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Ohhh, gotcha, you can write this as
a^(b) ≡ a (mod b).
Other than that, great! This is actually a version of Fermat's little theorem. I would recommend becoming familiar with mod notation before getting too far into it, but the proof for this is quite neat. More generally, it states that:
a^(p-1) ≡ 1 (mod p), where a is any number relatively prime to p, i.e. a can be greater than p if you wish. Props to you for finding this yourself, modular arithmetic is hella cool.
You just figured out the formula for the sum of the first n positive integers! The formula is n(n+1)/2 and you realized that the sum is equal to the last number(n) multiplied by the middle number((n+1)/2).
The Brower fixed-point theorem seems, to me, at once intuitive and completely inexplicable.
A simple demonstration of the theorem can be done with two pieces of graph paper. Since it's graph paper, we can imagine that there's a coordinate system on the pages: the bottom left square is (0,0), through real numbers, all the way to the top right which might be (8.5, 11.0).
Say we take one page and crumple it up. Then, we lay the other page flat and drop the crumpled ball onto the flat page.
Both pages have the same coordinate system. Let's think of the crumpled page as C(x,y) and the flat page as F(x,y).
Brower's theorem tells us that there's at least one point in the coordinate space of the crumpled page that has mapped directly over the same coordinate on the flat page. That is, there's at least one (x,y) such that C(x,y) is directly over F(x,y).
It's totally obvious when you -- no way, dude. But, wait, I guess that ...
Bump.
Been feeling so fucking dumb lately. Was trying a new brain training website brilliant.org, and boi was I fucking dumb.
I actually felt so dumb, that I instantly quit just so end the misery. I'm 24 btw.
There are a few steps skipped there. Consider the periodic table of the elements, or the periodicity of trig functions, or the way birthdays happen periodically. The common thread is cycles and things happening cyclically.
cc: /u/ViciousPuppy
Alcumus. It covers a range from pre-algebra through geometry, including probability and number theory, with a lot of incredibly creative and difficult problems. The creators are probably the leading US group in preparing people for competition math, and they do a fantastic job of presenting what is compelling within math. It’s my single favorite learning-focused website in any subject.
You might also try brilliant.org, although that one has some costs attached and is a bit more corporate.
I don't know if this is such a good example. The Pythagorean theorem is extremely simple and elegant to prove geometrically. The algebraic proofs seem way more ad-hoc and complicated in this case.
https://brilliant.org/wiki/proofs-of-the-pythagorean-theorem/
See the proof by similarity and rearrangement on that page. Maybe you know a simpler proof than the algebraic one on that page but anyway, the geometric proofs are still great!
Based from the comments below I believe I can make a safe assumption that most people are given the impression from Brilliant.org that this website is a great way to learn mathematics on a deep level.
I am not one of those people. Based from the website's advertisement it's more of a fun learning course than a serious in-depth learning course of any particular topic. It's mainly used to challenge you to think differently in specially-designed puzzles.
I found links in your comment that were not hyperlinked:
I did the honors for you.
^delete ^| ^information ^| ^<3
Delta is a forward looking statistical estimate! While an estimate and not guaranteed it is something I use on every trade and have for many years of successful trading and expect most traders use it as well.
As it is a statistical estimate I think it is far more useful than watching for the chart to be up or down. When a stock is moving up or down it is just as likely to continue up or go down and there are no stats behind this but simply luck.
If you're a math person the link below shows how it is dervived.
drlizard is right
This is a typical case of gambler's fallacy. The jumbo cactpot numbers are independent events, the outcome A happening has no influence on the outcome of B happening, A in this case being the first number and B being the second.
Checkout this to learn more: https://brilliant.org/wiki/probability-independent-events/
Or better yet, can someone just explain this?
Edit: Perhaps: https://brilliant.org/wiki/what-is-00/
The difference between theory and practice, is that in theory, the theory is correct while in practice, it is not.
> this approach is only applicable in theory and cannot be shown practically as both the temperature and pressure at a given point on Earth keep on changing continuously at variable rates.
In the real earth we live in, the earth is lumpy, with differing weather patterns etc ....
> this assumes the temperature varies continuously.
OP's post is what happens when folks confuse math and physical reality.
Edit : I might have to eat humble pie here
> this result doesn't care at all about being a perfect sphere, uniformity, weather systems, insolation, cloud cover, land masses, or anything else.
If you want a general method:
https://brilliant.org/wiki/linear-diophantine-equations-one-equation/
diophantine equations are a standard thing in number theory.
But, in this case, it might actually be quicker to just do it brute force.
5x+8y=(x,y)=38
[I don't know what you mean by "probability tree." So I'm probably just doing what you did.]
There are only six possibilities for y and three of those are very obvious and the other three are simple. So you can go through those very quick.
0.y=/=0 because 38 ends in "8" not 0 or 5.
1.y=1 implies x=6
This is a solution.
y=2 implies
5x+16=38
x=(38-16)/5=4.4
2. y=/=2
y=3
5x+24=38
x=(38-24)/5=2.8
3.y=/=3
y=4
implies
x=(38-32)/5=1/2
4.y=/=4
The only other possibility is all fives but that can't be correct.
So there's only one solution.
