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When you're talking about volumes and hypervolumes, your set needs to be equipped with a measure function. The technicalities are too much to go into in a single post, but see the wiki link if you're interested. The standard measure used on the Reals is the Lebesgue measure where the length (in one dimension) is just the difference of the endpoints. So, if we're looking at the length of a point p, we need to just take p-p, which is, of course, 0.

It depends on the kind of classical physics and the complexity. If you want to understand basic toy kinematics and dynamics at the high school level, you could get away with going up to the first order differential equations. You'll be doing stuff like taking the derivative of a displacement equation to get a velocity equation. If you want to understand more complex problems with more modern paradigms, then you'll need multivariable calculus. You'll be doing things like solving the Euler-Lagrange equations and integrating Lagrangians to find the least action. With waves, electromagnetism, and "Modern Physics", that table of contents specifically tells me you'll likely need the multivariable parts of the calculus text as well. But it might be simplified for like an introductory course. You'll have to look at the physics book. If it has [math]\partial[/math] or an upside-down triangle,you know you'll need the multivariable parts. TL;DR: That physics book's topics suggest you'll need the entirety of the calculus book Edit: ! Moderator Note Topic moved from Lounge to Classical Physics

It should be pointed out that this is an uncountabe infinity. It is waaaaaaaaaaaaaaay bigger than the infinite cardinality of the natural numbers Infinities come in different sizes

Well thank you. It took some doing to fix my bbtex mistakes on the phone with autorendering

It's good that you asked in Linear Algebra And Group Theory, because we're going to need some algebra you likely have never seen (unless you went to college) to answer it. Multiplication isn't one thing. What multiplication is depends on what you're multiplying. Algebra is how we define this. In Algebra, there are a handful of different kinds of structures. Here, we're interested in Groups, Rings, and Fields. Rings and Fields are kind of made of Groups, so we'll start there. Say we have a set (the lay concept of set will work fine for our purposes), and we'll call it S. On this set, we need to define a rule called a "binary relation" that takes any two things in the set and gives some output. We want this set to be closed under this relation, so the relation can only give us things that are already in the set. For this combination of set and relation (for now, we'll use ? to denote the relation) to be a Group, they need to have the following properties: 1) Associativity: for any three things a, b, and c in the set, the relation doesn't care about where the parentheses go. a?(b?c)=(a?b)?c 2) Identity: there is a special thing in the set (traditionally denoted by e when talking abstractly) where, for any other thing a in the set, a?e=e?a=a 3) Invertability: for any thing a in the set, there is another thing in the set a* where a?a*=e=a* That's enough to be a Group. But we want a special kind of Group, called an Abelian Group. That's just a regular Group that has an extra property: 4) Commutativity: for any two things a and b in the set, a?b=b?a Tradition dictates that for Abelian Groups, + is used in place of ? and 0 is used in place of e and -a in place of a*. If a Group is not Abelian, we often use × (or nothing at all) in place of ? and 1 in place of e and 1/a in place of a*. If S is the set, we write (S, +) or (S, ×) for the group, but we often just write S if it is clear from the context that we're talking about a group. This is enough to let us build a Ring. With Rings, we still have a set S, but we have two relations. (S, +) is an Abelian Group, but × is a bit more lax. × only has to satisfy two properties: 1) Identity, and 2) Distributivity: for any three things a, b, and c in S, a×(b+c)=(a×b)+(a×c) Like how (S, ?) is a Group, (S, +, ×) is a Ring. If a Ring is commutative and has inverses for each relation, then the Ring is called a Field. There are four particularly important facts mentioned above that are important to why a negative times a negative is a positive: 1) a+(-a)=0, 2) a+0=a, 3) a×0=0 (not mentioned above, but still important), and 4) a×(b+c)=(a×b)+(a×c) Proof -a×-b = a×b: Let a and b be positive numbers in our field (S, +, ×). 0=b+(-b)=a(b+(-b)) -a×-b = (-a×-b) + a(b+(-b)) = -a×-b + a×b + a×-b = -a×-b + a×-b + a×b = -b×a + -b×-a +a×b = -b(a+(-a)) + a×b = a×b TL;DR: It's because Fields are commutative, have identities, and are distributive

How many coin flips does it take to make a free choice?

