correct, a right triangle cannot have a hypotenuse of length zero, the shape implied by the variables in OPs picture is a line, and the length "i" is also 1.
For the side to have length i it must have length 1 and be rotated 90°, so it is on the same line as the side of length 1 therefore the hypothenuse is 0.
In order for a metric to be well-defined, you need to have that d(x,y) > 0 for any x=/=y. Is i > 0?
>In order for a metric to be well-defined, you need to have that d(x,y) > 0 for any x=/=y.
you missed the part where d: MxM->R being M the metric space and R the real numbers >Is i > 0
according to what order relation?
you guys have to study more before pretending you know math
https://en.wikipedia.org/wiki/Order_theory#Basic_definitions
https://en.wikipedia.org/wiki/Metric_space#Definition
Complex numbers form a Kolmogorov-Asimov-Rutherford-Eisenstein-Michelin quasisemipseudometricoidal space that lack all the properties you will rise to object.
Abstractly, yes, however, negative lengths are unfounded in math/physics. The negative sign is useful to indicate direction (opposite of that which is agreed to be the positive side, almost always left), but the length is modulus wherever, whenever.
That's using the complex numberline, anon, and even so the distances are modulus of whatever quantity is chosen. So far as I have come to understand it's interesting to use complex numbers because with them you can represent rotations algebraically, I'm sure it must be more profound than that, but I'd be lying If I said I knew.
If you assume i > 0 then either you get -1 = i^2 > 0 or you consider that a > 0 and b > 0 doesn't imply ab > 0. In both cases you lose some property that you'd like to have.
The image has pixels.
The length can be computed by counting the number of pixels.
Labelling two sides as length one is only an assigned value.
It doesn't represent reality.
When an object is made up of a easily countable number of real things, approximations are not neccessary.
Pythagorean's theorem holds for lengths. Lengths are always nonnegative real numbers, so there's no such thing as a side of a triangle having length $i$
This is actually correct. If you plot this in a 2d cartesian space then it makes no sense but if you plotted this out in a Riemann surface it would make perfect sense.
Imaginary numbers are not imaginary, they are very real. The need for the complex plane arises as basically a work around for fundamental flaws in standardized math (ie. taking the square root of a negative number). This is because negative number themselves dont exist (you cant have a shape with a "negative" side) but again were introduced (after some controversy) because they can be useful in some cases and it allows math to be explored further without the need to rethink the rest of mathematics. It was only a matter of time before this workaround would cause issues and imaginary numbers were created as another workaround.
The arithmetics is correct.
i^2 + 1^2 = 0^2
-1 + 1 = 0
0 = 0
But it is senseless to assign a nonpostive number to a distance. Geometrically this is nonsensical.
Don't mind me now, just trying to relearn how to use latex and edit my text (haven't been here in a LONG while).
Have never heard that, and still I'm more inclined to think that the negative sign does not represent a value less than zero but rather direction (like vectors) or something else, perhaps time dialation in this case, but again, I got no clue on that.
Does at any point in high level physics or math a negative distance (not indicating direction) is axiomatically accepted?
It's imaginary numbers, not negative numbers. Compute the distance between two points in spacetime that are actually separated by a time, and you'll get an imaginary distance. Similarly, compute the time between two points in spacetime that are actually separated by a distance, and you'll get an imaginary time.
2 years ago
Anonymous
which has nothing to do with assigning a nonpositive number to a positive distance
2 years ago
Anonymous
>It's imaginary numbers, not negative numbers
Lowkey they seem negative. I mean, it pretty much revolves around the fact that there's a number, convetionally called "i", that when squared equals negative one. And I gotta agree with the other anon, I still see no relation between this and assigning nonpositive numbers to positive distances.
lol I don't know if this is a joke or not but this is actually a better visual representation of what's actually going on. Better yet would be to make this a 3D cartesian space and have the side of length i on the Z-axis.
no it's the other way around
the green hypotenuse is the 45 degree angle, which is supposed to be precisely 0 no matter the quadrant, while the i length is the Y axis
Yes (i)^2 + (1)^2 equals (0)^2, but you can't have side with length zero or length i. It's inbreds like you who can't think beyond basic shapes who held back math for thousands of years
Not a valid metric of distance
>Not a valid metric of distance
correct, a right triangle cannot have a hypotenuse of length zero, the shape implied by the variables in OPs picture is a line, and the length "i" is also 1.
>the shape implied by the variables in OPs picture is a line
correction, a line segment.
For the side to have length i it must have length 1 and be rotated 90°, so it is on the same line as the side of length 1 therefore the hypothenuse is 0.
>length i
that doesn't even make sense do your argument is invalid
You are moronic if that doesnt make sense.
calling me moronic doesn't make your argument less wrong
you just exposed yourself as mentally moronic
>In order for a metric to be well-defined, you need to have that d(x,y) > 0 for any x=/=y.
you missed the part where d: MxM->R being M the metric space and R the real numbers
>Is i > 0
according to what order relation?
you guys have to study more before pretending you know math
https://en.wikipedia.org/wiki/Order_theory#Basic_definitions
https://en.wikipedia.org/wiki/Metric_space#Definition
Complex numbers form a Kolmogorov-Asimov-Rutherford-Eisenstein-Michelin quasisemipseudometricoidal space that lack all the properties you will rise to object.
but it is rotated 90 CCW with respect to X. so it is already where it needs to be. why do you want to rotate it back to 1? then it would be 1 not i.
