70% is fine isn't it? Our Linear Algebra classes have a ton of people failing. With 70% you'd probably get an A. I am a first semester at college and linear algebra 1,2 and Calculus (idk... 3,4) is required for our comp sci major.
you probably aren't practicing older material enough. I find that if I exchange a majority of my studying time before exams to solving problem sets from some of the older chapters that the test is using I do better.
You should work to understand the whole concept of your test subjects a week before the test, and only be reviewing/practicing the week leading to the test. I learned this during my more extreme classes, some concepts are too difficult to just learn a day or two or three before the exam.
I have a love hate relationship with math. I love learning new concepts, but I hate it when you think you understand something, and then the next problem throws a wrench in it. Looks like the ones in college magnify that
In some ways, I think the opposite is true. Up through the mathematics I learned in (engineering) undergrad, but especially in high school and earlier, abstraction was mostly avoided for fear of confusion and lost interest. But proper abstraction is exactly the way to keep wrenches out of your math. The wrenches that used to frustrate me ("why is that true?" -> "well, to understand that you need to know more about X,Y and Z") are removed when you see each theorem derived _as a result of axioms/assumptions_. In properly abstracted mathematics, you always know exactly how much you understand, and how much you don't. So, I think there's hope.
Funny I was looking at the course list and thinking "hey this looks exactly like the courses offered at my school" only to find out it is my school lol. Cal Poly stand up
1:50 "This is intimately connected to what I call the “ladder myth”- the idea that mathematics can be arranged as a sequence of “subjects” each being in some way more advanced, or “higher” than the previous. The effect is to make school mathematics into a race- some students are “ahead” of others, and parents worry that their child is “falling behind.” And where exactly does this race lead? What is waiting at the finish line? It’s a sad race to nowhere. In the end you’ve been cheated out of a mathematical education, and you don’t even know it. " - Paul Lockhart
I think there are many other reasons why most people who get cheated out of a math education, get cheated. And I think they’re far more apparent than the structure of the curriculum. I like this structure. I’ve learned so much from my classes as a result of this structure outside of simply “math aptitude”. I’ve learned how to be a far more effective student than I could have been without it. through this structure I’ve acquired a baseline level of competence in something. This competence has instilled way more confidence in my intellectual ability. I did kind of view this structure as a “race”, perhaps as a result of the structure, but after maturing a bit I’ve found it to be way more complicated than that because life is complicated. Honestly regardless of structure people are naturally competitive and anyway you try to structure a curriculum some want to be part of the 1% of exceptional students that are “ahead”.
Really enjoyed watching this. I earned my Ph.D in chemical physics 4 years ago. While your video is very informative, I don't think you realize how much abstract fields in mathematics play into chemistry in physics. Graph theory is heavily utilized in attempting to understand covalent bond network in organic chemistry, especially when you introduce molecular quantum theory. Group theory, from abstract algebra, helps tremendously in calculations of electron density of complex molecules. Topology, well, it's used in theoretical chemistry in helping define quantum mechanical wavefunctions assignments to bond types. Matter of fact, part of my research was applications of topological analysis on quantum mechanical wavefunctions.
Congratulations on ur phd and thesis. Im doing physics and hope to do a phd in a theoretical topic (i have literally no idea which area). is the dover book on intro to abstract algebra a good book? if u havent read it; what would you reccomend as an intro?
@@fernando8246 Electron density is how electron dense different regions of molecules are, which allows you to predict their interactions with other molecules of interest. Chemical interactions are entirely dependent on what electrons are doing within systems. Maybe you're interested in curing diseases or making better solar panels, these kinds of things require that you understand electron density of molecules.
Isn't the entire electron cloud orbital shapes based purely on quantum mechanical calculations? I got out of O Chem as fast as possible so I could focus on Biochem and Human Phys.
0:49 Calculus and Linear Algebra 1:53 Math is a Ladder, Then A Tree 4:37 Geodesics, Differential Geometry Complex Analysis 6:02 Numerical Analysis 7:48 Abstract Analysis And Pure Mathematics 8:43 Topology Topologically The Same 9:13 The Möbius Loop 13:03 14:15 Engineering Computer Science
I am 98% sure you wrote this, so in future when you will come back, you will know where to go. Actually, I'm freshman and I would like also to timecode everything important. Thank you, bruh!
Its a real shame that most kids don't ever get to see the real beauty in mathematics. Our school system is so devoid of any of the passion that originally belonged in the subject, that 90% of the population believes math is just calculating numbers.
Eh, how do I put this, most of the beauty of maths comes after a lot of necessary foundational knowledge, and unfortunately a large part of the reason children don’t get to see that beauty is not because the school is bad but because they are genuinely just too dumb to get a good grasp on even those foundations
@@vilmospalik1480lol no it’s because many teachers are garbage in grade school and schools in general don’t take math seriously. Being in that environment sucks the joy out of learning, especially math
I love how you portray college as an opportunity to pursue your passion for knowledge and expand your horizons, versus just going to get a good job on the other side. That’s how college should be, and if you don’t think that way you’re probably better off going into a trade, which is probably financially smarter anyways.
This is bullshit. The school he used as an example is absurd and unique. Most colleges will have those MINOR requirements as their end goal MAJOR requirements. The stuff he says is for a math major are graduate school courses...
Awesome video! This brings back sweet (and equally frustrating) days while I studied for my major in physics and my minor in mathematics. I really enjoyed calculus 3, linear algebra (I remember it being easy, till the second half of the book when it got into 4 dimensions. ACK!), and complex numbers. Haha, I remember sometimes doing math that was really complex (for me anyway) that at the end, I had no idea what the answer meant. Anyways, keep up the great work and I wish all you math majors and minors the best! :-)
I like the way, your videos now follow a red line from the conclusion. You said instead of taking technical courses like DSP in senior year, you would have taken some physics or high level math courses and you now take the time to explain what it is about
Zach, consider doing a video on some of the awesome books that give great overviews for math enthusiasts, such as _Mathematics: Its Content, Methods, and Meaning._ These types of books can be great stepping stones to more involved texts.
I certainly enjoyed the high-level descriptions of these vast and nourishing mathematical realms. Given my electrical engineering and physics academic background, I've already dwelled upon most of these topics in varying extents and definitely agree with the approach and the examples. This is a really great guide for scientific newcomers to get a hint on what their preferred niche is. Much appreciated Zach, and I'm certain this single piece alone will do wonders for many undergrads. Kudos^2!
I've always been strong in math and will be attending university this fall for Mechanical Engineering. However, I have not exactly loved math until recently. In our school system, we do not learn how to find the area under a curve in calc until university. So, I asked a teacher at my school with a major in mathematics about it and he gave me a great textbook to learn it from after a brief example. At this point, I had never gotten involved with calc yet. So I taught myself Lim of functions, derivatives and finding area under a curve. This may sound boring but learning it all was actually somewhat beautiful. I went through these subjects in 2 weeks, in comparison to the 5 months our school would devote a 75-minute class to every day. Because I enjoyed being able to do things that seemed almost out of reach a few weeks earlier. And now I am considering a minor in mathematics once I go to university so that I can learn to love and understand more of math!
That's what the proofs class was for us (although computer scientists did have their own discrete math course just for them). I did multiple videos on discrete math so didn't go into much depth on it.
At my university, math students also don't have Discrete/Combinatorics required, but cs students do. Well, it's a part of math that is probably more important to them than to us.
Duuuude I had to take that class a second time to pass. Maybe it was my professor but screw that class. Though, I can see how one can get nerdy on combinatorics and I found problems involving Fibonacci numbers quite fun too.
