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[fleet glacier]

+close

[calm vortex]

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[nova crescent]

That diagram looks... arbitrary.

[fleet glacier]

I mean this

This is what I meant to do

But I guess it doesn’t work like this?

[nova crescent]

...that's still arbitrary.

[fleet glacier]

I don’t know what you mean

What do you mean arbitrary

[nova crescent]

Why do you think it's a square?

[fleet glacier]

Because it’s 4 roots I guess

So I thought of a square

[nova crescent]

"Roots"?

[fleet glacier]

Yes roots

[nova crescent]

...there aren't any "roots".

[fleet glacier]

We use the roots of unity technique to solve it

[nova crescent]

And four roots don't make a square, they make a quartic.

[fleet glacier]

That’s what was taught in the exercise

[eager bane]

tf is correct?💀

[fleet glacier]

This is what they want me to do

But I’m trying to solve using trig and geometry

I just don’t know how

Because I ain’t doing allat

[nova crescent]

Okay, so.

We can think of the legs of the ant's journey as vectors, right?

[fleet glacier]

Yes

[nova crescent]

The first vector we can say has magnitude 1 and angle pi/2, moving in the pure positive y direction.

[fleet glacier]

Yes

[nova crescent]

Then the second vector has magnitude 1 and angle (pi/2 - 2pi/9).

Can you extrapolate the third and fourth vectors from this pattern?

[eager bane]

There's a very easy way to illustrate this is that it's 2pi/9 4 times, which you can imagine 4 consecutive sides of nonagon

[fleet glacier]

I don’t understand the second bit

😭

[nova crescent]

The ant turns right by 2pi/9.

That is, turns clockwise.

Which, in polar coordinates, is a negative angle.

[fleet glacier]

Wait this makes more sense

To me

[nova crescent]

I don't think it does.

[fleet glacier]

I haven’t done polar yet so I don’t understand the negative angle thing

[nova crescent]

For one thing, the interior angle of a regular nonagon has measure 7pi/9.

[fleet glacier]

Wait I don’t even understand why this is pi over 2

[nova crescent]

Have you taken trigonometry?

[fleet glacier]

Yes

[nova crescent]

Do you know the unit circle definition of trigonometric functions?

[eager bane]

my bad

[fleet glacier]

I’m meant to. But um remind me

Magnitude is 1

And like angles sin theta cos theta?

Or smth

The the radius is 1

[nova crescent]

Okay, so. We begin by drawing the unit circle, which is the circle in the Cartesian plane with radius 1 centered at the origin.

[fleet glacier]

Ok

[nova crescent]

Then we draw the angle in what's called standard position.

[fleet glacier]

Sure

[nova crescent]

Standard position is when one ray of the angle coincides with the positive x-axis, the vertex coincides with the origin, and the angle is measured counterclockwise.

[fleet glacier]

Wait lemme try draw this

[fleet glacier]

Like this

?

[nova crescent]

Exactly like that.

Then the second ray of the angle intersects the circumference of the unit circle at a point, right?

So if the angle has measure theta, that point is defined to be (cos(theta), sin(theta)).

[fleet glacier]

Ok I see

[nova crescent]

In any case, the important point of that lecture was the point about the standard position of an angle.

Because what's a standard position angle of measure pi/2?

[fleet glacier]

The like y axis?

[nova crescent]

The entire y axis?

[fleet glacier]

Oh

So we represent the ant walking forward 1 unit as it walking along the y axis

[nova crescent]

That was my idea.

[fleet glacier]

So as you said it’s a vector magnitude 1 as that’s what the ant has walked through and angle (pi/2-2pi/9) because it’s turning negatively away from the y axis?

[nova crescent]

It's turning to the right, which is clockwise, which is the negative direction of an angle in standard position.

One thing your lesson is right about is that complex numbers are a convenient way to represent vectors.

[fleet glacier]

I see. So ok back to the treating it as vectors…. The first vector is mag 1 and pi/2

Angle

That is the ant walking forward 1 unit?

As you said

[nova crescent]

Right.

[fleet glacier]

Why would the second one have the same magnitude

I understand that the angle of it would be pi/2-2pi/9

[nova crescent]

Because the problem says so.

The ant keeps walking "forward" one unit at every step, where "forward" is the direction it happens to be facing.

[fleet glacier]

Ah I see… so then the pattern is it walks forward 1 unit you do pi/2-2pi/9 it walks forward another 1 unit and you do ((pi/2-2pi/9)-2pi/9)

?

[nova crescent]

I think that's correct.

Not how I would've chosen to represent it.

[fleet glacier]

Now the part I don’t get is how to put that into an answer in the form that’s wanted

[nova crescent]

Well, that's not "the answer", that's just the description of the situation.

[fleet glacier]

Well I’m not sure how to what I said to form it into what they want

[nova crescent]

They want you to find the distance of the ant from its initial position.

[fleet glacier]

I could draw 3 triangle. And maybe work out the sintheta

And add them up?

[nova crescent]

No, that's not necessary.

See, that's the beauty of expressing the legs of the ant's journey as vectors.

