Why are imaginary numbers taught in high school and basic astronomical navigation not...

Why are imaginary numbers taught in high school and basic astronomical navigation not? Imaginary numbers would only be more helpful than something like astronomical navigation if the individual were to pursue a career in mathematics or physics, outside of that its virtually useless.

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en.wikipedia.org/wiki/Fourier_optics#4F_Correlator
en.wikipedia.org/wiki/Quantum_computing
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Whenever I steal money from the register at my job as a Walmart cashier I use imaginary numbers to cover the discrepancy.

Works every time.

>imaginary
>not complex

how?

Serious question, what are you on about?

complex numbers are imaginary + reals

>Walmart.
huh, that's what they do at Wallstreet too

The best way to rob a bank is to own one.
-William K. Black

imaginary numbers are my waifu

Banks are hard businesses to start though, so you can't.

seriously how tho.

Assuming schooling is for excellence criminal activity.

Because imaginary numbers are useful across a multitude of disciplines while astronomical navigation is a pretty narrow topic

elaborate...

Knowledge of complex numbers is pretty much required for most STEM degrees, wheras I can't think of any discipline that uses astronavigation outside of sea based jobs. Sounds like something that would be taught at naval academies

elaborate. specifically how. and that's assuming that all high school student will be STEM majors (And for those majoring in a field that requires it they'll have to relearn it in their course.). But no matter what you decide to go into, astronomical navigation will always be actively useful.

Because quadratic equations are taught. What do you do for x^2 + 1 = 0? Imaginary numbers come up relatively organically. Because astronomy isn't taught in school, astronomical navigation wouldn't (not to mention that it is also useless).

why are you required to learn quadratic equations is you're passionate about foreign affairs?

Help a layman here. What the fuck is the use of an imaginay number?

Practically, we use complex exponentials in physics and engineering to represent waves all the time. Would you like to know more?

also, imaginary numbers make trig 100x easier to remember.

I know some scraps of calculus and physics from high school (legit enjoyed doing calculus), but I abandoned that when I finished high school - not sure how much I wanna know, just curious.

Tbh imaginary numbers sound like, completely useless and non-sensical to me, as if mathematics was inherently flawed (which I actually think is the case).

Also your explanation made no fucking sense to me, sorry user.

Not the guy who asked, but is this true? Like, identities and derivs and such? Cuz I'd straight up suck your cock for an easy way to remember those.

not only is it easier, for each complex exponential identity, you get two trig identities.

For instance, take summing sines:

[math]sin(\theta + \phi)=sin(\theta)cos(\phi) + cos(\theta)sin(\phi)[/math]

That's a bitch to remember.

Take multiplying complex exponents:

[math]e^{i\theta}e^{i\phi}=e^{i(\theta+\phi)}[/math]

expanding the exponents using Euler's thingy

[math](cos(\theta)+isin(\theta))(cos(\phi)+isin(\phi))=cos(\theta+\phi)+isin(\theta+\phi)[/math]


[math]cos(\theta)cos(\phi)+icos(\theta)sin(\phi)+isin(\theta)cos(\phi)-sin(\theta)sin(\phi)=cos(\theta+\phi)+isin(\theta+\phi) [/math]

Separating the real and imaginary parts, you get

[math]cos(\theta)cos(\phi)-sin(\theta)sin(\phi)=cos(\theta+\phi)[/math]
[math]cos(\theta)sin(\phi)+sin(\theta)cos(\phi)=sin(\theta+\phi) [/math]

Look about right?

The imaginary numbers are what 'complete' algebra. There are a few types of number similar to them (quarternions and octernions), but that's it. They are very special, and very necessary for theoretical, and thus practical and applied development.

As you can see, there is a close connection between trig (periodic functions) and complex numbers and complex exponentials. Waves are described using trig (because they are periodic), and thus it should be plausible that complex numbers are useful for describing waves.

Whoa.

All right. Present cock.

>Tbh imaginary numbers sound like, completely useless and non-sensical to me, as if mathematics was inherently flawed (which I actually think is the case).
Fun fact: it's possible to calculate a complex Fourier transform of a 2D signal with a laser, a lens, and something to modulate (attenuate and delay) light. en.wikipedia.org/wiki/Fourier_optics#4F_Correlator

>thats a bitch to remember

maybe if you are retarded

Ever heard of AC electricity?
How about radars?

[math] \mathbb{C} - [/math] vector spaces (despite one isn't calling them like this in high school) is the first counter-intuitive algebraic form one encounters. So in order to proceed to more abstract thinking -absolutely necessary for one's life-, teaching complex numbers becomes a necessity.

Imaginary numbers are important for anything related to waves and periodic motion aka half of engineering and physics.
You use imaginary numbers every day when your phone does a fast fourier transform on the wifi/cell signals it receives to tell it how much of the signal is at each frequency. We have no way of doing this without imaginary numbers.

They were also required to design the circuits on all of the AC electronics you interact with every day.

Astronomical navigation on the other hand is pretty useless in the era of google maps,

>We have no way of doing this without imaginary numbers.
>They were also required to design the circuits on all of the AC electronics you interact with every day.
Slow down here. Anything that can be done with complex number while keeping the result purely real can be done with trigonometric functions and differentiation. For example Fourier initially formulated his series in terms of sines and cosines alone. And it sometimes has to be done this way, because taking the real part of a complex expression doesn't commute with nonlinear operations. It's just much easier to use complex number when they are applicable.

and yet you posted this from an electronical device. ever done some of even the most basic EE calculations? hint: you're going to need a certain type of number.

e^2

how do i write like that

...