Analysis classes have made my lab reports a lot more rigorous cause I learned how to build up from what I had and deduce stronger conclusions. Making links between results and the discussion part was a lot easier after this, although I failed one of them and cursed them all the way through X)
For mechanical engineering, differential equations and matrices should be a must, understanding how those two work will make some of your upper division course a breeze
in my undergrad I was a double major in psychology and mathematics and was floored by the amount of mathematics I saw in my senior level psych classes when learning about cognition and robotics/ artificial intelligence (things like fourier series, graph theory, game theory, group/ set theory, category theory, number theory, and a lot of abstract algebra)
I think this video did a good job of separating the proof classes from the applied class, however I’d just like to point out that all of the applied classes are based on rigorous proofs and at the graduate level include proofs. Therefore, depending on the school, an applied class the video listed may be more proof based. For example, most universities’ topology classes are very proof based. Also, I think this university’s math minor is very proof light. Most schools require at least intro to real analysis, and proof classes require a very different set of skills than applied math classes. I know several engineers who thought it’s only three/four classes more for a math minor and take analysis and then drop their minor. So if you’re an engineer, take the proof courses seriously.
You're full of shit. Most schools do not even OFFER real analysis. That is grad school shit. At least in america, those "minor" reqs are major reqs at most schools, with the addition of abstract algebra
nothing First of all, there is no need to be rude. Real analysis is a requirement in almost every math major (with the exception of some liberal arts colleges that shouldn’t even offer a math major if they are not going to require any actual math) and most math minors. I can send you the requirements for my school’s math major and minor if you’d like and I have looked at dozens of other schools’ requirements and I can send you theirs as well. Real analysis is not “grad school shit” and if you don’t know what you are talking about, you don’t have to be rude to people who do.
First several school major requirements that popped up in google after searching for: bachelor's in mathematics requirements. math.arizona.edu/academics/undergrads/requirements/mathematics catalog.unc.edu/undergraduate/programs-study/mathematics-major-bs/#requirementstext catalog.unc.edu/undergraduate/programs-study/mathematics-major-ba/#requirementstext www.sas.rochester.edu/mth/undergraduate/ba-requirements.html coursecatalog.syr.edu/preview_program.php?catoid=3&poid=807&returnto=253 catalog.buffalo.edu/academicprograms/mathematics_ba_requirements.html mathstat.umbc.edu/files/2014/04/advising-Math-BA.pdf mathstat.umbc.edu/files/2014/04/advising-Math-BS.pdf At first glance, only one of these schools even offers an analysis class. They almost all follow the standard curriculum: Calc 1-3, Differential equations, Statistics, Linear algebra, Abstract algebra, intro to CS + electives.
I was introduced to the moebius strip in a magic, math and illusions book. "Make the strip. Now try to draw a line down the middle. The "edge" is "twice as long" as before joining the ends.
I graduated in computer science, but fell in love by engineering maths. I took diff. eqs., Fourier analisys, complex analysis, numerical analysis. However, I had to take also number theory, graphs algebra for computer science degree. I realized that I have only few disciplines to get a math major.
I perform adequately in mathematics, essentially, "pure math". Invariably, I struggle to apply this material to real world problems. As an engineering student i'm considering switching to mathematics. Does anyone else struggle with this? Any advice on switching from engineering to pure mathematics? Thank you and great video!
I don't know about switching majors and what challenges you might face but prepare for a nasty surprise when you finally get your maths degree. I got one of those and I got so frustrated looking for jobs that I became a freelancer. Unless you'll stay in academia you will not have a lot to do with pure maths at all. Most of the jobs I applied for actually wanted a programmer and the other 10% could be done with basic maths skills.
"Complex Variables" by John W. Dettman is a great read: the first part covers the geometry/topology of the complex space from a Mathematician's perspective, and the second part covers application of complex analysis to differential equations and integral transformations, etc. from a Physicist's perspective.
Integral Equations: Fourier & Laplace Transforms, Green Functions, etc. These are used a lot to solve PDEs (Jackson's "Classical Electrodynamics"!) and model systems rather than differential equations.
No lie, having a minor can’t hurt, getting my nursing degree and having a Chemistry minor and taking Calc 1-3/physics (NOT required) for nursing out of curiosity, it helped me secure a slot in CRNA school. When my admissions counselor did the interview, he was like my, my, you certainly are an ambitious one sir! lol! I only took those classes because I love science, nothing else, had nothing to do with climbing the career ladder. I do despise folks like that however, because they don’t care about anything, to include the job they are paid to do.
1:20 the easy way: fill up the 32-gallon jug by completely filling up the 6-gallon jug 6 times this will leave you 4 gallons in the 6-gallon jug. Then pour out the 32-gallon jug and pour the remaining 4 gallons into the 4-gallon jug, finally fill the 6-gallon jug and pour it into the 32-gallon jug.
i went from only being able to do very basic algebra equations (like barebones) 4 weeks ago , and now i’m headed into algebra 2 and trig in my rush to linear algebra i gotta say, math is pretty damn cool
even u are not studying the above subject I recommended linear algebra, caculas and statictics since we have utube, Udacity and open course anyone can learn any Maths subject very easily
Electrical Engineering does that. It did the same to me. It is awesome to see how mathematics manifests in something tangible and physical applications that follow from it.
My daughter only needed 1 class to add a Math minor to her Honors CS degree - Real Analysis. Honors CS required MVC/Vector, Linear, and a CS proofs class. To get a Math major is only four more - Elements of Algebra, DiffEqs, Linear 2, and Complex Analysis. Among many of the “pick 2 courses from this list” for Math are several CS courses she already needs to take (Numerical Methods, Algorithms, Theory of Computation). She’s still contemplating this.
I had been to school most all the time, and could spell, and read, and write just a little, and could say the multiplication table up to six times seven is thirty-five, and I don't reckon I could ever get any further than that if I was to live forever. I don't take no stock in mathematics, anyway.
I’m majoring in math and minoring in computer science. Thank you for this informative video as to what extra classes I need to take in terms of math and computer science together to be needed for my future job, which is cryptographer.
Even though I received my undergrad in Math many years ago I was curious to hear what you would say about Game Theory, the course which caused more pain and suffering than any other math course I ever took.. you almost made it sound pleasant for that half a second, but don't be fooled fellas haha.
I honestly cheated in Algebra I in eighth grade, and I harmed myself from there. Once I realized how many foundational skills I did not know, I decided to study my algebra foundations. I’m now in Pre-Calculus, and I love the class. Math is fun once you change your perspective and think of math as a game with rules you have to follow.
Glad to see I'm not the only 8th grader watching this. I am in my 8-9th grade summer break right now. I took Algebra 2 and Precalculus (at same time) and didn't do the homework for Algebra 2 because I was so busy in Precalculus, but in my school this year there was a new grading system which didn't count homework and no points were taken off on late (In my highschool, it would just be the regular grading system). I am planning to take Calc BC for freshman (because I want to learn about infinite series). I'm not sure if I'll do good, but I've always had a passion for math, biology and chemistry. Any tips would be nice
Counterintuitive fact: the thing you got when you cut the Mobius strip in half ( 10:33 ) is indeed not a Mobius strip, but it is actually (homeomorphic to) a completely untwisted thing (a cylinder). It's true that you can't deform this to an untwisted thing (it's twisted), but you could actually untwist it if the ambient space was 4D. (At the very least, I'm sure you could untwist it in 5D; I bet it's also true in 4D, but I'm too lazy to think to prove it.)
Wanted to point out that abstract algebra has application in computer science. For example, category theory is a core foundation of functional programming paradigm, and group theory principles can be applied in some algorithms
I am on a maths journey at 37 years old at pre-calculus, liner algebra and calculus level I want to study more advance maths topics after khan academy calculus's subject, I also use cosmo learn and youtube, I want to start when I am now but after calculus 3 does anyone know any advance, really advance pure maths, advance or higher pure maths topics I really want to get into it, because I am into physics now.
