/sci/ - Science & Math

>>11549700
What the fuck would a side length of i even be. Sure i^2 + 1^2 = 0^2, but you can't have a side length of i because it wouldn't even make any sense.

I can think of two ways to answer this:

1. The Pythagorean theorem uses lengths of the sides, not the values of the sides. This distinction doesn't matter in the real numbers (except, one could argue, for negative numbers)

The lengths of both sides are 1, therefore the length of the hypotenuse is $\sqrt{2}$.

2. The length of a complex number is defined as an operation called the modulus. Say we have a complex number x. The modulus of x is
$$
|x| = \sqrt{x x*},
$$
where $x*$ is the complex conjugate, which is calculated by reversing the signs of all imaginary numbers in x.

The number we have is the sum of 1 and i, or 1+i. Using the modulus function, we get
$$
|1+i| = \sqrt{(1+i)(1-i)} = \sqrt{2}.
$$

>>11549700
>distance measured in complex numbers

>>11549753
Apparently latex doesn't work the way I thought. Let's try that again:

I can think of two ways to answer this:

1. The Pythagorean theorem uses lengths of the sides, not the values of the sides. This distinction doesn't matter in the real numbers (except, one could argue, for negative numbers)

The lengths of both sides are 1, therefore the length of the hypotenuse is [math]\sqrt{2}[math].

2. The length of a complex number is defined as an operation called the modulus. Say we have a complex number x. The modulus of x is
[eqn] |x| = \sqrt{x x*}, [eqn]
where [math]x*[math] is the complex conjugate, which is calculated by reversing the signs of all imaginary numbers in x.

The number we have is the sum of 1 and i, or 1+i. Using the modulus function, we get
[eqn]|1+i| = \sqrt{(1+i)(1-i)} = \sqrt{2}.[eqn]

>>11549700
The numbers don’t satisfied the triangle inequality.
*box* qed

>>11549766
Frick. Again!

I can think of two ways to answer this:

1. The Pythagorean theorem uses lengths of the sides, not the values of the sides. This distinction doesn't matter in the real numbers (except, one could argue, for negative numbers)

The lengths of both sides are 1, therefore the length of the hypotenuse is [math]\sqrt{2}[\math].

2. The length of a complex number is defined as an operation called the modulus. Say we have a complex number x. The modulus of x is
[eqn] |x| = \sqrt{x x*}, [\eqn]
where [math]x*[\math] is the complex conjugate, which is calculated by reversing the signs of all imaginary numbers in x.

The number we have is the sum of 1 and i, or 1+i. Using the modulus function, we get
[eqn]|1+i| = \sqrt{(1+i)(1-i)} = \sqrt{2}.[\eqn]

>>11549700
imaginary numbers just correspond to y-coordinates in the cartesian plane. the "i" just signifies that we are referring to a y-coordinate, and doesn't represent a distance.

"i" i.e. "0+1i" on the plane is just another way of writing (0,1).

>>11549771
Click on the TEX button on the top left newfriend.

End me.

>>11549753
I can think of two ways to answer this:

1. The Pythagorean theorem uses lengths of the sides, not the values of the sides. This distinction doesn't matter in the real numbers (except, one could argue, for negative numbers)

The lengths of both sides are 1, therefore the length of the hypotenuse is [math]\sqrt{2}[/math].

2. The length of a complex number is defined as an operation called the modulus. Say we have a complex number x. The modulus of x is
[eqn] |x| = \sqrt{x x*}, [/eqn]
where [math]x*[/math] is the complex conjugate, which is calculated by reversing the signs of all imaginary numbers in x.

The number we have is the sum of 1 and i, or 1+i. Using the modulus function, we get
[eqn]|1+i| = \sqrt{(1+i)(1-i)} = \sqrt{2}.[/eqn]

>>11549782
I did it!
[math]\mathbb{EQXQUISITE}.[/math]

>>11549723
>>11549753
>>11549762
>>11549767
>>11549773
>>11549784
Seething mathlets.

>>11549858
Getting (you)s from retards is my calling in life

>>11549700
It's not a [math] \mathbb{R} [/math]eal triangle.

Your request is based upon an invalid premise, faggot.

>>11549753
>except, one could argue, for negative numbers
That would just mean it goes in the opposite direction, and it still works, so negatives are not an exception, do you even think before you post my friend?
>>11549762
Problem?