When it comes to tutoring, it does not really matter what you use to spread knowledge - textbook, Brilliant.org, whatever. The only thing that matters is You. A great tutor who is excited to teach their subject is a gem in education, regardless of what they use, so keep that in mind. Good luck in your tutoring journey!
> Can any of the people downvoting me explain why?
I upvoted you, but I will explain why others are downvoting.
I suspect most people's intuition matches mine here, that since the hyperloop tube is held at (near) vacuum that a sonic boom can't happen, or if it does that the force would be negligible because of the extremely low pressure.
That said, after some research I've found that you're probably right, even a small amount of air pressure puts some serious limits on the speed a capsule could travel through a tube.
It really comes down to a combination of how you're presenting your information and most people's intuition (regardless of if that intuition is correct). Specifically, you start out by immediately going against intuition, "Hyperloops can't go faster than the speed of sound." and then going on the attack, "Why are people posting and upvoting random ignorant tweets.". You would be better served by starting with the information relevant to your claim, then making your claim, and leaving out the attack entirely. External references would also help. Perhaps like this:
> Hyperloops aren't planned to be full vacuum, so there are some factors limiting their top speed, such as the speed of sound, which can have a significant impact despite the low pressure and limits a hyperloop capsule to subsonic travel speeds.
Lastly to actually answer your question of why, it's because these people lack some of the knowledge you do, or have not taken the time to apply critical thinking to the knowledge they have. Mocking people (intentionally or not) for their ignorance doesn't fix it, since most people, when feeling attacked, attempt to defend. Education does fix ignorance, but effective education is a lot more than just throwing facts at people and telling them what they know is wrong.
1st: Developers can manually refund, so the charge is still refundable
2st: Always innocent until proven guilty, so this might have been a simple glitch; if I were you I would go ahead and contact support of Brilliant.org, and if not resolved move on to google support and explaining your issue. If this is resolved through support, or if the developers shows full sincerity when refunding, I would then go ahead and remove your one-star review.
I found this article about it, which goes kinda in depth about the debate in mathematics and whether they 0 base always equals zero, or 0 exponent always equaling 1 takes precedent. From what it looks like, there’s no super clear correct answer and it varies by mathematician and their preferred definitions.
I checked out Brilliant a couple years ago as it was getting started, and at the time it didn't impress me, but it did seem like it was getting ready to take off. I haven't tried it lately but I'd be interested to hear how you like it if you do.
Sure! I can tell you only in how I see it; I may be misusing some words but I'll give you the idea.
I excelled at history/analysis because to me, if you explain the WHY something happens, I'll remember it forever-- because then it makes *sense* how one thing flows to the next. Same for how I approach process improvement at my current job. "What is the need this process/procedure this is filling?" and go from there, as my brain can then sometimes see a better way to get that need met without being bogged down by "it must be done like this".
Deductive reasoning is (or again, what I struggled with) a set of facts that you can then draw conclusions from. I tell you three things, and from there, you can infer three more, none of which are necessarily explicitly stated. Once someone tells me out loud what I was supposed to be able to reason out on my own? It makes total sense. I just... really struggle to make those connections.
Here are some examples of where I just turn into a pile of question marks:
In my somewhat dark sense of humor, I love this sort of thing. It’s a Smug Wonka meme.
>You’re having trouble counting calories as an adult?
>Let me tell you about 9 year olds that can plot the effective half life of medication close enough to precisely neutralize a slice of pizzas carbs.
If you want to help them, introduce them to Fermi Estimates, it’s basically a way to use landmark things you remember to to do estimates.
I realized I do this for carbs CONSTANTLY a few years back. It’s surprisingly good so long as you have good landmarks/reference points.
Not everyone can guess “how sweet an orange is compared to a slice of bread” or figure out how to pull out other comparable points to go against.
Fun fact: Fermi was so good at doing this, he was able to estimate the force of the first nuclear blast within 40% of its end value... using paper... that he was tearing into pieces and watching fall to the ground.... as the blast went off. He straight up was using litter to guess at the force of THE FIRST nuclear blast and was within 40%. That means he had no good and clear frame of reference, he was effectively making it up off the numbers.
It’s insane
So you don't know how base representation works? If so, then I suggest reading this article. In any base, there will always be numbers with more than one representation. For instance, in base-10, "0.4999..." and "0.5" both represent the same number. In base-3, "0.222..." and "1" both represent the same number.
It's not ultimately because of anything the developers have done. It's because of the mathematics:
https://brilliant.org/wiki/coupon-collector-problem/
You would have the same problem for units except for the fact that focus banners exist.
Some of the questions on the math portion would confound someone who was a junior or senior in college.
Check this out. It is like the most difficult calc II problem you could possibly still call calc II.
https://brilliant.org/problems/a-calculus-problem-by-nick-lee/
In my opinion, sounds like all you need to do is make sure you work hard once you begin the course. Currently I'm a junior math undergrad, but I had a late start to my math education. I never was "good" at math in grade school, always in the average math class. I didn't take one senior year in high school. I actually failed college algebra TWICE my freshman year of college, because I was skipping class. After that, i decided that I actually love math and just started to work hard, that was all, i just put in a lot of effort.
