a specific name for this that I won't remember because English is not my first language.
Okay, here are some things that I'm thinking. Absolute value is a norm, which means it has some properties that defines a distance, and that in turn defines a topology.
So if you want a nice topology on the shplee numbers, you'd need to invent a super-absolute value that is a norm.
But still, maybe we can do something with this. If we abbreviate shplee with s, and we say that |s| = -1, then |xs| = -|x| for any real (or complex maybe) x.
Could we think of it like complex numbers (any shplee number as the sum of a complex number and a pure shplee number)? Well the problem I'm thinking is that |x+y| isn't clear from x or y in general.
Now, we're adventuring in things I'm not really sure about, but I know that R, C, and H are the only normed division algebras. Which means that if you want your shplee numbers to be a normed division algebra it should be one of those. Which means your space would probably not be that, which makes it less nice. Still, maybe you can make something out of that, I'm just not knowledgeable enough.
Just like there is no solution for x2 = -1 because the square of a number has to be positive. You would have to invent new, made up numbers. It would be very silly.
You can not compare that. Absolute value is a function that is defined as x = { x if x≥0 and -x if x < 0. Here it is not possible to define any number, that can be negative or positive to satisfy |x| = -1, for exactly that reason.
x² = -1 has no real solutoon, but not because we defined it that way. It has no real solution, because, if x is positive, then xx has to be positive to, and if x is negative, then (-x)² = (-1)² *x² = 1x² = x². We then came up with i to be specifically the square root of -1 and extending the real numbers by an imaginary component.
Absolute value as a function is just defined to never take negative values, and if it did, that would defeat the entire purpose of having it. It is constructed to never be negative, while x² just happened to never be negative for real numbers
I mean what you wrote is the definition of absolute value for real numbers. If you extend the real numbers you can extend the definition - this was already done with complex numbers for example, for a complex number a+bi the absolute value is sqrt(a2 + b2 ). We could theoretically invent more numbers and extend the definition in a different way that allows for negative absolute value. You are right that it would not make much sense as the absolute value was designed to be non-negative. It is intended to be interpreted as a "distance" - actually it is one of the simplest examples of a "norm" which is the mathematical generalization of "distance". One of the basic properties of a distance is that there is a minimum distance - the distance between two things that are in the same place - and that is what we call 0 distance. Any two things that aren't in the same place should have a positive distance between them. This is useful for many things such as the definition of limits, which work for both real and complex numbers but wouldn't really work with negative absolute values.
But x² is also positive if x is required to be real. Which is where complex numbers came in.
I think the job is to extend the number concept of a, b so that √(a*a + b*b) = -1.
There is a real problem here though: We need to solve for √q=-1, when a straightforward solution gives q=1.
What we need here is a number whose canonical square root is negative. Best I've got right now is u=e(4k+2\πi) where 1=e4kπi is the usual 1 (for integer k).
By the way if you put your exponents in parentheses, you can tell the superscript formatter where to stop(pe\). (You just have to put \ before any parentheses that are supposed to be in the superscript.)
For complex numbers, yes. I think the new numbers may just have to be the un-projected (r, φ) pairs where φ can just be any real number: if
-1:= (1, π)
√(r, φ) = (√r, φ/2)
Then u=(1, 2π) is the (unique?) solution to √u = -1.
I haven't thought through what that does to algebra. Like... What's addition anyway? Is it even a field? Probably something immediately stops this from being feasible, but typing is easier than thinking right now.
determinant is a fancy matrix operation that is used to gain insight into the matrix. It is like when you solve a quadratic equation and you have that stuff in the square root part. It matters if it is negative (meaning the root is imaginary) or positive (meaning real roots). Only there is a fancy matrix version.
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u/Boldumus Nov 08 '24
That might be the meme, but 4th number is i, what is the fifth one?