I am also studying myself math for fun I finishieed undergrad abstract algebra and galois theory have some topology and finishing measure theory my next goal is differential geometry I m proving everything I do
Seriously, people have got to stop saying "Calculus 1, 2, 3, etc". In the pure field of Mathematics, there's just Calculus. The topic is never divided into "1, 2, 3" nonsense. That's just a system used by schools and colleges to organize their subjects. No such grouping is used in the pure field of Mathematics.
Look at the class descriptions of universities, they spell out a lot of the topics math students learn within each class, their names alone might be enough to research more on your own. As for resources that help understanding post-Calculus concepts, the RUclips creator 3blue1brown has amazing videos on just about everything (my favorite is the one on Fourier analysis). Or maybe pick up a textbook online
@Steven Lee I don't think it's ever to late. My dad got his degree in electrical engineering at 34. Then he went on to get his masters. And you know what? He did just fine. It's never too late.
1:27 fill the 6 gallon from the 32 gallon until the 32 gallon has 2 gallons remaining. 32 - 26 - 20 - 14 - 8 - 2. Refill the 32 gallon. Fill the 4 gallon from 32 gallons with 2 gallon remained to fill 6 gallon. Now 32 gallon has 28 gallons. Repeat filling 6 gallon from 32 gallon. 28 - 22 - 16 - 10. Now have 10.
The small loop showing up kinda makes sense if you realize that each side is being taped to the opposing side the second time round. Since the middle third is just attached to itself, it gets cut off from the other portion, meanwhile the left third connects back to what was the right third and the right third connects back to what was the left third. I didn’t predict it correctly, but the reasoning seems fair enough as an explaination, for anyone who didn’t quite get what was going on with those rings. Basically, splitting the loop into three segments basically isolated that center loop, which then disconnected.
i am really confused, i've never seen a math lecture without proofs, even in something very applied like numerics. How in the world would you go through a class in Calculus (let alone 3) or linear Algebra without being able to proof even the simplest of statements? isn't the point of those classes to proof statements in Calculus and linear algebra? I mean what do you learn instead? Also someone studying Physics for example would also need to know about proofs and technical stuff to be able to survive right? how does that work?
Lorch95 at my university classes that are not proof based still show the proofs in class to explain how something work but the students themselves don’t have to write proofs and the in class proof are usually less formal
Lorch95 demonstrating how theories are used and why they work (or giving a special case "proof" that shows it is true) doesn't often require actually formally proving it. In Calculus in my uni at least, it's more of the former as a formal proof class is not a prerequisite and proving a lot of calculus theories supposedly involve higher level math. Though I suppose one could make the argument that the solution to *any* mathematical problem is a proof, formal or not.
my calculus 1-4 classes leaned more towards mechanically calculating expressions involving derivatives and integrals, usually through real-world applications. we would either need to demonstrate or simply assume that statements were true, formal proofs were few and far between
That Möbius Strip demo has me feeling smart because I guessed both questions right... but then also because I got it right it also doesn’t feel major 🧐
@1:24 it is totally possible. Let's say you have a bucket that you want your 8 gallons to end up in. Thus if I say pour into bucket, I mean pour the water currently in the jug into the bucket. If I say empty it, I just mean pour it down the drain or something. Step 1: fill the 32 gallon jug Step 2: use the 32 gallon jug to fill the 6 gallon jug Step 3: pour out the 6 gallon jug into the bucket Step 4: use the 32 gallon jug to fill the 6 gallon jug Step 5: empty the 6 gallon jug Step 6: repeat steps 4-5 exactly 4 times Step 7: Empty the remaining 2 gallons in the 32 gallon jug into the bucket
for the gallon problem, fill the 32 gallon bucket with the 6 gallon bucket. After it's completely full, the 6 gallon bucket will have 4 gallons of water in it. Empty out the 32 gallon bucket and pour the 4 gallons into the 32 gallon bucket. Then fill the 6 gallon bucket and pour it into the 32 gallon bucket.
Pure Mathematics (algebra/topology/geometry etc.) is pretty boring for me personally. But I'll be honest, it's partly because it's difficult. Applied Mathematics is way more interesting but that's because I enjoy applications of Differential Equations, Fourier analysis, Boundary Value Problems etc. (Analysis).
This video gave me a really pleasant nostalgia over how unintuitive higher knowledge always feels at first…I love that reality gets more wild the more sense it comes to mKe
Graph theory at my school was ridiculously difficult for me. Unlike differential equations, they’re mostly proof based, abstract, and no straightforward “‘methodology” or simple approach to problems. Each problem is so open ended and there’s nothing that hints at which proof method will work best, and not only that, but sometimes you CAN prove something using one technique, but it may be 10x harder than another technique that you hadn’t considered. Not to mention the mental strain and focus you NEED to keep your logic in order. Probably my hardest math class ngl
So at 5:35 I saw you talking about the proofs to some of the trig identities inclusive of the sum and difference identities. I just took precalculus (so I'm still quite low level math) but in class we actually proved the sin and cosine sum identity using a triangle inscribed in a square where one corner of the triangle is also a vertex of the square. Hence, we have angles α, β, and 90-(α+β).
There should be another specific set of math courses offered as standalone courses for these majors / minors. These advanced courses should be specifically based on Fractals both numerically and geometrically. Another mathematics course that should also be in this list is Boolean Algebra.
One advantage of a minor compared to a double major is that you don't need to do extra GE classes. At my school, math is in the school of arts and sciences and computer science is in the school of engineering. Arts and sciences and engineering have different GE requirements, so a math double major would be mostly GE. Minor actually lets me take more math.
if anyone was wondering about the 32g 6g jug one, what i came up with was filling the 32 with 5 6g scoops, for the last scoop to fill it all the way save the excess now you start with a 4/6 filled 6g or 4gs now empty the 32g and fill it was the 4 and another 6
What do you mean with unsolved? It has (several) solutions. The only thing that's not clear is whether there are any polynomial time solutions (is P=NP?), but very probably not.
If you’re reading this as an engineer thinking about a math minor in telling you it’s absolutely rough. My experience is different than most but I am currently taking my higher level engineering courses and about to graduate but doing a minor in applied math and at my school it requires applied statistics, discrete math OR, the straight linear algebra course (on top of the diffeq linear algebra hybrid you take for engineering) I would take discrete math if you are thinking about it because it helps greatly with understanding proofs and I unfortunately did not take that and the linear algebra course at my school had proofs but they wernt on the level of rigor that I was ran into after, so after that you need a “rigorous” proofs course which as I mentioned kindof leaves off from discrete and I really struggled in this class because it was so unlike the engineering classes. Currently I’m in the last class which is advanced calculus which some universities call “real analysis” but that term should be reserved for graduate level analysis because apparently it’s another beast. My prof told me that this class in undergrad math is commonly called “baby analysis”. I also found out that this is regarded as the toughest class in a math undergrad. Currently taking 4 classes and 2 labs this included and working and I am absolutely swamped and struggling but surviving (I hope). He has a video about self teaching pure math and funny enough we are using the same textbook “Elementary Analysis” by Kenneth Ross. I’m the only person in the class not going for a math degree and most of my peers seems to lighten their workload when they take this class but not me lmao. If you do what I do I suggest you self teach and get ahead by reading the textbook BEFORE the semester starts and actually work out problems and make sure your foundations in proof techniques are solid. Direct proofs, proofs by contradiction, induction, and predicate logic, things like negating compound statements etc. this class requires a ton of tricks and of course a very solid algebra toolkit because this class relies on these “tricks” for proofs. Brush up on Calc 1 and 2, sequences especially but also stuff like series tests from Calc 2 and you will be okay. Absolute values and sequences and the triangle inequality are heavily used in analysis.