I went to a different school and just went to trig because of test scores, got an A. Then went back to my university, and have gotten A's in Calc 1-3, Linear Algebra, and Complex Analsysis. Literally only because I just worked really hard. I found a study partner and we did many hw assignements together and studied for exams together which helped too.
So it already sounds like you are 5 steps ahead of where I was in your position. I think you shouldn't worry as long as you do each homework, study for exams, and think about what you are doing in class conceptually. Talking it out with someone is a great way to retain that info, and explaining it to someone is even better. I got a job at the tutoring center at my university so it would force me to have to explain the math to other students, maybe that is an option at your school you could look into.
edit: spelling
edit 2: i guess this wasn't directly answering your question of resources that you could use to help you preapre. I guess in my mind you sound quite prepared, and I am just trying to ease your worries about univeristy calc, its not so bad! It's actually great fun :) Check out brilliant.org for some fun exercises
I think it might be interesting to dispel some myths about infinity, such as the common misconception that infinity is just a really large number (like when kids say "Infinity plus one!")
Brilliant has a good write-up on infinity that isn't too advanced, i.e. is perfect for a casual informative speech. https://brilliant.org/wiki/infinity/
The unit step function doesn’t really have that much of an impact on the time invariance of a system because of how nice of a function it is. It basically acts like a switch that turns on at t=0 and stays on as you approach infinity.
Time invariance is concerned with getting the same output signal from the input no matter when the input is applied. With the unit step function, we know it will always be magnitude 1 so anything being multiplied by that signal will remain unchanged.
Additionally, the unit step function can be shifted very easily in the time domain allowing for the following to be true; y(t-T) = x(t-T) where x(t)=cos(t)u(t) for example.
Here is a link to a basic example of testing for time invariance:
https://dsp.stackexchange.com/questions/23194/proof-of-time-invariance-of-continuous-time-system
Here is a link to an article about the properties of LTI systems:
https://brilliant.org/wiki/linear-time-invariant-systems/
Hope this helps!
Starting at the "bottom".
A class is an Object Oriented Programming concept:
> a class is a blueprint for creating objects (a particular data structure), providing initial values for state (member variables or attributes), and implementations of behavior (member functions or methods). The user-defined objects are created using the class keyword. https://brilliant.org/wiki/classes-oop/
A module is a grouping of things, the could be classes, they could be functions, or both or other stuff. They should be related. For example there is the datetime module that contains classes and functions for working with datetimes.
A library is a grouping of modules. There is a datetime library built in to python that contains the datetime module above but also a time module for working with times and a dates module for working with dates.
Specifically for mathematics, I don't like how it is taught to us in a sort of mechanical, non-intuitive manner (except probably geometry). We are taught how to solve problems through a set of rules and we are expected to follow it in good faith, without even understanding why these rules work.
Most of us then tend to limit our problem solving ability using only these methods, without realising that there's multiple ways to solve such problems. Try solving the daily math problems at brilliant.org and having a look at the discussions section afterwards, some didn't even need to do any calculations to come out with an elegant solution. Or check out Art of Problem Solving and Better Explained.
This is a classic Bayes Theorem question.
Let's name the urns: Urn A (Balls 1-10) and Urn B (Balls 1-100). I think you're asking, if you draw a ball with #1 on it, what is the probability that you drew it from Urn A (or Urn B).
So the probability that you drew it from Urn A is: P(A | #1) = P(A and #1) / P(#1) = P(A and #1) / [P(#1 and A) + P(#1 and B)] = (0.5)(1/10)/[0.5(1/10) + 0.5(1/100)] = 10/11 = 0.91.
On the other hand, the probability that you drew it from Urn B is the complement: 1-P(A|#1) = 1-10/11 = 1/11 =0.09.
For your generalization question, the value of k is irrelevant. If you change the problem above to be about drawing the ball with a #2 on it, nothing at all changes. For urns with 1->n and 1->m, just substitute the "10" and "100" in the above calculations with n and m. That is P(A | #1) = (0.5)(1/n)/[0.5(1/n) + 0.5(1/m)] = (1/n)/(1/n + 1/m).
Barbara Oakley's book A Mind For Numbers and accompanying TED talks and open online courses on "Learning How To Learn" specifically address this topic. She is a professor of engineering who started out as an Army linguist who wanted to see if she could retrain her brain to get better at math and science and seek out jobs in a STEM field. Olympia LePoint is a former aerospace engineer who also writes and gives TED talks about overcoming "math phobia."
Also, if you are having serious difficulties learning or understanding specific types of material it doesn't hurt to get a full neuropsychological assessment for learning disabilities such as dyscalculia.
Brilliant.org and Khan Academy should also have a lot of free resources for explaining difficult concepts and getting better at math.
Brilliant.org is really good, but you have to pay. You can find discounts on Wendover's channel on YouTube. I feel their math stuff is more succinct and easier to follow than Khan Academy, but to each his own.
If you flip a coin and get heads-tails-tails-tails-tails-tails is something wrong with the coin?
