I'm doing math again and my head hurts

[Malacandra]

@Kimmy_Gibbler

Well, if you said a radius of 1 length-unit, then you would have a diametre of 2 length-units, a circumference of 2π length-units and an area of π square length-units.

Let’s review:

A circle C of radius r is the locus of all points in a plane which is at a distance r from the center of the circle, O.

A diametre of the circle is a straight line from a point on C and through O and terminating at the point on the circle opposite the starting point. It has a length of 2r.

The circumference of the circle is the arc length of one complete revolution along the edge of C. It is 2πr. This gives rise to the radian angle measure and π appears in a welter of situations apparently unrelated to the geometry of circles (the Gaussian distribution, Euler’s Formula – famously, exp(πi)=-1, etc.).

Well, absolutely, and you can approximate π to any desired accuracy by taking the series 1 - 1/3 + 1/5 - 1/7 + 1/9 … to a sufficiently large number of terms, and then multiplying by four. It takes quite a while to get anywhere, though, even on a computer, simply because after you’ve processed the first half a million terms you still haven’t nailed it to six decimal places.

eleanorigby, front and centre.

Since triangles have such darn useful properties, we make use of them all over the place both in real life and in mathematics. Structurally, you get the maximum rigidity for the minimum of materials if you build things out of triangles - that’s why pylons look the way they do, and roof-trusses too. It’s just inconvenient that triangular spaces are not always a handy shape for fitting things into.

Because of this direct relationship between sides and angles, we take one particular special case - where one of the angles is a right angle - and define the ratios between the sides using some special terms. Looking from one of the non-right-angles, there is an “opposite” side and an “adjacent” side, and we say:

• The length of the opposite side divided by the length of the hypotenuse is the sine of the angle
• The length of the adjacent side divided by the length of the hypotenuse is the cosine of the angle.

Both of these magic numbers vary between 0 and 1 for an angle between 0 and 90 degrees. But they have important implications not only for triangles but for circular motion too: If you mark a spot on the rim of a wheel that is rotating, then mark a vertical and horizontal line through the hub, then the position of the spot from the vertical and the horizontal can be found from the radius of the wheel and the cosine and sine of the angle the wheel has rotated through. And if the spot on the rim represents an unbalancing weight, then you can use sines and cosines to explore how wobbly the wheel is going to feel. (This is handy when you’re designing a car engine, say, which contains a number of unbalanced wheel which, hopefully, you’re going to play off one against the other so that the engine doesn’t vibrate uncomfortably.)

[eleanorigby]

@Malacandra

Well, absolutely, and you can approximate π to any desired accuracy by taking the series 1 - 1/3 + 1/5 - 1/7 + 1/9 … to a sufficiently large number of terms, and then multiplying by four. It takes quite a while to get anywhere, though, even on a computer, simply because after you’ve processed the first half a million terms you still haven’t nailed it to six decimal places.

This is the part where I admire the native dancing and nod my head, but am totally clueless as to what is going on. PLEASE don’t explain it anymore.

eleanorigby, front and centre.

<groans>

Since triangles have such darn useful properties, we make use of them all over the place both in real life and in mathematics. Structurally, you get the maximum rigidity for the minimum of materials if you build things out of triangles - that’s why pylons look the way they do, and roof-trusses too. It’s just inconvenient that triangular spaces are not always a handy shape for fitting things into.

That makes sense, she said, cautiously.

Because of this direct relationship between sides and angles, we take one particular special case - where one of the angles is a right angle - and define the ratios between the sides using some special terms. Looking from one of the non-right-angles, there is an “opposite” side and an “adjacent” side, and we say:

• The length of the opposite side divided by the length of the hypotenuse is the sine of the angle
• The length of the adjacent side divided by the length of the hypotenuse is the cosine of the angle.

Could I pretend we call them Harry and Fred instead? Are you trying to slip calculus in here in disguise? I didn’t get to calculus. No, no thank you–I don’t want to get to calculus. Really.

Both of these magic numbers vary between 0 and 1 for an angle between 0 and 90 degrees. But they have important implications not only for triangles but for circular motion too: If you mark a spot on the rim of a wheel that is rotating, then mark a vertical and horizontal line through the hub, then the position of the spot from the vertical and the horizontal can be found from the radius of the wheel and the cosine and sine of the angle the wheel has rotated through. And if the spot on the rim represents an unbalancing weight, then you can use sines and cosines to explore how wobbly the wheel is going to feel.

Been there, done that. It was a bent rim and made my tire have a slow leak. One afternoon at the mechanic’s and voila! rim fixed, tired inflated and I’m outa here…

[Malacandra]

@eleanorigby

This is the part where I admire the native dancing and nod my head, but am totally clueless as to what is going on. PLEASE don’t explain it anymore.

Not a problem, I don’t expect you to understand Gregory’s Series, and to explain how it’s arrived at would need a couple of years of intensive study on your part. Mostly I trotted it out to demonstrate to Kimmy Gibbler that if there’s a dick-measuring contest going on, I ain’t through unbuttoning my fly just yet.