No, because if the side were of length 1, then it would stay where it is, but the side is of length i, so it is rotated relative to where it is.
Abstractly, yes, however, negative lengths are unfounded in math/physics. The negative sign is useful to indicate direction (opposite of that which is agreed to be the positive side, almost always left), but the length is modulus wherever, whenever.
That's using the complex numberline, anon, and even so the distances are modulus of whatever quantity is chosen. So far as I have come to understand it's interesting to use complex numbers because with them you can represent rotations algebraically, I'm sure it must be more profound than that, but I'd be lying If I said I knew.
In order for a metric to be well-defined, you need to have that d(x,y) > 0 for any x=/=y. Is i > 0?
sqrt(n) is always positive. Thus sqrt(i^2) must be positive.
How do you define > on C rigorously?
If you assume i > 0 then either you get -1 = i^2 > 0 or you consider that a > 0 and b > 0 doesn't imply ab > 0. In both cases you lose some property that you'd like to have.
>How do you define > on C rigorously?
I don't. I'm too busy fricking your dad.
Are you assuming my algebraic properties?
>How do you define > on C rigorously?
Define a bijection g from C to R. Then for x, y in C define x>y if g(x)>f(x)
plus
In order for a function to be a metric*
Who said that? You can define a metric to be whatever you want. You could have a complex metric
You multiply i by its negative, therefore its 2
Refute this
Imagine believing in irrational numbers
> irrational numbers
Painfully moronic
The word "irrational" is LITERALLY in "irrational numbers"! Mathematicians have taken us for absolute fools...
The image has pixels.
The length can be computed by counting the number of pixels.
Labelling two sides as length one is only an assigned value.
It doesn't represent reality.
When an object is made up of a easily countable number of real things, approximations are not neccessary.
Pythagorean's theorem holds for lengths. Lengths are always nonnegative real numbers, so there's no such thing as a side of a triangle having length $i$
I see no problem here.
You're treating an imaginary number as a real number. Guess what, you can't.
This is actually correct. If you plot this in a 2d cartesian space then it makes no sense but if you plotted this out in a Riemann surface it would make perfect sense.
Imaginary numbers are not imaginary, they are very real. The need for the complex plane arises as basically a work around for fundamental flaws in standardized math (ie. taking the square root of a negative number). This is because negative number themselves dont exist (you cant have a shape with a "negative" side) but again were introduced (after some controversy) because they can be useful in some cases and it allows math to be explored further without the need to rethink the rest of mathematics. It was only a matter of time before this workaround would cause issues and imaginary numbers were created as another workaround.
I refute your image
Define the objects that you are measuring, explicictly define the distance and prove that it satisfies the definition of distance
>uses imaginary numbers in real geometery >gets surprised when shit breaks
The arithmetics is correct.
i^2 + 1^2 = 0^2
-1 + 1 = 0
0 = 0
But it is senseless to assign a nonpostive number to a distance. Geometrically this is nonsensical.
Don't mind me now, just trying to relearn how to use latex and edit my text (haven't been here in a LONG while).
[math]sqrt{-1}^{2} + 1^{2} = 0^{2}[/math]
>But it is senseless to assign a nonpostive number to a distance. Geometrically this is nonsensical.
It makes sense in the theory of relativity.
where in "relativity" is a nonpositive number assigned to a positive distance
Have never heard that, and still I'm more inclined to think that the negative sign does not represent a value less than zero but rather direction (like vectors) or something else, perhaps time dialation in this case, but again, I got no clue on that.
Does at any point in high level physics or math a negative distance (not indicating direction) is axiomatically accepted?
It's imaginary numbers, not negative numbers. Compute the distance between two points in spacetime that are actually separated by a time, and you'll get an imaginary distance. Similarly, compute the time between two points in spacetime that are actually separated by a distance, and you'll get an imaginary time.
which has nothing to do with assigning a nonpositive number to a positive distance
>It's imaginary numbers, not negative numbers
Lowkey they seem negative. I mean, it pretty much revolves around the fact that there's a number, convetionally called "i", that when squared equals negative one. And I gotta agree with the other anon, I still see no relation between this and assigning nonpositive numbers to positive distances.
I refute it with a ruler.
Fixed it.
lol I don't know if this is a joke or not but this is actually a better visual representation of what's actually going on. Better yet would be to make this a 3D cartesian space and have the side of length i on the Z-axis.
homosexual beat me to it
You are on your way to discovering hyperbolas, OP.
Unit hyperbola:
[math](x)^2+(iy)^2 = 1[/math]
You are on your way to discover hyperbolas, I am on my way to discover hyperborea. We are not the same.
alternative angle
no it's the other way around
the green hypotenuse is the 45 degree angle, which is supposed to be precisely 0 no matter the quadrant, while the i length is the Y axis
Yes (i)^2 + (1)^2 equals (0)^2, but you can't have side with length zero or length i. It's inbreds like you who can't think beyond basic shapes who held back math for thousands of years
Geometric Algebra expresses blades as essentially hypercomplex numbers