Thanks to De Morgan's laws,
(NOT A) AND (NOT B)
is equivalent to
NOT (A OR B)
So in the original post,
username != "root" && password != "root"
is the same as
!(username == "root" || password == "root")
Now this expression is to test if the authentication has failed (as it is used in the code to determine if the "Invalid authentication details" message should be shown). This means that the rules for valid authentication credentials are the opposite of that expression, or
!(!(username == "root || password == "root"))
Since both 'not' operators cancel out, this is equivalent to
username == "root" || password == "root"
Have you looked at all into https://brilliant.org yet? Not only are some of the problems challenging in terms of some of the mathematics involved (although you can search by subject), but many will require a lot of thought about what types of mathematics are required (or not) for a problem.
Is this what you're looking for when you say you're looking for challenging math problems?
It is complex. It also works. My problem here is that it is simply not made for a market that is that one sided. it is a mathematical formula that calculates an options value based on statistical characteristics of the price curve. But i seriously think it breaks apart when the market is so one sided.
THAT SAID: this is not a bad deal for both sides.
Buyer: You get a lot of upside with a lot less downside. Combine an option with 60% of a bitcoin and it is a decent bet for the price of one bitcoin.
Seller: This makes sense IF you have plenty of capital and do not mind owning bitcoin when it fails. Basically you get 2250 from the buyer and buy a coin on the market ;) Financing is 5500 (roughly, i round) for a year. If it fails you STILL got the coin cheaper than buying it now. If the price goes up, you get another 10000 - nearly 100% profit. Now, owning the coin IS better, but if you are a miner, for example, or a very rich individual that already has a stash - that is not that bad a potential income. Also in December you may buy futures to offset the risk ;)
THAT SAID: Still: black sholes? I really would love to see their input parameters. Which time period did they plug in?
Btw., the math is here:
https://brilliant.org/wiki/black-scholes-merton/
History: This was the first more complex formula in wide use in finance: First time computers took over calulation of what was guessed before.
brilliant.org has a seriously good page on this:
AA HL is highly recommended for students who has any intention of doing anything maths-related after high school (including physics and other sciences/engineering, but also finance, economics, architecture, etc).
I suggest people to do some Khan Academy exercises. You can start with the course Getting Ready for Algebra 2. Move up or down depending on your progress.
AA HL is a demanding subject even if you are very good at maths. If it's important to you to do well, you can learn ahead on Khan Academy, or Brilliant.org (the separate, paid, independent, GUCCI version of KA).
Remember Maths is more or less an acquired skill. I highly doubt any so-called prodigy can perform at the same level without any guidance. The question is then how well and how fast you acquire and build this skill.
I found links in your comment that were not hyperlinked:
I did the honors for you.
^delete ^| ^information ^| ^<3
It's an entertaining video, but such a totally imprecise and misleading way of explaining it that it honestly becomes unhelpful.
Summability methods for divergent series are such dramatic generalizations of normal additions that they are essentially redefinitions entirely.
So, I wouldn't recommend at all for beginners to watch the Numberphile video; watch the Mathologer video instead or read the Brilliant article.
In general, for a set of n items there are n! ways to rearrange it. This set has the special fact that there are duplicates, which reduces that number by a lot. here is a brilliant.org article that explains your problem.
Luckily, with physics (beyond theoretical physics), you can observe and test things yourself.
If you're really interested in knowing more, and have the money for a subscription:
https://brilliant.org/courses/dynamics-bootcamp/
https://brilliant.org/courses/astronomy/
I genuinely think these interactive courses would help. It's not a class or anything, you learn and do problems as you please.
Otherwise, you could probably learn the same thing from r/askscience or something.
It would be helpful if you elaborated on what kind of math you currently do. Mathematics is also a very broad field so its difficult to give specific advice for understanding the whole of mathematics better
That being said, if you are trying to better understand mathematics, i would recommend just trying to read stuff about mathematics wherever you want and when you get to a point where you don't understand something, look it up. Wikipedia, Wolfram Mathworld, brilliant.org, and The Art of Problem Solving are all good resources for looking stuff up when you get confused.
You might also try learning more about proofs if you haven't already done so because proving theorems is really what pure mathematics is about. Unfortunately i don't know any great resources on learning to prove stuff, specifically, but math textbooks written for math majors will almost certainly include proofs as a major part as opposed to focusing on applications and computation.
Finally, if you ever find yourself using math for whatever you are doing professionally, try and learn more about the theory behind what you are doing
Hope that helps Tl;dr read about math a lot
> This means that we can view x as a possible proof and s as a possible counter-proof. The formula is true iff there exists a proof that ‘defeats’ every possible counter-proof.
This is kinda the reverse of the epsilon-delta definition of limit, isn't it? There, for every challenge ϵ, there is a response δ that defeats it.
(I might be talking nonsense.)
I agree that Brilliant.org is a solid spot, but I'd recommend a maths olympiad. Find some problems, if they're too tough, drop down a grade, too easy, jump up one.
Outside of Australia it'd probably be hard to find, but there's some really nice problems in the NSW Year 12 Extension 2 Maths final example bundles, e.g. In the 2006-2016 book one of the final problems on the exam would break down the Basel Problem into a dozen smaller questions, guiding you towards a proof.
Hvad fanden hedder den der tjeneste som Youtube bombarderer mig med... Den virker til måske at være lige det du leder efter... Øhm... Yes, kom i tanke om det! https://brilliant.org/
Har aldrig prøvet det, men giv det et skud :) Og fortæl mig gerne om det er noget jeg selv burde bruge, hah.