Because then we can just sum the vectors.

And then the ant's distance from its starting position is just the magnitude of the sum.

This is perhaps where the thing about expressing vectors as complex numbers can help the most.

Because you recall how to represent a complex number in polar form, right?

[fleet glacier]

z = r(costheta + isintheta) where r is magnitude and theta argument

Of the complex number z

[nova crescent]

I mean, I was looking for z = r e^(i theta).

[fleet glacier]

Oh I also know this

[nova crescent]

That would be step one of the answer to my next question, which is how do you convert polar form into rectangular form.

[fleet glacier]

Hmmm I’m not sure what you mean by rectangular form

[nova crescent]

z = a + bi

[fleet glacier]

Ahh

[fleet glacier]

I’d use this

Sub in theta

[nova crescent]

Right.

[fleet glacier]

And r

[nova crescent]

So then we want to add our four vectors together.

[fleet glacier]

Hmm ok let me try

I got this?

[nova crescent]

...where'd you get that?

[fleet glacier]

4 from the 4 times walking one unit

And the angles from -2pi/9 each time

And I just made the negative ones positive

[nova crescent]

That's... not how that works.

When you add vectors, you add them componentwise.

[fleet glacier]

Oh

[nova crescent]

That is, <x_1, y_1> + <x_2, y_2> = <x_1 + x_2, y_1 + y_2>.

That's why complex numbers work well as vector representations.

[fleet glacier]

I think I understand now. Lemme try again

Like this?

[nova crescent]

...no.

I repeat.

Use complex numbers.

And where's the one with angle pi/2?

[fleet glacier]

I think I’ve sort of got myself confused. Not really sure how this works

I understand the other questions in this topic but it’s because they sort of told you the methods to do

Like number 5 for example I was able to work through easily

[nova crescent]

Okay, just take it from the top.

The legs of the ant's journey are vectors.

The total journey is the sum of those vectors.

[fleet glacier]

Ok I’ll try again from the top

[nova crescent]

The sum of two vectors is the vector whose components are the sum of the corresponding components of the vectors being added.

[fleet glacier]

Surely there’s none with Pi/2 if the ant moves forward 1 unit the magnitude is 1 and it has turned from 2pi 2pi/9 to the right so wouldn’t the first vector be the cos component of that add the sin one

[nova crescent]

That is, <x_1, y_1> + <x_2, y_2> = <x_1 + x_2, y_1 + y_2>

[nova crescent]

...a vector isn't "add" anything.

A vector is two components.

The components are separate.

[fleet glacier]

Sorry just how my teacher explained it 😭

[nova crescent]

That's why complex numbers are a good representation.

Because the real part and the imaginary part are separate.

And when you add complex numbers, you add real to real and imaginary to imaginary.

Just like when you add vectors, you add x-component to x-component and y-component to y-component.

[fleet glacier]

Yeah I’m struggling to see what the components are

I think that’s the issue

I know the magnitude is 1

[nova crescent]

A vector is an arrow pointing from the origin to a point on the Cartesian plane.

The vector is therefore defined by the point to which it... points.

[fleet glacier]

So why isn’t the first vector just 1

That’s the magnitude

And it comes from the origin

[nova crescent]

Because vectors have direction.

[fleet glacier]

Could you go through what the components for the first journey would be

And maybe the second. And I might figure it out from there

I think I’m just fundamentally misunderstanding what you’re saying

[nova crescent]

So leg 1 has magnitude 1 and angle pi/2.

We need to define both the magnitude and the direction because that's what makes a vector.

[fleet glacier]

Yup

[nova crescent]

We can equivalently describe this vector by its components, its vertical and horizontal displacement.

[fleet glacier]

Yup I understand that

[nova crescent]

Which is equivalent to the coordinates of the point to which it points if its base is on the origin.

[fleet glacier]

Yup

[nova crescent]

...are you familiar with the complex plane?

[fleet glacier]

Yup

Perpendicular to the real axis

[nova crescent]

...no.

[fleet glacier]

That’s what I was taught from my teacher

The complex axis

[nova crescent]

No.

[fleet glacier]

The imaginary axis?

[nova crescent]

Yes.

[fleet glacier]

Perpendicular to the real one

Oh

Complex is a+bi

Imaginary is just bi

[nova crescent]

...yes, and the complex plane is... y'know... the plane. Defined by both of those axes.

[fleet glacier]

Yup ok I understand

[nova crescent]

...and the complex plane is equivalent to the Cartesian plane.

Or at least analogous.

[fleet glacier]

Yup

[nova crescent]

...if you don't need me to explain this stuff to you, what do you need explained? Because everything you seem to not be getting is just quite a simple logical consequence of everything you're casually accepting.

[fleet glacier]

I think I just sort of like an explanation of the formation of the horizontal and vertical vectors

So say start with the first journey of the ant

And simply just say what the horizontal and vertical components would be

And I was hoping I’ll just sort of figure it out from there from everything you’ve said

I don’t understand how to form those I think this is the most confusing part of it for me

[nova crescent]

You have the vector in polar form.