Acid Plug If you don’t understand what’s going on, see the professor during office hours. Math is hard, but it’s like learning the piano. When you start off with learning the song twinkle twinkle, you think it’s really hard but eventually you move onto learning harder songs and think playing twinkle twinkle is easy. Keep it up and work hard and you’ll be looking back at college algebra as easy! Good luck!
Definitely go to office hours if you struggle with math. But I also want to point out something else that a lot of students seem to do. They tend not to _think_ when doing math. They view it as a very algorithmic process where you just have to memorize a bunch of random stuff and repeat it. That's not a good way to do mathematics. You really need to think about what's going on and what everything means because everything in math _makes sense_ - there's a reason _why_ everything is the way it is. Sometimes that reason may be too complicated for you to understand, but most things you encounter should be in your grasp as soon as you see it. You'll definitely need to learn some things on your own, but if you are really struggling with a concept, ask _why_ something happens. Here's an example of something that confuses people: graphical transformations of functions. Let's say you have a function f(x). It's generally not too difficult to see that f(x)+2 shifts the graph of the function _up_ 2 units, and that f(x)−2 shifts the graph of the function _down_ 2 units. But what about f(x+2) and f(x−2)? Most people initially think that f(x+2) should shift the graph of the function right 2 units, and f(x−2) should shift the graph of the function left 2 units, but it's actually the exact opposite: f(x+2) shifts the graph of the function _left_ 2 units, and f(x−2) shifts the graph of the function _right_ 2 units. Most students at this point are content to just memorize that: if you change the inside of a function, it does the opposite of what would happen when you change the outside. But if you want to understand math, don't be content with just knowing this information: try to think of _why_ this is true. So what you can start to try is to look at some simpler functions that you kind of understand. Like f(x) = x. It's a nice line passing through the origin with a slope of 1. Now look at f(x+2) = (x+2) (replacing x with x+2). Now try to graph this function next to f(x) = x, and notice how it is shifted 2 units to the left from f. Now look at f(x−2) = (x−2) (replacing x with x−2) and try to graph it next to f(x) = x and notice how it is shifted 2 units to the right from f. And maybe try a more complicated function like f(x) = x^2 or f(x) = |x|. Try graphing each of these and then try graphing the versions where you replace x with x+2 and with x−2. A nice way to sketch a graph is to try a couple different x-values and find the corresponding y-values and plot all of those points. And after plotting these, maybe you'll start to get a sense of why this is true in general. The point here is that f(x+2) and f(x−2) are technically new functions, right? So let's treat them like new functions. Rename f(x+2) as g(x). In other words, g(x) = f(x+2). Now, for example, if we plug in x=0, we get g(0) = f(0+2) = f(2). Or if we plug in x=5, we get g(5) = f(5+2) = f(7). And no matter what value for x we plug into g, we get that its input is 2 numbers to _the left_ on the number line for the same output on f. So you've shifted f to the left: you get the same output as f, but 2 units earlier! And similarly, let h(x) = g(x−2). Plug in x=0, and you get h(0) = f(0−2) = f(−2). Or plug in x=5, and you get h(5) = f(5−3) = f(2). And no matter what value for x we plug into h, we get that its input is 2 numbers to _the right_ on the number line for the same output on f. So you've shifted f to the right: you get the same output as f, but 2 units later! What you notice here is that g(x) = f(x+2) and h(x) = f(x−2) is that you're affecting the _input_ value of the function. So you could think: If I start with g, how can I get _back to_ f? What do I have to plug into f in order to get the same value as g(3)? Well, notice that g(3) = f(3+2) = f(5). So you need to plug 5 into f in order to get the same value as plugging 3 into g. So notice that in order to get _to_ f _from_ g, you have to shift two units to the right. So in order to go backwards _to_ g _from_ f, you have to shift two units tot he left. So that's the sort of thing you can do if you don't understand something in math. Try a few examples and really think about _why_ things are true. Try thinking backwards too (like what I did at the end there: instead of thinking about what you have to do to get from f to g, think about what you have to do to get from g to f). If you can figure out the reason on your own, you'll really remember and understand it much better! But sometimes it's really tricky to think in the "right way" to make the reason clear. That's when you can ask in office hours or ask someone else who may understand the topic. Yes, this is time consuming, so you won't be able to do this for every topic. But if you want to understand math, this is the sort of process you should strive to do as much as possible.
Honestly I think the main issue with math is that children are given problems and not much real world applications to it. I had the same issue in high school but once I started taking physics everything started making sense to me. And once I started calculus in college I went in a bit confused from precalculus but I felt that calculus just made everything easier to understand.
Additionally in some cases mobius strips are used in belts for drive systems. This is becasue they have one side and can provide uniform wear (material removal of a surface due to friction).
Abstract algebra is the foundation of functional programming which has significantly increased in market use in the past few years which means it can definitely be useful for computer science.
I love math, my problem is i only finally understand the whole concept AFTER the exam when I get a 70%
70% is fine isn't it? Our Linear Algebra classes have a ton of people failing. With 70% you'd probably get an A. I am a first semester at college and linear algebra 1,2 and Calculus (idk... 3,4) is required for our comp sci major.
Yeah this is a good grade still
you probably aren't practicing older material enough. I find that if I exchange a majority of my studying time before exams to solving problem sets from some of the older chapters that the test is using I do better.
You should work to understand the whole concept of your test subjects a week before the test, and only be reviewing/practicing the week leading to the test.
I learned this during my more extreme classes, some concepts are too difficult to just learn a day or two or three before the exam.
We all do. Don't ever feel discouraged by evaluations. You've even said it yourself: now you get it.
How hard was it to reject the line of girls after showing them the Mobius Strip tricks?
i'm dead lol
Honestly i would be in that line
This actually made a girl crush on me in high school.
Trans Tetris oof
easily the best comment of 2018
I have a love hate relationship with math. I love learning new concepts, but I hate it when you think you understand something, and then the next problem throws a wrench in it. Looks like the ones in college magnify that
haha I think many people would agree
Figuring out how to remove the wrench is what the actual mathematics is.
In university, it's all wrenches.
That's what progress feels like. The more you know, the more you realize you don't know.
In some ways, I think the opposite is true. Up through the mathematics I learned in (engineering) undergrad, but especially in high school and earlier, abstraction was mostly avoided for fear of confusion and lost interest. But proper abstraction is exactly the way to keep wrenches out of your math. The wrenches that used to frustrate me ("why is that true?" -> "well, to understand that you need to know more about X,Y and Z") are removed when you see each theorem derived _as a result of axioms/assumptions_. In properly abstracted mathematics, you always know exactly how much you understand, and how much you don't.
So, I think there's hope.
Hahahahahaha an ad for antidepressants popped up. They certainly know I'm a math major.
Duh who are you
Joseph Sekavec gottem
Now that's a bruh moment
Smae but an ad for pampers came up
Funny I was looking at the course list and thinking "hey this looks exactly like the courses offered at my school" only to find out it is my school lol. Cal Poly stand up
1:50 "This is intimately connected to what I call the “ladder myth”- the idea that mathematics can be arranged as a sequence of “subjects” each being in some way more advanced, or “higher” than the previous. The effect is to make school mathematics into a race- some students are “ahead” of others, and parents worry that their child is “falling behind.” And where exactly does this race lead? What is waiting at the finish line? It’s a sad race to nowhere. In the end you’ve been cheated out of a mathematical education, and you don’t even know it. "
- Paul Lockhart
I think there are many other reasons why most people who get cheated out of a math education, get cheated. And I think they’re far more apparent than the structure of the curriculum.