I completely feel you but you should also understand that not everyone has the same definition of a "good riddle" as us. For many people, especially those who don't have an advanced knowledge in maths, this "riddle" can be considered good or fun.
If you are really looking for good logical riddles and puzzles then reddit is definitely not the right place. Instead I recommend you Brilliant which is an awesome website for logical questions related to maths and science.
Have you tried https://brilliant.org/?
It's mostly science and math focused, so you can say that they have a much narrower field than ANY of the others, but I find that they are more solid in those than any other too.
Not my students, but the mathematician Gauss had a great reaction to some busywork his teacher assigned him when he was 10 years old (in the late 1700s). His class was told to add the numbers from 1 to 100. He supposedly figured out the answer immediately, astonishing his teacher.
Many of the "basic" problems on brilliant.org are accessible to bright 8th graders and often hit on more interesting mathematical topics.
I was intrigued by the free material, including the problems and their wiki pages, like this one:
https://brilliant.org/wiki/cauchy-schwarz-inequality/
A question I have for people who signed up--are people fine that they display your name and age next to your profile? Here's an example:
https://brilliant.org/community/home/featured-members/
Most math communities you can stay a bit more anonymous, so just curious if this mattered to people.
Wikipedia page on factorial discusses gamma function and pi function as ways of extending the idea of factorials to non-integer arguments. The pi function for n does yield n! for integer values of n.
Are you maybe doing binomial expansion with a fractional power? That does not require factorial of nonintegers.
https://brilliant.org/wiki/fractional-binomial-theorem/
Edit: Part b in your problem is precisely this. Read the link.
I would try exercising, eating well and getting more sleep. And above all reducing your stress in any way you can. Find out what works for you to reduce stress. It could be as simple as taking half an hour every sunday to plan out what you have to do each day of the week, or doing 30 minutes of exercise or reading or computer games each day.
As for the concepts, I recently just discovered a website called brilliant.org. It has short, fun little lessons and quizzes about all kinds of physics, chemistry, maths, logic, philosophy, and more. It's really simple, and I've re-learned all kinds of concepts that I thought I had long forgotten.
There are probably more websites like this too. It's a great way to see some achievement in a low-stress and fun way, while learning back all your old concepts.
I think you should go with online courses on edx and/or coursera for more advanced stuff, and for basics you could try Khan Academy, if you want to learn formally. If you need more challenges then I recommend Brilliant. For gaining some math intuition you should check out 3blue1brown channel. I never done olympiad in math, but from my experience in computer science olympiad you should get a tutor (in group or individually) and join your local high school olympics club (if there is one), and just give it a try
Do you have a decent understanding of special relativity? Are you comfortable computing stuff such as relative time dilation between different systems, etc?
If not, tackling general relativity will be hard. And you should figure GR first, because black holes are just an outcome of GR theory. And frankly, GR from scratch is hard.
But if you just want to do simple computations based on known equations, then sure, there is a lot of material out there.
https://en.wikipedia.org/wiki/Schwarzschild_metric
https://brilliant.org/wiki/black-hole/
https://www.nasa.gov/sites/default/files/atoms/files/black_hole_math.pdf
When you square x,x, it becomes positive no matter what it was before; then, when you take the square root, it's still positive. Therefore, the answer is just |x|∣x∣, not ±x.
From brilliant
Getting a subscription to the proctored, hands-on STEM MOOCs would be great. Something like https://brilliant.org is far superior to what you can find on Khan Academy, but you might want to start with the latter for a third grader.
- Modafinal is a schedule-H drug, it is psychotropic and will affect your brain chemistry ~ you will end up fucked and worse than you were when you began. DOn't beleive me, just go and read stories of people who used it for too long and they'll tell you.
- CAT exam is actually based on what is known as Discrete Mathematics ~ it is a bastardized version of topics from number theory, discrete probability, recursion, combinatorics and luckily you can learn it (Discrete math) from several well taught anf free courses on NPTEL as well as MIT-OCW.
- The main fear of tackling this exam people have is the ridiculous uncontested crowd wisdom that you require aptitude to crack it. No, you need to know how to finish the exam in time and what you're preparing. Apart from discrete math, you need some DI skills but again, someplace like brilliant.org or several test series can help you. YOu need to develop confidence and for that you need to know/understand what is happening, no amount of modafinal can guarantee that because the performance needs to be consistent.
- What are you going to do when you find out that the IIM curriculum is actually ridiculously more difficult? If you want to join banking try european nations quant program. They cost the same as IIM but are a lot more specialist and well rounded, they will train you much more thoroughly in quantitative methods if you opt for their financial engg course and an MBA in comparison will pale(yes even from the IIM). France has been consistently providing some of the best quant traders and analysts in the world, the IIM quants/traders can't hold a candle to them because MBA curriculums are never deep enough ( same goes for several world renown MBA programs)
Brilliant's Logic course is used by many teachers, you can share it for free with your students once you get your free educator account.
Anyone have experience with Brilliant?
They are a sponsor of a youtube channel I've been watching lately, specifically Theories of Everything
Here is a simple example https://brilliant.org/wiki/multivariate-regression/.
The same principle holds for other regression techniques.
There many other modelling tools and approaches that can handle multiple responses. Just search for multivariate + desired method.
