How can we use imaginary numbers when you can't take the square root of -1

How can we use imaginary numbers when you can't take the square root of -1

The whole idea of imaginary numbers is that you come up with an arbitrary number i that is the square root of -1. i is not a real number, it's just the square root of -1. Most of the "uses" or applications of imaginary or complex numbers involve finding situations where i is squared, and you end up with -1 again. Fundamentally though "you can't take the square root of -1" is only true for real numbers; with complex numbers you can take the square root of -1, and its value is i.

>The whole idea of imaginary numbers is that you come up with an arbitrary number i that is the square root of -1.

No it's not. i^2 = -1 but sqrt(-1) is undefined.

imaginary numbers is proof that contemporary people don't care about their educations. They're worthless garbage that just dumb the lesson down, and in the process, the person's understanding of the lesson.

They help with understanding dimensions and things like fibration

Well, you're right that I was wrong about it, but sqrt(-1) is either i or -i for complex numbers, not undefined.

I believe that the real numbers themselves are far more controversial then constructing the imaginary numbers from the rationals.

In later case you just define a few things in the other you need a shit ton of construction.

ja visst

>mathematicians tell your stupid young ass that you can't take the square root of -1
>then, hoping you've become smarter and more open minded tell you there is actually a way to do that
>b-but you can't do that, it is known, complex numbers are imaginary, we can't use them

You can't. The idea of "complex numbers" is entirely arbitrary and non rigorous by definition.

Wrong. The square root by definition is always positive. That is sqrt (x^2) =|x|
sqrt(-1) is defined as i. I don't see the problem.

If we can just say that root -1 is i why can't we do the same with something like ln(-1)?

It's all about breaking the 3rd dimension.
We are currently limited to the 2d plane of numbers when squaring or rooting.
Using i opens up another dimension.

You can, but there's no need for it.
i was defined in order to ensure that every nth degree polynomial has n roots.

|i| =/= i

If you're really uncomfortable with the idea of i, just work with ordered pairs of real numbers. Define "addition" as (a,b)+(c,d) = (a+c, b+d) and "multiplication" as (ac - bd, ad + bc).

That would be i*pi*(2k + 1/2) for any integer k.

i is x+ in the extension field R[x]/ of R.

1x10^-2
sqr root = ^.5 = -1^.5

You can, but you usually leave the left side of the real axis when you define ln, so that it can be continuous (and differentiable).
Instead of that you can define a new ln on the whole plane minus any simple curve starting from the origin and ending at infinity.

Anything you want to represent using complex numbers can be equivalently represented in simpler terms which do not rely on imaginary numbers or the complex number plane.

but then it won't do the cool Euler's formula thing in the complex plane, that's important to me

complex differentiation cannot be expressed in simpler terms with other methods
if you use multivariable differentiation to emulate complex differentiation, then you have to have to shoe horn the Cauchy-Riemann equations separately

or you can just look whether [math]d f \in \mathbb{R} \times SO(2)[/math]

Help me, these faggots confused me and i'm brainlet. Can anyone clarify their points?

>abstract concepts are for fags

what does it mean for a complex number to be "positive" ?

Imaginary numbers are one of the worst naming decisions in the field. Call the complex numbers like those of us who actually need to consider them for real world applications.

What I mean is that sqrt(-1) is always i. That's by definition.
You're thinking of
x^2 = -1
Which has solutions i and -i

The coefficient being positive in this case. So |b|i

the 'defnintions' are a shortcut that fail to help the person in the long run. They can't extrapolate any of the lessons because they demanded to be taught the most short-sighted formulatoin possible, one that does not allow them to actually understand the lesson, just find the answers.
they do, but the person should just learn the hard way so they can better use their skill
>huuuuuuuuuuuuuuu

>They can't extrapolate any of the lessons because they demanded to be taught the most short-sighted formulatoin possible, one that does not allow them to actually understand the lesson, just find the answers.
I don't think anyone can really appreciate complex numbers and how they work until they deal with electromagnetics. Although stuff like taylor series and how a calculator computes trig stuff should give enough appreciation on their own.

Because the square root of negative one exists, imaginary and complex numbers must exist.

pretty much how i think about it. call me a brainlet, but imaginary numbers are strange imo

>you can't take the square root of -1

I really wish we could just permaban anyone who starts a thread with a frog picture. It's always a shitpost.

literally this.

This. Euler's Identity is the most beautiful thing that exists anywhere forever.

Imaginary numbers can be used to find every possible Pythagorean triple

indeed, many tattooed math majors have informed me of this.

Remember when teachers in elementary school told you you can't subtract bigger number from a smaller, and later you were introduced to negative numbers? Same with complex numbers, first teachers tell you you can't take a square root of a negative number, but later you're introduced to complex numbers

Not true, if √-1 weren't defined then i wouldn't be either

Is 1-i positive?

retard

Same goes for so called real numbers

You still can't subtract a bigger number from a smaller one, unless you create an arbitrary context of reference that allows it. In fact you always have a subtraction limit, you write about subtracting a bigger number from a smaller but you are not doing it, you are either subtracting a smaller from a bigger new pair of numbers or subtracting and adding what is left. No relation with the root of -1 except the fact that a teacher told you couldn't and another told you could do it. Ridiculous.

Are you retarded?

negative numbers aren't real like how can you have negative 2 apples lmao, negative apples don't exist and neither do negative numbers

Yes
Square root my ass = -1
Because my ass is so amazing.