Alright, so complex numbers are basically the most natural way to represent *rotations* mathematically, at least in 2 dimensions. Ordinary real numbers have a magnitude; complex numbers have a magnitude and an orientation , and unlike vectors, they're more 'number-like', making them easier to manipulate mathematically.

This makes them very useful for representing waves - light waves, quantum mechanics, voltage and current in AC circuits - because it lets you represent not only the amplitude of that wave, but its phase.

One of the reasons this is convenient is because multiplying two complex numbers is equivalent to a rotation - if you think of complex numbers as vectors, multiplying by -1 is a 180° shift, by i is a 90° shift, and multiplying a complex number by cos(30°) + i sin(30°) will rotate it by 30°, and converting between rectangular and polar is trivial with Euler's identity. It's often much simpler to work with than vectors.

Quaternions, their more complicated cousin, are the equivalent for 3D rotations. Vector notation has mostly replaced them, but they're still the best way to specifically represent rotations and orientations in 3D space without any rotation matrices or awkward special cases, and so computer graphics programs and game engines use them internally.

en.wikipedia.org/wiki/Quantum_computing
imaginary numbers are used in quantum mechanics

What this guy said
between [math] [?math] where "?" is a "\"
or [eqn] [?eqn] for in paragraph shit.

it ends with [/math]
not [\math]

>herp derp WALL StREETZ! LMEG!
berniefags must leave

shut up hilldog, not everyone kisses w.s. ass

i^4 ;) And then recount the echolon again
With no motivationals.

... I can explain lot of mathematics behind this binnari code :D

What about...

atom is whole,

you are a whore.

>Applications are the only measure of a subject's worth.
Oh, you.

>But no matter what you decide to go into, astronomical navigation will always be actively useful.

What the fuck. Astronomical navigation is useless unless you're a sailor without technology.

imaginary is a complex number whose real part is 0, so all imaginary numbers are really complex numbers

And, more significantly, a real is just a complex number whose imaginary part is 0.

Any time you learn math in school using real numbers, they're basically giving you the babby version. When it comes to grown-up math, the reals barely exist; you have integers and you have complex numbers. The case where the imaginary part of a complex number just happens to be zero seldom warrants separate treatment.

I just spent all day working on a program that's full of complex arithmetic.
I'm an intern at an engineering company.

Being enrolled in high school, historically, was for the purpose of gaining admission to college. This is why you learn mostly academic things in high school rather than vocational, real life stuff.

The only "real world" application of complex numbers is for expressing the roots of irreducible polynomials (polynomials that have as many real roots as the size of the degree, but cannot be algebraically solved without running into square roots of negative numbers) in terms of its coefficients with additions, subtractions, multiplications, divisions, exponentiations, and roots. However, it is pointless here because you don't even get the decimal approximation of the roots, and you could just use Newton's method to directly calculate the roots without imaginary numbers. Not only that, but imaginary numbers are not even required in electronics, because it is just an algebraic shortcut for changing (cos(a),sin(a)) to (cos(a+b),sin(a+b)); you could do the exact same action by just paying attention to the trigonometry. The only other thing important about imaginary numbers is their use in elliptic functions, which only have practical significance in that they can be used to logically catalog many different versions of the three elliptic integrals, so that people trying to solve elliptic integrals can change them to canonical form, which will allow them to use elliptic integral tables to lookup an approximation of the solution.

Considering you can't see stars in any major city in the entire world, anything besides "the Big Dipper points north" is just stupid.

Complex numbers are in my opinion a very important topic.

The way they were taught to me was
>Hey, remember how we previously could not find solutions for certain polynomials? Well, we can build a new system of mathematics in which we can get solutions for those polynomials and also have those solutions be meaningful, even if it is impossible to get a numerical value for them.

If you ask me that is important knowledge for a new person to have. A person needs to know that mathematics is a tool they can use in their daily life to model problems and then find meaningful solutions. HS math before complex numbers is all very 'obvious'. Things follow immediately from another because all real numbers are things we can see in our calculators so for a kid they obviously exists but who says that you can simply say that a number exists so that it is the square root of -1?

Nobody says that, but also nobody says you can't so you can do it if you wish. Can you find a set of problems for which that number is meaningful? Yes you can, so you do.

For people who don't major in mathematics, complex numbers are the only idea that they get introduced to that teaches flexible mathematics. The idea that you can use math to model anything and as long as you get meaningful results it is a valid idea. And that is pretty good, much more useful than any other thing they could possibly teach you.

It was a joke. Imaginary numbers = Imaginary money.

the need for astronomical navigation would come in handy mainly when electrical grids fail. so there also would be little light. not to mention you could navigate your way through a city by land marks. my point is that in either situation imaginary numbers are useless for practical everyday use and only become particularly useful if you choose to follow a certain career..

so why arent you taught how to read a circuit board in highschool

but thats not the purpose of high school, its the purpose of particular college courses.

>worth
>*Importance

Why should you be required to learn astronomical navigation if you're passionate about foreign affairs?

Wtf is this. Thanks for the fun fact? lol.

Imaginary numbers are just like real numbers, it's just that plebs like you don't get to use them ever, but that's not a reason to want everybody to be an incompetent moron.