I like this structure. I’ve learned so much from my classes as a result of this structure outside of simply “math aptitude”. I’ve learned how to be a far more effective student than I could have been without it. through this structure I’ve acquired a baseline level of competence in something. This competence has instilled way more confidence in my intellectual ability. I did kind of view this structure as a “race”, perhaps as a result of the structure, but after maturing a bit I’ve found it to be way more complicated than that because life is complicated. Honestly regardless of structure people are naturally competitive and anyway you try to structure a curriculum some want to be part of the 1% of exceptional students that are “ahead”.
Really enjoyed watching this. I earned my Ph.D in chemical physics 4 years ago. While your video is very informative, I don't think you realize how much abstract fields in mathematics play into chemistry in physics. Graph theory is heavily utilized in attempting to understand covalent bond network in organic chemistry, especially when you introduce molecular quantum theory. Group theory, from abstract algebra, helps tremendously in calculations of electron density of complex molecules. Topology, well, it's used in theoretical chemistry in helping define quantum mechanical wavefunctions assignments to bond types. Matter of fact, part of my research was applications of topological analysis on quantum mechanical wavefunctions.
Congratulations on ur phd and thesis. Im doing physics and hope to do a phd in a theoretical topic (i have literally no idea which area). is the dover book on intro to abstract algebra a good book? if u havent read it; what would you reccomend as an intro?
I don't know why you'd want calculate the density of an electron and stuff like that for a living but sounds cool
@@fernando8246 Electron density is how electron dense different regions of molecules are, which allows you to predict their interactions with other molecules of interest. Chemical interactions are entirely dependent on what electrons are doing within systems. Maybe you're interested in curing diseases or making better solar panels, these kinds of things require that you understand electron density of molecules.
Is chemical physics the physical properties of chemistry?
Isn't the entire electron cloud orbital shapes based purely on quantum mechanical calculations? I got out of O Chem as fast as possible so I could focus on Biochem and Human Phys.
Sometimes you are just sitting there solving math, and thinking about how beautiful it is.
And difficult too
Is it just me who wants to take *ALL* of these classes?
No, me too
dude, me too. Is it possible?
Same!
Arturo Gonzalez You could go to mit ocw
Sadly wanting or being able to take them are two different things lol
0:49 Calculus and Linear Algebra
1:53 Math is a Ladder, Then A Tree
4:37 Geodesics, Differential Geometry
Complex Analysis
6:02 Numerical Analysis
7:48 Abstract Analysis And Pure Mathematics
8:43 Topology
Topologically The Same
9:13 The Möbius Loop 13:03
14:15 Engineering
Computer Science
I am 98% sure you wrote this, so in future when you will come back, you will know where to go. Actually, I'm freshman and I would like also to timecode everything important. Thank you, bruh!
Is there any best book for this high level mathematics?
Damn
Its a real shame that most kids don't ever get to see the real beauty in mathematics. Our school system is so devoid of any of the passion that originally belonged in the subject, that 90% of the population believes math is just calculating numbers.
Eh, how do I put this, most of the beauty of maths comes after a lot of necessary foundational knowledge, and unfortunately a large part of the reason children don’t get to see that beauty is not because the school is bad but because they are genuinely just too dumb to get a good grasp on even those foundations
@@vilmospalik1480lol no it’s because many teachers are garbage in grade school and schools in general don’t take math seriously. Being in that environment sucks the joy out of learning, especially math
Dollar store?! Take it easy with the insults, man. I don't make engineering money, yet.
Dante Falls I READ THIS RIGHT AFTER HE SAID THAT LMAO
I think the world needed this video!
I love how you portray college as an opportunity to pursue your passion for knowledge and expand your horizons, versus just going to get a good job on the other side. That’s how college should be, and if you don’t think that way you’re probably better off going into a trade, which is probably financially smarter anyways.
This is SO COOL. I like math and I’m ok at it, but this makes me want to learn some of these classes. Job well done.
I do not excel at maths but this is amazing. Makes me want to study more and more! Thank you for that.
that topology tutorial was so cool haha
Waited a long time for this. Thank you
This is bullshit. The school he used as an example is absurd and unique. Most colleges will have those MINOR requirements as their end goal MAJOR requirements. The stuff he says is for a math major are graduate school courses...
Awesome video! This brings back sweet (and equally frustrating) days while I studied for my major in physics and my minor in mathematics. I really enjoyed calculus 3, linear algebra (I remember it being easy, till the second half of the book when it got into 4 dimensions. ACK!), and complex numbers. Haha, I remember sometimes doing math that was really complex (for me anyway) that at the end, I had no idea what the answer meant. Anyways, keep up the great work and I wish all you math majors and minors the best! :-)
I like the way, your videos now follow a red line from the conclusion. You said instead of taking technical courses like DSP in senior year, you would have taken some physics or high level math courses and you now take the time to explain what it is about
Nice information. The world needs more math majors (minors). Have a great day!
You deserve way more views. Amazing video as always. Keep up the work! 👍
The mobius strip was the first time my mind was blown! 4D analysis is crazy
Yeah bout to start studying math
@@metachirality nice vihart reference
Zach, consider doing a video on some of the awesome books that give great overviews for math enthusiasts, such as _Mathematics: Its Content, Methods, and Meaning._ These types of books can be great stepping stones to more involved texts.
Why am I watching this? I struggle with algebra lol.
Wow
No shame! It just takes practice!
Because your curiosity is not limited by your current state of knowledge/ignorance.
[edit] Unlike, for example, "My beautiful and amazing Princess."
I hate math and have difficulty with it too, and I dont care about it
it's never too late
I certainly enjoyed the high-level descriptions of these vast and nourishing mathematical realms. Given my electrical engineering and physics academic background, I've already dwelled upon most of these topics in varying extents and definitely agree with the approach and the examples. This is a really great guide for scientific newcomers to get a hint on what their preferred niche is. Much appreciated Zach, and I'm certain this single piece alone will do wonders for many undergrads. Kudos^2!
5:10 Differential geometry is quite useful in computer graphics.
8:00 Abstract algebra is applied in cryptography as well.
Number theory is 90% abstract algebra. Specifically, ring theory.
I've always been strong in math and will be attending university this fall for Mechanical Engineering. However, I have not exactly loved math until recently.
In our school system, we do not learn how to find the area under a curve in calc until university. So, I asked a teacher at my school with a major in mathematics about it and he gave me a great textbook to learn it from after a brief example.
At this point, I had never gotten involved with calc yet. So I taught myself Lim of functions, derivatives and finding area under a curve.
This may sound boring but learning it all was actually somewhat beautiful. I went through these subjects in 2 weeks, in comparison to the 5 months our school would devote a 75-minute class to every day. Because I enjoyed being able to do things that seemed almost out of reach a few weeks earlier. And now I am considering a minor in mathematics once I go to university so that I can learn to love and understand more of math!
It's interesting that Discrete Math isnt shown. That's required for CS Majors at my school.
That's what the proofs class was for us (although computer scientists did have their own discrete math course just for them). I did multiple videos on discrete math so didn't go into much depth on it.
At my university, math students also don't have Discrete/Combinatorics required, but cs students do. Well, it's a part of math that is probably more important to them than to us.
Duuuude I had to take that class a second time to pass. Maybe it was my professor but screw that class. Though, I can see how one can get nerdy on combinatorics and I found problems involving Fibonacci numbers quite fun too.
I fucked hated that class. God bless I managed to get a C my first goaround
Joseph G. Hotto That's what Graph Theory and Combinatorial Math are...