Physics of the Everyday or Gravitational Physics . You would need to create a free teacher account at brilliant.org/educators , add the courses and student to your classroom. The student can then work at their own pace at no cost. Normally is $25 a month, but is free for Educators.
Others have given you great answers, but I want to touch on the crazy Pythagoreans. Rather than type out all the goodness, I'm just going to link you to a great page about the history of irrational numbers:
https://brilliant.org/wiki/history-of-irrational-numbers/
It's well worth reading this. The history of mathematics is pretty damned interesting.
100% recommend brilliant.org. Doing a couple calculus level 4-5 problems from the Community every couple of days while I was teaching high school kept me up with calculus enough to do passably on the math subject GRE without much additional studying. (I am NOT referring to the "math section of the standard GRE")
This is a 'depressed cubic'. See brilliant.org for substitutions that reduce the problem to a quadratic equation. You should find three integer answers, one of which is indeed 9.
The combinations formula you have allows for some students not to receive any pens. For this situation you need (n-1, r-1).
Here is a link
https://brilliant.org/wiki/identical-objects-into-distinct-bins/
I found links in your comment that were not hyperlinked:
I did the honors for you.
^delete ^| ^information ^| ^<3
Nah, this reeks of venture capitalism. You can't be operating at a loss since your inception and be so consistently useful to damn-near everyone while still relying on venture-capital.
What does Uber do that Lyft doesn't do? It's sneaky money that doesn't make any sense.
This is a publicly traded company. They need to hit positive at some point, and they have to compete with Lyft. All they do is put cars from one place on a map to another. A naive programmer could write that code up in a weekend. It's CS 201 - Floyd's algorithm. Their whole company is based off an an algorithm that was invented in 1962.
Maybe there's been a translation error on some end. https://math.wikia.org/wiki/Cardinality
tl;dr natural numbers and odd natural numbers both have a cardinality of aleph null and thus they are the same "size".
Here's yet another source on the matter. https://brilliant.org/wiki/cardinality/
I don't think this is exactly ELI5 (if I knew how to ELI5 regex, I would), but it's a great answer.
Wanted to add in Chomsky hierarchy: https://brilliant.org/wiki/regular-languages/#relationship-with-other-computation-models
In fact, the underlying reason you can't just throw regex at HTML (reference: https://blog.codinghorror.com/parsing-html-the-cthulhu-way/ -- but the image is the best bit https://blog.codinghorror.com/content/images/2014/Apr/stack-overflow-regex-zalgo.png) is because HTML is a context-sensitive language and you can't parse a context-sensitive language with regular expressions.
All great suggestions here. I would say also that one of the most enjoyable aspects of physics/math is solving interesting problems. You can find lots of great problems on brilliant.org that are free(along with some really high quality content if you are willing to pay). I would also suggest you check out alcumus which has a lot of great problems to work though. You can adjust the difficulty and I find these problems to be a lot more enjoyable then the khan academy ones.
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Solving problems is also the best way to really get good at a topic. You learn math by doing math so keep in mind as you are watching videos/reading to solve lots of problems also.
Zeno’s paradox of Achilles and the Tortoise?
The paradox is basically “how can we skip past something that increments infinitely? It keeps growing!”
Check out Brilliant's practice problems and wiki. They have a good amount of problems for all of the topics you outline. You could also look into MIT OCW and Khan Academy for their practice problems.
Try to approach math with an intuitive mindset. That way, not only you will be able to derive theorems and other mathematical objects from scratch but develop a deeper understanding of mathematics.
There are plenty of resources on the web to learn math without confining yourself entirely to memorization - although some of it is required. In my opinion, visiting the following websites frequently may improve your mathematical skills:
- 3Blue1Brown - want to learn math in a fun way? If the answers is positive, this YouTube channel might be what you are looking for.
- Brilliant - in there, you'll find challenging problems and brainteasers.
- Khan Academy - great tool for learning and reviewing concepts.
You might also enjoy using Brilliant. I am doing the same thing, trying to refresh and extend what I learned in school. Math was one of my favorite subjects and somewhat regret allowing it to get rusty. Brilliant combines a recreational approach (it uses puzzles) with tailored tutorials. There is also an active community discussing each problem.
Whatever experiment you're thinking about doing, I would advise against it. Do a lot more research before experimenting with electricity. Start by reading some simple websites and books about simple circuits.
https://www.allaboutcircuits.com https://brilliant.org/wiki/simple-circuits/ http://www.electrical101.com/basic-electricity.html
I see what you're getting at, but you're not explaining it well enough. It's all about specificity/sensitivity.
The problem is that "91% accuracy" is a meaningless statistic when it comes to comparison of two non-equally-distributed populations. Look at the two extreme cases of what this might mean:
1) Straight people are identified with 100% accuracy while gay people are identified at 82% accuracy
2) Vice versa
In case 1, when applied to a real population of 10% gay people, the accuracy on the population as a whole is 100%*90% + 82%*10% = 98.2% accurate. In this case, all that happens is some gay people don't get identified as such, so the technical term for this kind of test is "specific"
In case 2, you swap the detection percents around and get 82%*90% + 100%*10% = 83.8%. In this case, you identify all of the gay people, but a ton of straight people are also incorrectly identified as gay. This is a "sensitive" test.
