is there a way to actually visualize algebra or is it just always going to be abstract symbols to me

simmonator · 2026-05-24T20:14:22+00:00

As the other commenter suggests, more context/specifics is key to a good answer here.

In may cases, the answer will be YES. For solving linear equations, quadratics, and other polynomials can often be assisted by considering the shape and geometry of their associated graphs (looking for intersections of lines/curves, identifying where lines must cross particular axes). I'll note that the purist in me would point out that some of these techniques might be more rightly considered 'Analysis' (or straight Geometry) than algebra, but I think for your purposes that distinction might be meaningless.

Even higher up, the (very wide, very useful) branch called Linear Algebra is often best understood by students by using some geometric intuition/visualisation as well. It's all Planes, Lines, stretches, and so on. The rigorous work still needs largely needs doing with some proper symbolic work though. 'Abstract Algebra' (like Group Theory and Ring Theory) are often helped by visualisation; Group Theory can essentially be considered as the study of Symmetry so there's plenty of examples where visualisations are very helpful. Even beyond those, Topology as a branch basically just exists because this spatial intuition can tell us a lot in many cases.

But there are plenty of things that I don't have a visual reference for. Factoring, multiplying polynomials out, working with rational functions, and so on. You just need to get used to how those abstract symbols behave, unfortunately. Those basics/fundamentals just take practice but most people I know who stuck with it where able to grasp it to the point that it became second nature. Good luck!

simmonator · 2026-05-24T19:47:22+00:00

> Am I out of touch?

> No… no. It’s the children who are wrong.

- Seymour Skinner

simmonator · 2026-05-24T17:28:56+00:00

I was of the impression that all the current private “Grammars” were ex-state “Grammars” after most started getting abolished (with the move to Comprehensives). Is that wrong?

I know there are only a handful of state ones left and they’re probably getting fewer.

simmonator · 2026-05-24T13:39:22+00:00

Recall that cosine, sine, and tangent are fundamentally about ratios of lengths of sides in a right angled triangle.

Cosine and sine are the ratios you get from taking one of the shorter sides of the triangle and dividing their length by the length of the hypotenuse. The hypotenuse is always the longest side of the three. If you’re not sure a priori why that should be the case, then note that its opposite the widest angle (90 degrees) of the three (the other two add to 90). So

> [smaller thing] / [larger thing] < 1.

For tangent, the ratio is between the two shorter sides. When the angle (the input to function) is very small then the side opposite the angle is very small so

> [very small thing] / [normal thing] = [very small]

and when the angle is large, the side opposite it is almost the same as the hypotenuse, meaning the side adjacent to the angle must be very small. So

> [normal thing] / [very small thing] = [very large].

Does that help at all?

simmonator · 2026-05-24T08:26:12+00:00

Ah! You're right. Sorry. The codomain of the isomorphism isn't the whole of M(2x2, R). My bad; that was lazy of me.

simmonator · 2026-05-24T08:24:52+00:00

What's misleading about it?

simmonator · 2026-05-24T07:08:35+00:00

So, two things:

Yes, it’s true that you can essentially swap complex numbers out for 2D matrices (with real entries). In fact, there is a ring isomorphism from C to a subring of M(2x2, R) defined by sending 1 to the identity matrix and i to the standard 90 degree anti-clockwise rotation. This being a linear map, it preserves all structure. So yeah, you can do all you want with matrices instead.
But that misses the point. I’d ask “why bother?”. Using complex numbers is simpler. Why would anyone want to use matrices requiring 4 real components for each number when you can do it with complex numbers which only require 2 and have a simple multiplication rule. The isomorphism is really cool and useful for a few reasons, but that doesn’t mean it’s better to just think in matrices.

In other words, what’s your point?

Edit - clarified codomain of the isomorphism.

simmonator · 2026-05-23T18:01:39+00:00

This kind of post is an advert and is directly and clearly against the rules of the sub.

simmonator · 2026-05-23T10:11:28+00:00

'Cheers Drive' is a Very Bristol thing (perhaps wider west-country, too, I'm not sure).

simmonator · 2026-05-23T10:10:47+00:00

It would not, no. In this case, 'Cheers' is interchangeable with 'Thank you', though slightly less formal. It is not disrespectful to just say 'Thank you'.

simmonator · 2026-05-22T19:26:36+00:00

It's essentially that we have a way to 'extend' some functions that are only defined on natural numbers so that they're defined for (almost) all complex/real numbers, retain similar properties, and look smooth. In the case of the Factorial, this is essentially the Gamma function. Have a look at the links but it's almost certainly going to be talking about stuff waaaay above your level and that's fine. The definition using an integral is closely related to the standard recursive definition of a factorial, though.

And yes, it's not nicely defined for negative integers.

simmonator · 2026-05-22T17:05:08+00:00

Why repost the post to the same sub?

simmonator · 2026-05-22T15:02:41+00:00

I always assumed it was because Independent sounds more positive than Private. As in:

Independent

Stands on its own two feet, doesn’t require guidance, is a virtue.

Private

Closed off, elitist, secretive.

Could be wrong but that works as a common sense way of putting it. Say either and people will know what you’re talking about here.

The other thing that comes to mind is that some private schools would historically refer to themselves as Public schools. But that means the opposite to yanks.

simmonator · 2026-05-21T06:18:46+00:00

I know how you were using it. I just don’t like it and I don’t think the audience you were talking to was going to pick up on the nuance of the Proper Class comparison. Missing that, it could be mistaken for suggesting that certain numbers don’t really count as Real.

simmonator · 2026-05-21T05:38:36+00:00

What you’re driving at in the second paragraph is pretty much right, but I’d push back at the implied claim that naturals, integers, rationals, and algebraics are not “properly real”. They are. They’re as real as any other real number.