Analysis classes have made my lab reports a lot more rigorous cause I learned how to build up from what I had and deduce stronger conclusions. Making links between results and the discussion part was a lot easier after this, although I failed one of them and cursed them all the way through X)
For mechanical engineering, differential equations and matrices should be a must, understanding how those two work will make some of your upper division course a breeze
Idk why but my heart beats 100x faster when I come in contact with math
As in crippling anxiety or excitement?
Excitement, of course. Gotta pump blood to my... organs.
in my undergrad I was a double major in psychology and mathematics and was floored by the amount of mathematics I saw in my senior level psych classes when learning about cognition and robotics/ artificial intelligence (things like fourier series, graph theory, game theory, group/ set theory, category theory, number theory, and a lot of abstract algebra)
Too much theory not enough application
@@huey1153 what? they ARE applying the theories
I’m barley passing calc 1
Why am I here.
love the content bro!
Great enjoyable teaching and I really appreciate the enthusiasm and joy you bring to difficult subjects thank you 🙏🏾
ill comment just to make videos like this a fraction more popular on youtube
4:46 Say that again for the flat earthers in the back who didnt hear you.
best comment award. please come pick
honest
Hi, please do a video about physics and its subfields please!
I will!
Excellent video!! Enjoyed it immensely especially on the Mobius strip - Wow just blew me away. Thanks!
I think this video did a good job of separating the proof classes from the applied class, however I’d just like to point out that all of the applied classes are based on rigorous proofs and at the graduate level include proofs. Therefore, depending on the school, an applied class the video listed may be more proof based. For example, most universities’ topology classes are very proof based. Also, I think this university’s math minor is very proof light. Most schools require at least intro to real analysis, and proof classes require a very different set of skills than applied math classes. I know several engineers who thought it’s only three/four classes more for a math minor and take analysis and then drop their minor. So if you’re an engineer, take the proof courses seriously.
You're full of shit. Most schools do not even OFFER real analysis. That is grad school shit. At least in america, those "minor" reqs are major reqs at most schools, with the addition of abstract algebra
nothing First of all, there is no need to be rude. Real analysis is a requirement in almost every math major (with the exception of some liberal arts colleges that shouldn’t even offer a math major if they are not going to require any actual math) and most math minors. I can send you the requirements for my school’s math major and minor if you’d like and I have looked at dozens of other schools’ requirements and I can send you theirs as well. Real analysis is not “grad school shit” and if you don’t know what you are talking about, you don’t have to be rude to people who do.
Dude, I have a fucking bachelor's in mathematics!
nothing so do I and I’m a PhD student. It takes all of two seconds for anyone reading this to google math BS requirements or math minor requirements.
First several school major requirements that popped up in google after searching for: bachelor's in mathematics requirements.
math.arizona.edu/academics/undergrads/requirements/mathematics
catalog.unc.edu/undergraduate/programs-study/mathematics-major-bs/#requirementstext
catalog.unc.edu/undergraduate/programs-study/mathematics-major-ba/#requirementstext
www.sas.rochester.edu/mth/undergraduate/ba-requirements.html
coursecatalog.syr.edu/preview_program.php?catoid=3&poid=807&returnto=253
catalog.buffalo.edu/academicprograms/mathematics_ba_requirements.html
mathstat.umbc.edu/files/2014/04/advising-Math-BA.pdf
mathstat.umbc.edu/files/2014/04/advising-Math-BS.pdf
At first glance, only one of these schools even offers an analysis class. They almost all follow the standard curriculum: Calc 1-3, Differential equations, Statistics, Linear algebra, Abstract algebra, intro to CS + electives.
I was introduced to the moebius strip in a magic, math and illusions book. "Make the strip. Now try to draw a line down the middle. The "edge" is "twice as long" as before joining the ends.
I graduated in computer science, but fell in love by engineering maths. I took diff. eqs., Fourier analisys, complex analysis, numerical analysis. However, I had to take also number theory, graphs algebra for computer science degree.
I realized that I have only few disciplines to get a math major.
Currently a math major at cal poly, I didn't know you went there which is super cool
I perform adequately in mathematics, essentially, "pure math".
Invariably, I struggle to apply this material to real world problems.
As an engineering student i'm considering switching to mathematics.
Does anyone else struggle with this? Any advice on switching from engineering to pure mathematics?
Thank you and great video!
I don't know about switching majors and what challenges you might face but prepare for a nasty surprise when you finally get your maths degree. I got one of those and I got so frustrated looking for jobs that I became a freelancer. Unless you'll stay in academia you will not have a lot to do with pure maths at all. Most of the jobs I applied for actually wanted a programmer and the other 10% could be done with basic maths skills.
Yeah that’s because pure maths rarely has applications in irl problems
@@umartheguy4612 I think mathematics is applied in irl but it's usually elementary level.
@@umartheguy4612 exactly
"Complex Variables" by John W. Dettman is a great read: the first part covers the geometry/topology of the complex space from a Mathematician's perspective, and the second part covers application of complex analysis to differential equations and integral transformations, etc. from a Physicist's perspective.
Alright! I’ve been waiting for this
Integral Equations: Fourier & Laplace Transforms, Green Functions, etc.
These are used a lot to solve PDEs (Jackson's "Classical Electrodynamics"!) and model systems rather than differential equations.
No lie, having a minor can’t hurt, getting my nursing degree and having a Chemistry minor and taking Calc 1-3/physics (NOT required) for nursing out of curiosity, it helped me secure a slot in CRNA school. When my admissions counselor did the interview, he was like my, my, you certainly are an ambitious one sir! lol! I only took those classes because I love science, nothing else, had nothing to do with climbing the career ladder. I do despise folks like that however, because they don’t care about anything, to include the job they are paid to do.
1:20 the easy way: fill up the 32-gallon jug by completely filling up the 6-gallon jug 6 times this will leave you 4 gallons in the 6-gallon jug. Then pour out the 32-gallon jug and pour the remaining 4 gallons into the 4-gallon jug, finally fill the 6-gallon jug and pour it into the 32-gallon jug.
You are the best man!
i went from only being able to do very basic algebra equations (like barebones) 4 weeks ago , and now i’m headed into algebra 2 and trig in my rush to linear algebra
i gotta say, math is pretty damn cool
even u are not studying the above subject
I recommended linear algebra, caculas and statictics
since we have utube, Udacity and open course anyone can learn any Maths subject very easily
Well. They have the resources to learn it. Just because you have resources doesn’t mean you can completely understand it
Finally got all the achievements for Math
It is refreshing to find an engineer that sees the beauty of math.
Electrical Engineering does that. It did the same to me. It is awesome to see how mathematics manifests in something tangible and physical applications that follow from it.
My daughter only needed 1 class to add a Math minor to her Honors CS degree - Real Analysis. Honors CS required MVC/Vector, Linear, and a CS proofs class. To get a Math major is only four more - Elements of Algebra, DiffEqs, Linear 2, and Complex Analysis. Among many of the “pick 2 courses from this list” for Math are several CS courses she already needs to take (Numerical Methods, Algorithms, Theory of Computation). She’s still contemplating this.
I had been to school most all the time, and could spell, and read, and write just a little, and could say the multiplication table up to six times seven is thirty-five, and I don't reckon I could ever get any further than that if I was to live forever. I don't take no stock in mathematics, anyway.
I’m majoring in math and minoring in computer science. Thank you for this informative video as to what extra classes I need to take in terms of math and computer science together to be needed for my future job, which is cryptographer.
hey
if I want to become a data analyst, should I major in math or computer science?
Even though I received my undergrad in Math many years ago I was curious to hear what you would say about Game Theory, the course which caused more pain and suffering than any other math course I ever took.. you almost made it sound pleasant for that half a second, but don't be fooled fellas haha.
i love this channel so much, i regret not finding it sooner
I honestly cheated in Algebra I in eighth grade, and I harmed myself from there. Once I realized how many foundational skills I did not know, I decided to study my algebra foundations. I’m now in Pre-Calculus, and I love the class. Math is fun once you change your perspective and think of math as a game with rules you have to follow.