In order for a the test to give you any meaningful information, you need to know how specific/sensitive it is. Testing positive on a sensitive test doesn't mean much, nor does testing negative on a specific one. But testing negative on a sensitive test means you very likely do not meet the conditions we're testing for, while testing positive on a specific tests means you very likely do.
EDIT: This is a well-known fact related to Bayes' Theorem. And here is a PBS crash course episode on it
https://brilliant.org Check this out for a variety of maths topics including combinatorics and discrete mathematics. Answer quizzes, don't feel bad about mistakes. Count on answers to clarify what you don't understand. Also, their CS topics are not bad.
You also like to believe in conspiracy theories too?
Answer me, which one is most likely?
AGI exists on earth
AGI exists on earth, and it's hiding from us
For an intuition, check this:
Perhaps. The thing is that I want it to be highly specific in the direction it takes. Some platforms like brilliant.org are interesting, but they have a very unstructured approach. I want this to be something where the Indian parent can trust their children to, eyes closed, without thinking, 'yeh toh theek hai, par skool ke padhai ka kya hoga?'. This needs to be unabashedly Indian and for Indian students.
Another thing is that the students , except for the prodigies, seem to be getting 'hooked' to learning for learning's sake far later than is ideal. I want them to have and retain that attitude from day zero, which means that the system needs to be able to nurture from the stage of day zero -- without clashing in the future with school's stuff.
As for unacademy, I fear this would be buried under the massive number of IAS and similar courses; the content of CDALI would be far too vast to not have its own platform.
Fun fact:
This can be done with any two relatively-prime sized jugs (that is, the jugs are m-gallons and n-gallons with gcd(m,n) = 1) for any (non-negative integer) amount of water you want, as long as one of the jugs can hold it.
Example:
Given a 4-gallon jug and a 9-gallon jug, we can get exactly 2 gallons of water by the following (using a similar naming scheme):
Fill 4GJ.
Pour into 9GJ.
Repeat 1 and 2 until 9GJ is full.
Empty 9GJ.
Pour the remaining water from 4GJ into 9GJ [at this point, this should be 3 gallons].
Do 3 again.
4GJ should now have 2 gallons.
This is a consequence of Bezout's lemma. The algorithm is basically the Euclidean algorithm.
It is pointless, but I had fun for a minute contemplating what the formula does and how you derived it. So I guess you could say it wasn't worthless.
You might enjoy learning about Fourier series.
Before I continue any farther I should probably label what a paradox even really is. A paradox is anything that would contradict itself and still seem like it should be true even if it's not. Paradoxes are quite real because they are merely concepts of how we process information.
The concept of a paradox also exists in more languages that English. Paradoxes date all the way back to Greece with the most famous being the "all powerful" making an "unliftable rock."
Mathematical paradoxes can prove this and there are plenty good ones that Brilliant.org has https://brilliant.org/wiki/introduction-to-paradoxes/ I'll let you take a look at that to see a few examples of what I'm talking about as well as a few anecdotes. They come in mathematical form, anecdotes, statements and even as optical illusion images.
Also in your example above it would have to be that one of the statements claimed the other to be true for it to be paradoxical. "Statement A is false" "Statement B is true"
It doesn't matter when the information is given or received even to determine how useful it is. Contesting validity of information in time doesn't work. Linear thinking is how computer brains function. There is never a need to restate information given to process. Visual paradoxes can show us just that: https://1millionmonkeystyping.files.wordpress.com/2013/11/currys.jpg
Oh also I owe you a big thank you for your patience and understanding. Internet debates are scary and many people are rude during things like these. I appreciate that we have kept it civil :)
You’re correct about division being multiplication by inverse. However, taking the inverse is a mathematical operation and to say “doesn’t have an inverse” isn’t quite meaningful.
Here’s my take:
Option 1: xa =x is simplifiable to a=1 by dividing both sides by x. What’s curious is that now you’ve established a value for a without ever using a value for x. Now if you plug a=1 into the original equation you get x=x which makes sense.
Option 2: If you don’t simplify the original equation xa=x, you can still solve for a: a=x/x. Now here we know that x/x =1 for all values except x=0.
If you followed me through that you will see we end up with 0/0, and that is an undefined number.
I think you're mixing up the idea of "problem" with the idea of an "exercise." An "exercise" is repeating a pattern or process you've already learned, in a context that is familiar to you already. Doing a page of factoring exercises is an example of what you're describing.
But a "problem" is a mathematical question that you haven't learned a process to solve, but you can use processes you've learned to solve it (if that makes sense). For example, if you've taken some simple geometry, this is a pretty decent problem that only requires you to know a few things, but it's how you put them together that takes creativity.
I'd agree with you -- once one type of "exercise" is familiar to you, there's not much point in doing a lot of them all at once. It might benefit you to go back to a couple similar problems a day later, a week later, and then a month later, to improve your memory, but not much more than that. (It's called "spaced practice" and improves how much you remember!)
But if you want to improve your mathematical thinking in general, doing a lot of problems is absolutely necessary! If you're 11, I'd strongly suggest puzzling yourself over the 100 day summer challenge this summer :)
If we were to definite a paradox as a seemingly impossible or counterintuitive formulation or concept, I think Grabriel's Horn is a good example. It is a geometrical object that resembles a horn (hence it's name) with infinite surface area but finite volume.