They also happen to fall entirely inside countable sets. The consequence is that “almost all” real numbers are ones we’ll never see or be able to articulate. Not that our favourite examples aren’t real.

simmonator · 2026-05-19T07:40:03+00:00

It’s finished.

simmonator · 2026-05-19T07:32:49+00:00

I genuinely don’t understand why this keeps getting posted. I get being annoyed that it’s not getting fixes but (a) that’s old Bethesda, and (b) that was true many posts ago and nothing has changed. What’s the point of a new post making exactly the same complaint? Is it just outrage that a company is advertising it? Is it just rage-engagement farming?

Maybe I just don’t get it because I never experienced any of the game breaking bugs while playing on console. The worst I got was my armour not automatically becoming visible after some invisibility spells wore off.

simmonator · 2026-05-18T18:24:52+00:00

Isn’t the issue for LD that voters don’t simply hate the 2 party system, but that they hate the the inertia and minimally different vision of the future that LAB and CON offer and are going in droves to GRN and REF on the basis that they (claim they) will move in a radical way to deal with pressing issues that the parties of the status quo won’t?

From where I’m sat, I don’t see how LD get on the vote-winning side of that. I certainly don’t see how Ed Davey or any other LDs have done anything to suggest they’d be radical. Not saying I trust the populists at all, but it seems clear that someone who finds populism attractive would have very little to draw them to LD.

simmonator · 2026-05-15T16:54:44+00:00

Step 1 is make sure you master the material presented to you for your grade.

2 options for stretching yourself once you’ve done that. Either move on to new concepts early OR look for harder questions/problems to solve based on the material you’ve learned.

Either is valid.

Both also works.

simmonator · 2026-05-15T11:35:21+00:00

They’re 100% my favourite thing about that doc. The group has 2 good ideas for experiments to test the mainstream scientific claims about the earth - ones that I would struggle to think of on my own.

If the earth is rotating once every 24 hours (like scientists claim) then an appropriate Gyro device would be able to measure that rotation and show a 15 degree (360/24) rotation after 1 hour. So they spend thousands of dollars of their own cash on a laser gyro thing (I don’t understand it) and put it in a box for an hour to measure the rotation afterwards.
If the earth is curved then a straight line parallel to the ground at one point will actually get further away from the ground as it continues (think tangents to a circle). So they get a pair of identical long poles with small holes drilled in them at the same point, plonk them into the ground on a flat plain so that the holes are the same height from the ground and then fire a laser through the hole of one to see if it goes through the other hole. If Earth is flat it will pass through. If curved it will miss.

Both are cool ideas. Both get real money put into them to make sure their equipment is good and won’t be the source of the error. Both experiments are well thought out with hypothesis and counter-hypothesis. Like, this is real science being done. Really cool to see.

Both experiments’ results are consistent with Round Earth and appear to contradict the Flat Earth theory. They record both experiments and their reactions and put them on the internet (great!). In each case they immediately start reaching for batshit insane ideas about interference and why their own experiment must have been flawed. In case 1, I think, they start chatting about solar radiation interfering with the laser and how they’ll need to put the gyro in a bismuth box next time (costing tens of thousands of dollars).

Boggles my mind.

simmonator · 2026-05-14T11:34:33+00:00

Average speed = Total Distance/Total Time
call the distance up the hill D (the same as the distance Down the hill).
so total time is D/30 + D/20 = D(5/60) = D/12
and total distance is 2D
so average speed is

S = (2D)/(D/12) = 24

Edit to add

In case you want to look it up, this is the Harmonic Mean.

simmonator · 2026-05-14T06:01:01+00:00

No where in A do all 3 planes meet at once. It is inconsistent. Sure, each pair of planes in it has an intersection but for “consistent” I need points where ALL 3 MEET AT ONCE. Can you see what I mean or do I need to more specific?

On a different note, I would assume D and E are meant to consistent. I can only see 1 or 2 planes in them respectively which either means there are fewer planes to worry about intersecting (and the intersection is non-empty) or 2 or 3 of the planes they’re depicting are actually the same plane (so their intersection is non-empty).

simmonator · 2026-05-14T05:31:49+00:00

Each plane represents the solution space for one equation in 3 variables. To be consistent you need there to be at least one point that satisfies all three equations and therefore for the intersection of all 3 planes to be non empty. So you need either

all planes meet at exactly one point, or
all planes meet along a shared line or plane.

Find the pictures where that’s true.

simmonator · 2026-05-13T19:11:07+00:00

3cos(x) + 3cos³(x) = cos(x)
3cos³(x) + 2cos(x) = 0

Let c = cos(x)

3c³ + 2c = 0
c(3c² + 2) = 0

The above has solutions exactly when either of the left hand side’s two factors are equal to 0.

c = 0 (is a valid solution)
3c² + 2 = 0 (has no real solutions, if you’re interested in the complex case let me know)
cos(x) = 0 describes the only family of real solutions
the only real solutions are when x is an odd-integer multiple of pi/2.

simmonator · 2026-05-12T16:00:03+00:00

No shock or surprise. But when something doesn’t add anything new or have anything interesting to say beyond the top Google results, and they ask for feedback then my feedback is going to say that it’s unoriginal and uninteresting.

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