Im in 8th grade right now, my teacher signed me up for advanced next year and i dont think i’ll survive lol.
Glad to see I'm not the only 8th grader watching this. I am in my 8-9th grade summer break right now. I took Algebra 2 and Precalculus (at same time) and didn't do the homework for Algebra 2 because I was so busy in Precalculus, but in my school this year there was a new grading system which didn't count homework and no points were taken off on late (In my highschool, it would just be the regular grading system). I am planning to take Calc BC for freshman (because I want to learn about infinite series). I'm not sure if I'll do good, but I've always had a passion for math, biology and chemistry. Any tips would be nice
Counterintuitive fact: the thing you got when you cut the Mobius strip in half ( 10:33 ) is indeed not a Mobius strip, but it is actually (homeomorphic to) a completely untwisted thing (a cylinder).
It's true that you can't deform this to an untwisted thing (it's twisted), but you could actually untwist it if the ambient space was 4D. (At the very least, I'm sure you could untwist it in 5D; I bet it's also true in 4D, but I'm too lazy to think to prove it.)
you should do a video on actuarial science
Bruce N. I wanted him to talk about combinatorics so bad 😢
Wanted to point out that abstract algebra has application in computer science. For example, category theory is a core foundation of functional programming paradigm, and group theory principles can be applied in some algorithms
Now i'm motivated to study math xD
Same!
Me too! I just got major in mathematics in university just by luck. I had not zeal to study it. But Zach Star has changed my mindset.
Today i start the pure math career, I'm very excited about this opportunity and your video gets me a major idea about career
I am on a maths journey at 37 years old at pre-calculus, liner algebra and calculus level I want to study more advance maths topics after khan academy calculus's subject, I also use cosmo learn and youtube, I want to start when I am now but after calculus 3 does anyone know any advance, really advance pure maths, advance or higher pure maths topics I really want to get into it, because I am into physics now.
Sorry I had myself at a wrong age lol
I am also studying myself math for fun I finishieed undergrad abstract algebra and galois theory have some topology and finishing measure theory my next goal is differential geometry I m proving everything I do
Seriously, people have got to stop saying "Calculus 1, 2, 3, etc". In the pure field of Mathematics, there's just Calculus. The topic is never divided into "1, 2, 3" nonsense. That's just a system used by schools and colleges to organize their subjects. No such grouping is used in the pure field of Mathematics.
Look at the class descriptions of universities, they spell out a lot of the topics math students learn within each class, their names alone might be enough to research more on your own.
As for resources that help understanding post-Calculus concepts, the RUclips creator 3blue1brown has amazing videos on just about everything (my favorite is the one on Fourier analysis). Or maybe pick up a textbook online
@Steven Lee I don't think it's ever to late. My dad got his degree in electrical engineering at 34. Then he went on to get his masters. And you know what? He did just fine. It's never too late.
1:27 fill the 6 gallon from the 32 gallon until the 32 gallon has 2 gallons remaining. 32 - 26 - 20 - 14 - 8 - 2.
Refill the 32 gallon.
Fill the 4 gallon from 32 gallons with 2 gallon remained to fill 6 gallon. Now 32 gallon has 28 gallons.
Repeat filling 6 gallon from 32 gallon.
28 - 22 - 16 - 10. Now have 10.
Oh my god I completely relate to how you feel with profs and geometry lol
The small loop showing up kinda makes sense if you realize that each side is being taped to the opposing side the second time round. Since the middle third is just attached to itself, it gets cut off from the other portion, meanwhile the left third connects back to what was the right third and the right third connects back to what was the left third. I didn’t predict it correctly, but the reasoning seems fair enough as an explaination, for anyone who didn’t quite get what was going on with those rings. Basically, splitting the loop into three segments basically isolated that center loop, which then disconnected.
i’m in 10th grade and all this seems terrifying.
Cool demo! I guessed incorrectly. You're one of my favorite math teachers. Thanks!
i am really confused, i've never seen a math lecture without proofs, even in something very applied like numerics. How in the world would you go through a class in Calculus (let alone 3) or linear Algebra without being able to proof even the simplest of statements? isn't the point of those classes to proof statements in Calculus and linear algebra? I mean what do you learn instead? Also someone studying Physics for example would also need to know about proofs and technical stuff to be able to survive right? how does that work?
Where I'm from everyone in a technical field has to do proofs at the very least in the basic maths classes.
Lorch95 at my university classes that are not proof based still show the proofs in class to explain how something work but the students themselves don’t have to write proofs and the in class proof are usually less formal
Lorch95 demonstrating how theories are used and why they work (or giving a special case "proof" that shows it is true) doesn't often require actually formally proving it. In Calculus in my uni at least, it's more of the former as a formal proof class is not a prerequisite and proving a lot of calculus theories supposedly involve higher level math. Though I suppose one could make the argument that the solution to *any* mathematical problem is a proof, formal or not.
my calculus 1-4 classes leaned more towards mechanically calculating expressions involving derivatives and integrals, usually through real-world applications. we would either need to demonstrate or simply assume that statements were true, formal proofs were few and far between
NO WAY! Did you go to Cal Poly SLO? I recognize the website layout for the courses that’s crazy
That Möbius Strip demo has me feeling smart because I guessed both questions right... but then also because I got it right it also doesn’t feel major 🧐
I can't wait to learn differential geometry
10:32 dis how CVS be making they long-ass receipts
@1:24 it is totally possible. Let's say you have a bucket that you want your 8 gallons to end up in. Thus if I say pour into bucket, I mean pour the water currently in the jug into the bucket. If I say empty it, I just mean pour it down the drain or something.
Step 1: fill the 32 gallon jug
Step 2: use the 32 gallon jug to fill the 6 gallon jug
Step 3: pour out the 6 gallon jug into the bucket
Step 4: use the 32 gallon jug to fill the 6 gallon jug
Step 5: empty the 6 gallon jug
Step 6: repeat steps 4-5 exactly 4 times
Step 7: Empty the remaining 2 gallons in the 32 gallon jug into the bucket
There is just too many to learn 😅
for the gallon problem, fill the 32 gallon bucket with the 6 gallon bucket. After it's completely full, the 6 gallon bucket will have 4 gallons of water in it. Empty out the 32 gallon bucket and pour the 4 gallons into the 32 gallon bucket. Then fill the 6 gallon bucket and pour it into the 32 gallon bucket.
Pure Mathematics (algebra/topology/geometry etc.) is pretty boring for me personally. But I'll be honest, it's partly because it's difficult. Applied Mathematics is way more interesting but that's because I enjoy applications of Differential Equations, Fourier analysis, Boundary Value Problems etc. (Analysis).
This video gave me a really pleasant nostalgia over how unintuitive higher knowledge always feels at first…I love that reality gets more wild the more sense it comes to mKe
When you have math in your mind too much that you get zeros in every other class
Lol specially language classes
Thanks for the video... Came at a good time for me!
It’s going to cut evenly in half (just a broad guess )
Graph theory at my school was ridiculously difficult for me. Unlike differential equations, they’re mostly proof based, abstract, and no straightforward “‘methodology” or simple approach to problems. Each problem is so open ended and there’s nothing that hints at which proof method will work best, and not only that, but sometimes you CAN prove something using one technique, but it may be 10x harder than another technique that you hadn’t considered. Not to mention the mental strain and focus you NEED to keep your logic in order. Probably my hardest math class ngl
This is why I cant wait to go to college
So at 5:35 I saw you talking about the proofs to some of the trig identities inclusive of the sum and difference identities. I just took precalculus (so I'm still quite low level math) but in class we actually proved the sin and cosine sum identity using a triangle inscribed in a square where one corner of the triangle is also a vertex of the square. Hence, we have angles α, β, and 90-(α+β).