The components are just the vector in rectangular form.

You know how to convert it.

[fleet glacier]

cos (pi/2) + isin (pi/2)

That’s just i

And then i just keep going

[nova crescent]

Right.

[fleet glacier]

Ok thank you i think I finally got it.

I think I understood this whole time but I think my brain was just conflicted because I’ve never seen such a context

Maybe it’s the fault of my teaching

I don’t know

But thanks a lot

[nova crescent]

Yeah, that'd be my assumption. You seem to have been taught a very rote, procedural method of calculation instead of a formal conceptual understanding.

[fleet glacier]

Yeah my teacher goes “don’t bother understanding this just do the method”

Lmao

[nova crescent]

That's literally not even teaching math.

[fleet glacier]

He literally scrunched up a piece of paper with the backbones of calculus

And threw it in the bin

And said just do the techniques

[nova crescent]

Without proof, math isn't math, it's just random nonsense.

[fleet glacier]

No need to understand

So I don’t think I’m used to like actually thinking in examples like this which go out of the comfort zone of what I’ve been taught

When I go to university I should probably relearn everything from scratch

[nova crescent]

That's kind of a problem all over.

[fleet glacier]

And the worst part… is I’m the top scorer of my class 98 average. So I think this shows that teaching styles needs and overhaul. Because the entire course is computational and requires no further understanding apart from the odd question here or there like this one which will never be asked

[nova crescent]

It's really depressing too, because the beauty of math is how structured and interconnected it is.

[fleet glacier]

If you have any suggestions to fix my deficiencies in maths I’d be glad to take them and do it over summer before uni

[nova crescent]

I mean, my biggest suggestion is just to seek out proofs.

[fleet glacier]

I wish I had that one teacher who explained everything is such a way like this

[nova crescent]

Look for proofs of everything.

Proofs are what make math math.

[fleet glacier]

But the course I do is high school level so by design teachers won’t teach like this

[nova crescent]

If there's anything you don't understand, anything you have the slightest doubt about, or even anything that you just kinda wonder why it's like that, look for the proof.

[fleet glacier]

Ok thanks a lot

[nova crescent]

There is literally a proof that 1 + 1 = 2.

[fleet glacier]

Wow

So would you say to key to improving in maths is understand the bare bone fundamentals

[nova crescent]

Like I said, it's to understand the proofs, and more generally to develop a precise and logical state of mind.

If I had to describe math in so many words, I'd call it the process of discovering what we already knew.

[fleet glacier]

OK cool thanks a lot

[nova crescent]

Because every true statement in math is an inevitable logical consequence of our axioms.

The reason we don't know every true statement in math is because we just don't know how the axioms combine to render those consequences.

[fleet glacier]

First I’ll get good at exams sit them. After I’ll try to become a real mathematician

[nova crescent]

I mean, a strong theoretical understanding will certainly help you in exams.

[fleet glacier]

I don’t really have the time unfortunately

1 month

I can only rely on how I’m being taught + intuition

[nova crescent]

That's understandable, but if you understand the techniques you're being taught instead of just memorizing them, you'll be able to adapt them on the fly for circumstances you weren't expecting.

[fleet glacier]

I can sort of do this for the regular math course I do but not the extra math course. It’s much newer and more complex pun intended (as it involves complex numbers)

[fleet glacier]

In any case thanks for helping me see math in a different light

[nova crescent]

No problem. I'm glad this perspective agrees with you, and I'm sad it wasn't taught to you by the people whose job it was.

[fleet glacier]

Well I suppose the school doesn’t see it as practical because we would never get anywhere if we stopped at everything to make sure everything understands it. I suppose the class is made for the average person in the class. And goes at that pace. Me and this other guy finish the work long before the rest but we aren’t really given more challenges ahead of that… it’s like we are forgotten within a class

[nova crescent]

...if school can't afford to make sure students understand the material, then... it's just fundamentally failing at its job, right?

[fleet glacier]

Yes and that’s why I believe the school system sucks. But then again I suppose it gets complicated because then you may have to split classes by “ability” or I suppose performance. And then it becomes a complex system

So to be fair to them I’m not sure how they’d get it to work to be fair to everyone

Being the learners who are slower to learn and the faster ones

[nova crescent]

I mean, the issue is just that they're not even teaching the subject they're claiming to.

They're just dropping random alleged facts on your head with no context. That's not how you teach anything.

[fleet glacier]

Yeah true… hopefully I’m not completely stunted for university because I’m really passionate about doing math

At uni

[fleet glacier]

I just wanted to say thank you for your help I really appreciate it

I had all the pieces but for some reason couldn’t put them together effectively at the time. I appreciate it. I’m going to close the post now but wanted to let you know I really appreciated you helping

[eager bane]

your vote is the biggest thank for him, so do +close if you're done and vote for him

[fleet glacier]

+close

[calm vortex]

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[fleet glacier]

I can’t vote : (

[eager bane]

every US citizen above 21 has right to vote 😭

[fleet glacier]

Nvm lol

[calm vortex]

@fleet glacier

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[fleet glacier]

+close

[calm vortex]

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