@11549879
Oh rly?

>>11549888
Ur invalid

>>11549773
nope
>>11549782
>having javascript enabled
top cuck

>>11549896
My trips say otherwise, faggot.

>>11549890
Doesn't obey any metric properties, the complex numbers are not orderable.

>>11549700
Dub zeros notwithstanding, what does a side length of 0 even mean? Or is that not the point?

>>11549919
Problem?
>>11549924
it means 1 plus i squared which is not actually 0 if we're being literal and looking at the complex plane!!!

>>11549700
abs(i)=1
the hypotenuse is sqrt(2)

>>11549936
No problem, math is about exploration. Go ahead and develop a geometry with complex valued distances.

>>11549700
If this were a triangle it would suffice the triangle inequality. Does it?

>>11549952
thanks :)

>>11549966
In your geometry what is length of a point?

>>11550082
muh dick

>>11549897
>nope
Massive brainlet detected. Complex numbers are literally just cartesian coordinates but with an extra rule that allows for useful vector operations.

>>11549858
indeed a constant problem

>>11549700
Pythagorean theorem is defined for positive reals. If you want to extend it to the complex plane, you need to talk about inner product spaces - this leads you to the complex conjugate

>>11549700
anon if you want to complex trig that's awesome just please don't be a retard about your high school is showing

>>11550166
the complex numbers and the real plane are isomorphic in the category of sets and as vector spaces, but their algebraic structure as rings and fields ([math]\mathbb[R}^2[/math] isn't even a field) is completely different

>>11550082
Planck or 0 depending if we're being quantum or mathematically theoretical

For every triangle with sides a, b, c:
a + b > c,
b + c > a,
c + a > b;

For your triangle (i, 0, 1):
i + 0 > 1
i > 1
0 + 1i > 1 + 0i
Comparing complex numbers is not defined, therefore ur triangle is not defined.

>>11549700
This is obviously bait, no one can be this retarded.

>>11549700
Triangle inequality.
0+i < 1 => i < 1
0+1 < i => 1 < i

Boom roasted

>>11549700
z*z

>>11549700
write that in polar coordinates and you'll find out just how wrong you are.

>>11549700
and what makes you think that i and 1 are orthogonal ?

Squaring complex numbers is simply multiplying angles and magnitudes, not giving a Euclidian distance.

This is because the complex VECTOR SPACE is one dimensional. We visually represent it in two dimensional space.

There are no 90 degree Euclidian angles in a one dimensional. Also, pythagoras was not around at the advent of complex numbers nor vectors space.

>>11551860
so i = 1?

>>11549700

>>11549700
sqrt(-1) isn't a "side length"

>>11552048
>This is because the complex VECTOR SPACE is one dimensional
This is wrong. The complex numbers are isomorphic to real 2x2 matrices, which assuming the standard basis, has dimension 2.

OK desu erm OwO bawsically [math] i = 1i + 0 [/math], and ummm desu, the awwctual lengthie is [math] \sqrt{1^2 + 0^2} = \sqrt{1} [/math] and ummm :3 dwen we habe [math] hyp = \sqrt{\sqrt{1}^2 + 1^2} = \sqrt{1+1} = \sqrt{2} [/math]
Ano ne senpai OwO desu

>>11552363
>using the complex conjugate definition
cringe

>>11552154
I don't get it. Does it loop back on itself?

>>11553382
The hypotenuse at any point after the split is always 0. They're the same track.

>>11549700
Well for a start you have to consider the fact that the angles of the legs have division by zero.
The top one is found by using [math]\cos\beta = \frac{i}{0}[/math] and the bottom one is [math]\cos\alpha = \frac{1}{0}[/math].

>>11553916
What a bullshit ass argument. Every normal triangle that "exists" runs into the same issue, the tangent of gamma being division by zero. Following that logic then no triangle should exist because tan(90) is sin(90)/cos(90) i.e 1/0.

I fucking hate your kind so goddamn much. You pull out the standard canned response just like the guy above spammed 4 posts of the same canned modulus reply yet pretend that the entire infrastructure that you've based these constructions on is flawed. The only missing post ITT is someone reflexively mentioning that complex numbers aren't ordered and we'd basically have an entire wikipedia article copy-pasted here with zero critical thought, as it usually happens in 99% of the threads here.