But there is an interesting fact attached that, again, we don’t need to go too deeply into - that you can add up an infinite number of steadily-decreasing numbers, and arrive at a finite result. The Ancient Greeks didn’t know that, though Zeno probably suspected there was something important being overlooked when he demonstrated that, logically, an athlete can’t overtake a tortoise if he gives it a head start. Now we know that whether a series converges (adds up to a finite number) or diverges can be mathematically determined - but unless you want any examples, I shan’t bother you with them.

@eleanorigby

<groans>

Don’t groan, dear, it’s unattractive in a young lady. Trust your Uncle Mal.

@eleanorigby

Could I pretend we call them Harry and Fred instead?

Well, you could, but it helps to speak the same language as everyone else. Much like the doctor saying “This patient’s got yukky eyes - give her some jollop” rather than “The patient has bacterial conjunctivitis, please see that she gets one drop per eye every four hours of chloramphenicol 1%, four times daily”.

@eleanorigby

Are you trying to slip calculus in here in disguise? I didn’t get to calculus. No, no thank you–I don’t want to get to calculus. Really.

And you’re best not to, until you have a firm grasp of algebra. But I will just teach you as much of calculus as you can learn while standing on one leg:

[ol]
[li]You must not divide zero by zero, but you can get very close to zero and still get a meaningful result[/li][li]Any sufficiently short section of a smooth curve is functionally indistinguishable from a straight line.[/li][/ol]

In both cases it helps if you can show that, however close you’ve got to zero, you could get a lot closer if you wanted. But on these two commandments hang all the Law and the Prophets that is calculus. The rest is commentary, but it does take a lot of learning.

[Kimmy_Gibbler]

@Malacandra

And you’re best not to, until you have a firm grasp of algebra. But I will just teach you as much of calculus as you can learn while standing on one leg:

[ol]
[li]You must not divide zero by zero, but you can get very close to zero and still get a meaningful result[/li][li]Any sufficiently short section of a smooth curve is functionally indistinguishable from a straight line.[/li][/ol]

In both cases it helps if you can show that, however close you’ve got to zero, you could get a lot closer if you wanted. But on these two commandments hang all the Law and the Prophets that is calculus. The rest is commentary, but it does take a lot of learning.

I will agree. Calculus is not really any more conceptually different than slopes (differential calculus) and areas (integral calculus). If you know how to find the slope of a straight line and the area of a rectangle, calculus is duck soup.

[eleanorigby]

@Malacandra

Not a problem, I don’t expect you to understand Gregory’s Series, and to explain how it’s arrived at would need a couple of years of intensive study on your part. Mostly I trotted it out to demonstrate to Kimmy Gibbler that if there’s a dick-measuring contest going on, I ain’t through unbuttoning my fly just yet.

Now, why didn’t any of my teachers go this route in trying to teach me math? Prof, I’m all eyes…

But there is an interesting fact attached that, again, we don’t need to go too deeply into - that you can add up an infinite number of steadily-decreasing numbers, and arrive at a finite result. The Ancient Greeks didn’t know that, though Zeno probably suspected there was something important being overlooked when he demonstrated that, logically, an athlete can’t overtake a tortoise if he gives it a head start. Now we know that whether a series converges (adds up to a finite number) or diverges can be mathematically determined - but unless you want any examples, I shan’t bother you with them.

You lost me with the tortoise and the runner…

Don’t groan, dear, it’s unattractive in a young lady. Trust your Uncle Mal.

Ok–I’ll make it a moan, if you’ll lose the “uncle”. My fantasies never include incestuous relations.

Well, you could, but it helps to speak the same language as everyone else. Much like the doctor saying “This patient’s got yukky eyes - give her some jollop” rather than “The patient has bacterial conjunctivitis, please see that she gets one drop per eye every four hours of chloramphenicol 1%, four times daily”.

Don’t cloud my mind with facts and pertinent details.

[LIST 1]
[li]You must not divide zero by zero, but you can get very close to zero and still get a meaningful result[/li][li]Any sufficiently short section of a smooth curve is functionally indistinguishable from a straight line.[/li][/LIST]

I don’t get why nothing divided by nothing cannot be done, but I completely understand the short section of a curve being indistinguishable from a straight line.

In both cases it helps if you can show that, however close you’ve got to zero, you could get a lot closer if you wanted. But on these two commandments hang all the Law and the Prophets that is calculus. The rest is commentary, but it does take a lot of learning.

I’ll take your word for it. And thus endeth the lesson!