That means that if you wanted to paint said object you would need an infinite amount of paint but if you wanted to fill it with water you could do it with a finite amount.
More info and the maths behind it here
Wilson's Theorem IIRC is 100% accurate and precise when it comes to prime detection. However, due to computational inefficiency, it's awfully worthless in that regard. It's more useful in things like Olympiad math problems or related problems involving modulus. It can be used alongside the Chinese Remainder Theorem to solve some pretty interesting problems.
We actually do some have content on abstract algebra, but it's still very much a work-in-progress. You can search for stuff using the search bar on the top of the main page (though for something like fields, the quizzes and problems will return a lot of Physics stuff on electric/magnetic fields).
Much of our wiki content tries to write from the perspective of why stuff works from an intuitive perspective, which is something that is often lost in the drier explanations on e.g. Wikipedia. I see this as the main difference between Brilliant and other math-content sites, along with the integration between wikis and quizzes.
It depends on what you mean by impossibility. For true "impossibility", you're usually talking about a logical contradiction (e.g. you've just shown that 1=2 or something), and we'd just say "and this is impossible, so blah blah" depending on what we could conclude. Basically, it means that your assumptions were wrong in some way. See this writeup about Proof by Contradiction for some examples of how we can use this.
However, in both of the cases you've mentioned (3/0 and sqrt(-11)) what you have is actually not a "mathematical impossibility" but instead a situation where the thing you've found isn't covered by the definitions you're working with. So in an Algebra 1 class, you'd say that sqrt(-11) is undefined because the square root has only been defined for positive numbers. However, if you were in slightly more advanced maths, you'd treat sqrt(-11) as a complex number and there wouldn't be any problem. This is simply because it becomes useful to define the square root of negative numbers in many situations.
(For 3/0, it pretty much always stays undefined, except for some specific situations in calculus where (with some additional context) that expression might reduce to some other value.)
Anyway, TL;DR: the thing you're looking for is "is undefined". That's the shortest/easiest way to describe the things you've asked about.
It's a website where users contribute mathematics, compsci, and science problems in a number of different areas open-source and users solve them in progression.
I think it could be a great tool for enhancing your basic mathematics skills, but it doesn't look like anyone's contributed many physics problems yet.
There's one interesting difference: Derren Brown had no influence over the race winners, whereas here there might be some feedback from the prediction to the predicted.
Another of my favourites is https://brilliant.org/wiki/k-level-thinking/
permutations. So from your link, thank you for that.
I have a set of 12, let’s say units of something in different colours For ease of understanding. In how many ways can I arrange these numbers? 12 x 11 x 10 …. X 1 = 479k combinations I want these to be actual outputs of possible combinations of these 12 values to be put into text. So I can analyse the data.
I still can't figure out what you want your output to actually be with 12, 12, 12, 12, in that list you've given. Are you looking for all possible combinations of 4 attacks or ?
Here take a look at this.
This problem can be solved with backtracking. You can use algorithms such as depth first search or breadth first search to solve it.
Here is an excellent writeup of how depth first search works. https://brilliant.org/wiki/depth-first-search-dfs
You may be wondering what mazes have to do with adding numbers to 100. Well, you can picture each operation (+, -, or concatenation) as an action, or a step in the maze, and you can go from left to right starting at 1 and ending at 9 choosing one of the three operations at each step. With some imagination this is like solving an abstract "maze" to reach 100 at the end. If you don't reach 100, you can undo the operations you chose (backtrack in the "maze") and try the other operations.
"Instantaneous rate of change" is a figure of speech and an oxymoron. I say if you find the phrase confusing just forget about it.
The derivative can be defined as a limit. Limits are easy to define in formal mathematical language ("for all epsilon greater than 0 there exists a delta greater than 0 such that...") but notoriously difficult to define in plain English. Your best bet might be to just hunker down and learn the epsilon-delta definition. Mathematicians didn't get this concept right until about 200 years after the invention of calculus so it's understandable if you find any of this tricky.
What Grade level?
First I will share my thing. Brilliant for Educators has interactive and engaging STEM curriculum for 6th-12th grade, with courses not often seen in schools like "Logic" or "Physics of the Everyday".
Teachers can apply for a 100% free account at brilliant.org/educators
Code.org has lots of free resources that are math adjacent. Mathsnacks.org has fun games for students that are math related. Would be happy to share more once I know the grade level.
No problem. These puzzles are very difficult. I've spent hours on some making incremental headway (like puzzle 21). Other's just click after a few days when I'm doing something else (like this one).
Take a look at this link: https://brilliant.org/wiki/pattern-recognition-visual-easy-2/
It takes you through some steps in pattern recognition that are used commonly.
Over time you'll build up a knowledge base of solution patterns that'll act like a tool box for solving other problems.
Good luck.
Eh. I don't want to have to type it all out, besides not having certain symbols at my disposal. But if you have any interest in learning how to find the cube root of any number without a calculator, it just requires a bit of memorization to start with, and a little mental juggling. But with practice you can find the cube root of just about any number with a trick you can read about here, at Brilliant.