! Moderator Note Topic moved to General Philosophy

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Algebra: Chapter 0

I take it you didn't actually read the OP. It's generally good practice to do that before posting in its thread.

If there's no relating the layers to each other, what's the point? If there is, then it's just modality with another name.

How does this differ from modal logic?

Learn some math. Even the really basic stuff can let you do really cool things. Many of the machine learning techniques aren't all that mathematically complicated. Neural networks, for examples, are little more than optimizing weighted averages of a step (or sigmoid or something like a sigmoid) function.

A common analogy is to comment like you're explaining it to a rubber duck.

I noticed that a lot of the programming questions here, while phrased in terms of coding in specific languages, are actually rooted in a broader context of programming itself. People are having problem with the ideas behind programming in general. If we help people with algorithm building, the language-specific syntax falls into place. So, this thread is for tips on programming in general. This isn't about programming in python or programming in java. This is about the fundamental heart of what one needs to do. If anything, it's tips on how to code in PseudocodeTM. I'll start: One of the biggest thing about programming is being able to break a problem down into chunks and then break those subproblems down. Keep doing that until you get a picture of what specifically you need to do to accomplish each major step. So, if your program needs to do things in three big steps, start even lower than pseudocode; start with a list. Step 1 Step 2 Step 3 Then, take each of those steps and do the same thing. Step 1: Step 1a Step 1b Step 1c Step 2: Step 2a Step 2b Step 2c Step 3: Step 3a Step 3b Step 3c You may or may not notice the indentation. That brings up my second tip. Use indentation, called "whitespace", in your programs and even pseudocode. Some languages require you to have whitespace indentation (like python and the never ending tabs v spaces war), but many do not. It is still good practice for writing though. Most languages have interpreters or compilers that ignore anything about whitespace other than whether there is at least one space between two letters, so it won't hurt the program but it will help you. Writing, or at least planning, with whitespace keeps ideas organized. Look at the example in the previous tip. You see each substep, and it's clear where in the logic that it belongs. It makes code easier to read and understand, even if it's not required. Speaking of readability and understanding, that's where my last two tips come in. Make good names. Names like "MyFunction" or MY_CONSTANT are bad names. They don't tell you what the function is doing or what the constant is for. Make names descriptive, and your code will be easier to understand and easier to read. Rename (or "refactor") if you need to. Many programming environments will even let you replace all instances of one name with instances of the new name, so it won't even be a long tedious thing. Renaming, if giving a new name, is a good thing. It promotes clarity. Comment everywhere. Comment to the point that you think you're overcommenting, and then comment some more. Future you will likely have no idea what your code is doing, so leave good notes for future you to use to figure it out. Your turn.

Unfortunately, it's currently unobtainable.

That's the incorrect answer. You can do it just marking out the possibility space with pictures of doors, cars, and goats, or you can do it the Bayesian way and you get the same answer. That answer is not 50/50. You can run actual trials (as Mythbusters did) and confirm. 50/50 is an incorrect answer. Horrible quality video to follow: https://www.youtube.com/watch?v=FAljAvR3L4s

You can apply to be a moderator via PM, but the process takes 8 years before the council votes on the appointment.

Questioning Physics has been banned for spam and disregarding warning points issued for spamming.

I did notice that we've got a lot of non-CS and non-techsupport computer threads. Perhaps it's time to reorganize the computer section to add subforums for programming and for hardware.

Iwonderaboutthings has kicked over the checkers board and gone home. We've kindly locked the door behind him.

fiveworlds is taking a week long vacation after plagiarism and publicly posting PMs without permission.

HRamjetTesla has been banned at his own request and for an abusive meltdown. And Soral has been banned as a sockpuppet of Frank_Baker.

Iwonderaboutthings is on a week long vacation following a core meltdown.

It's also something we've unfortunately had to explain several times to trolls.