[Malacandra]

@eleanorigby

Now, why didn’t any of my teachers go this route in trying to teach me math? Prof, I’m all eyes…

Sounds like one of those videos you can find on youtube (well, not “tube” exactly…)

@eleanorigby

You lost me with the tortoise and the runner…

Zeno said: suppose Achilles can run ten times as fast as a tortoise, but gives it ten yards start. By the time Achilles has run ten yards, the tortoise has run one. So Achilles has to run one, and then the tortoise has run another tenth, so Achilles runs that tenth, and the tortoise runs a hundredth…

Zeno presumably suspected that the conclusion “…therefore, no matter where Achilles gets to, he finds that the tortoise isn’t there any more” was wrong, but he couldn’t prove it wrong with the math available to him at the time. You know intuitively (and correctly) that Achilles can catch the tortoise easily - and we now have the math to prove it.

@eleanorigby

Ok–I’ll make it a moan, if you’ll lose the “uncle”. My fantasies never include incestuous relations.

Why, dear, I had no idea you were fantasising about me in the first place. But what you get up to in the privacy of your own head is no business of mine.

@eleanorigby

Don’t cloud my mind with facts and pertinent details.

You’ve been listening to Skald too much. I’m afraid math consists largely of harping on inconvenient facts.

@eleanorigby

I don’t get why nothing divided by nothing cannot be done,

Well, life gets a little too crazy if you allow it. Which is saying something, given that math has things like numbers with an infinite number of decimal places and numbers that are neither positive nor negative but mysteriously “perpendicular” to the positive-negative number line, but if you allow 0/0 then it’s but a short hop to proving that 2 = 1, which is just asking for trouble really.

@eleanorigby

I’ll take your word for it. And thus endeth the lesson!

Indeed. I’m happy to show you a glimpse of the snowy uplands and hint at what it’s like to walk on them, but you’re not ready to climb that mountain just yet.

[Johnny_L.A]

@Linty_Fresh

Another book that helped me out was Isaac Asimov’s Realm of Algebra. It’s a short theoretical treatise on the foundations of algebra for the layman, and I managed to read it before starting the whole math thing.

Based on this post, I decided to buy this book. (I must have gotten lucky, because the one I bought was less than half of the cost of the least expensive current offering.) Why? Why not? Actually, some things are coming together: I haven’t done Calculus in real life; just classes. And I’ve forgotten it. I thought I’d better refresh, just because. That means I have to refresh my Algebra. Like most people, my Algebra has been limited to the ‘everyday Algebra’ that everyone uses. And I’ve decided, as I’ve said, to learn Reverse Polish Notation. So I’ve got this calculator, and I need some input. Math has been on my brain for the past few weeks. May as well read up on some theory, eh?

I’ve only just started reading it. The first line is ‘Algebra is just a variety of arithmetic.’ Later it says, ‘But this is just arithmetic, you may be thinking. Exactly! And this is what I said at the very start. Algebra is arithmetic, only broader and better…’

Only a few pages into it, I think it’s a fun read. The only thing that would make it better is if it had DON’T PANIC! in large, friendly letters on the front cover.

Thanks for the recommendation.

[Linty_Fresh]

@Johnny_L.A

Based on this post, I decided to buy this book. (I must have gotten lucky, because the one I bought was less than half of the cost of the least expensive current offering.) Why? Why not? Actually, some things are coming together: I haven’t done Calculus in real life; just classes. And I’ve forgotten it. I thought I’d better refresh, just because. That means I have to refresh my Algebra. Like most people, my Algebra has been limited to the ‘everyday Algebra’ that everyone uses. And I’ve decided, as I’ve said, to learn Reverse Polish Notation. So I’ve got this calculator, and I need some input. Math has been on my brain for the past few weeks. May as well read up on some theory, eh?

I’ve only just started reading it. The first line is ‘Algebra is just a variety of arithmetic.’ Later it says, ‘But this is just arithmetic, you may be thinking. Exactly! And this is what I said at the very start. Algebra is arithmetic, only broader and better…’

Only a few pages into it, I think it’s a fun read. The only thing that would make it better is if it had DON’T PANIC! in large, friendly letters on the front cover.

Thanks for the recommendation.

No problem! I’m glad it’s helping. It was a very user-friendly laid back intro. Asimov was always great at that sort of thing, bringing the abstract and scary stuff down to earth so that we mere mortals could understand. I think his writing style in that book was what finally convinced me I could actually do math.

[ivylass]

You know, there seems to be a lot of time and energy devoted to defining lines, planes, triangles, squares, rhombuses (rhombi?) etc. After awhile all the congruent angles/sides/supplementary angles/alternate exterior/alternate interior start to blur together. I know what a fricking square is, why do I have to know it consists of congruent angles and sides?

I know, I know, if we didn’t know this, buildings would fall down. But I seem to get about three lessons done before my brain curls up into a little ball. (Seriously, what is that feeling? I actually feel like I’ve lost 20 IQ points and I’m fogheaded.)

[Johnny_L.A]

@Linty_Fresh

No problem! I’m glad it’s helping.

I thought I’d bust right through it, but I keep becoming distracted. There’s nothing I’ve read so far that I don’t already know; and thumbing through it, it looks like this will remain true. It’s still a fun read though, and I’m glad to have it.