Yeah let's watch this when I had to take geometry a like 4 times
There should be another specific set of math courses offered as standalone courses for these majors / minors. These advanced courses should be specifically based on Fractals both numerically and geometrically. Another mathematics course that should also be in this list is Boolean Algebra.
One advantage of a minor compared to a double major is that you don't need to do extra GE classes. At my school, math is in the school of arts and sciences and computer science is in the school of engineering. Arts and sciences and engineering have different GE requirements, so a math double major would be mostly GE. Minor actually lets me take more math.
if anyone was wondering about the 32g 6g jug one, what i came up with was filling the 32 with 5 6g scoops, for the last scoop to fill it all the way save the excess now you start with a 4/6 filled 6g or 4gs now empty the 32g and fill it was the 4 and another 6
"Saved you a trip to the dollar store"😂
Determining the shortest path is called the Traveling Salesman Problem and it's actually still unsolved.
What do you mean with unsolved? It has (several) solutions. The only thing that's not clear is whether there are any polynomial time solutions (is P=NP?), but very probably not.
Can you make a video about FEM (Finite element method)?
If you’re reading this as an engineer thinking about a math minor in telling you it’s absolutely rough. My experience is different than most but I am currently taking my higher level engineering courses and about to graduate but doing a minor in applied math and at my school it requires applied statistics, discrete math OR, the straight linear algebra course (on top of the diffeq linear algebra hybrid you take for engineering) I would take discrete math if you are thinking about it because it helps greatly with understanding proofs and I unfortunately did not take that and the linear algebra course at my school had proofs but they wernt on the level of rigor that I was ran into after, so after that you need a “rigorous” proofs course which as I mentioned kindof leaves off from discrete and I really struggled in this class because it was so unlike the engineering classes. Currently I’m in the last class which is advanced calculus which some universities call “real analysis” but that term should be reserved for graduate level analysis because apparently it’s another beast. My prof told me that this class in undergrad math is commonly called “baby analysis”. I also found out that this is regarded as the toughest class in a math undergrad. Currently taking 4 classes and 2 labs this included and working and I am absolutely swamped and struggling but surviving (I hope). He has a video about self teaching pure math and funny enough we are using the same textbook “Elementary Analysis” by Kenneth Ross. I’m the only person in the class not going for a math degree and most of my peers seems to lighten their workload when they take this class but not me lmao. If you do what I do I suggest you self teach and get ahead by reading the textbook BEFORE the semester starts and actually work out problems and make sure your foundations in proof techniques are solid. Direct proofs, proofs by contradiction, induction, and predicate logic, things like negating compound statements etc. this class requires a ton of tricks and of course a very solid algebra toolkit because this class relies on these “tricks” for proofs. Brush up on Calc 1 and 2, sequences especially but also stuff like series tests from Calc 2 and you will be okay. Absolute values and sequences and the triangle inequality are heavily used in analysis.
What do you recommend to do to understand mathematics? I like math, but sometimes I don't get what's going on so I plan on re-taking college algebra.
Acid Plug If you don’t understand what’s going on, see the professor during office hours. Math is hard, but it’s like learning the piano. When you start off with learning the song twinkle twinkle, you think it’s really hard but eventually you move onto learning harder songs and think playing twinkle twinkle is easy. Keep it up and work hard and you’ll be looking back at college algebra as easy! Good luck!
Izzy Explains thanks
Definitely go to office hours if you struggle with math.
But I also want to point out something else that a lot of students seem to do. They tend not to _think_ when doing math. They view it as a very algorithmic process where you just have to memorize a bunch of random stuff and repeat it.
That's not a good way to do mathematics. You really need to think about what's going on and what everything means because everything in math _makes sense_ - there's a reason _why_ everything is the way it is. Sometimes that reason may be too complicated for you to understand, but most things you encounter should be in your grasp as soon as you see it. You'll definitely need to learn some things on your own, but if you are really struggling with a concept, ask _why_ something happens.
Here's an example of something that confuses people: graphical transformations of functions. Let's say you have a function f(x). It's generally not too difficult to see that f(x)+2 shifts the graph of the function _up_ 2 units, and that f(x)−2 shifts the graph of the function _down_ 2 units.
But what about f(x+2) and f(x−2)? Most people initially think that f(x+2) should shift the graph of the function right 2 units, and f(x−2) should shift the graph of the function left 2 units, but it's actually the exact opposite: f(x+2) shifts the graph of the function _left_ 2 units, and f(x−2) shifts the graph of the function _right_ 2 units. Most students at this point are content to just memorize that: if you change the inside of a function, it does the opposite of what would happen when you change the outside.
But if you want to understand math, don't be content with just knowing this information: try to think of _why_ this is true.
So what you can start to try is to look at some simpler functions that you kind of understand. Like f(x) = x. It's a nice line passing through the origin with a slope of 1. Now look at f(x+2) = (x+2) (replacing x with x+2). Now try to graph this function next to f(x) = x, and notice how it is shifted 2 units to the left from f. Now look at f(x−2) = (x−2) (replacing x with x−2) and try to graph it next to f(x) = x and notice how it is shifted 2 units to the right from f.
And maybe try a more complicated function like f(x) = x^2 or f(x) = |x|. Try graphing each of these and then try graphing the versions where you replace x with x+2 and with x−2. A nice way to sketch a graph is to try a couple different x-values and find the corresponding y-values and plot all of those points.
And after plotting these, maybe you'll start to get a sense of why this is true in general.
The point here is that f(x+2) and f(x−2) are technically new functions, right? So let's treat them like new functions. Rename f(x+2) as g(x). In other words, g(x) = f(x+2). Now, for example, if we plug in x=0, we get g(0) = f(0+2) = f(2). Or if we plug in x=5, we get g(5) = f(5+2) = f(7). And no matter what value for x we plug into g, we get that its input is 2 numbers to _the left_ on the number line for the same output on f. So you've shifted f to the left: you get the same output as f, but 2 units earlier!
And similarly, let h(x) = g(x−2). Plug in x=0, and you get h(0) = f(0−2) = f(−2). Or plug in x=5, and you get h(5) = f(5−3) = f(2). And no matter what value for x we plug into h, we get that its input is 2 numbers to _the right_ on the number line for the same output on f. So you've shifted f to the right: you get the same output as f, but 2 units later!
What you notice here is that g(x) = f(x+2) and h(x) = f(x−2) is that you're affecting the _input_ value of the function. So you could think: If I start with g, how can I get _back to_ f? What do I have to plug into f in order to get the same value as g(3)? Well, notice that g(3) = f(3+2) = f(5). So you need to plug 5 into f in order to get the same value as plugging 3 into g. So notice that in order to get _to_ f _from_ g, you have to shift two units to the right. So in order to go backwards _to_ g _from_ f, you have to shift two units tot he left.
So that's the sort of thing you can do if you don't understand something in math. Try a few examples and really think about _why_ things are true. Try thinking backwards too (like what I did at the end there: instead of thinking about what you have to do to get from f to g, think about what you have to do to get from g to f). If you can figure out the reason on your own, you'll really remember and understand it much better! But sometimes it's really tricky to think in the "right way" to make the reason clear. That's when you can ask in office hours or ask someone else who may understand the topic.
Yes, this is time consuming, so you won't be able to do this for every topic. But if you want to understand math, this is the sort of process you should strive to do as much as possible.
MuffinsAPlenty thank you sooo much by the way and for taking the time to type all of that.
Honestly I think the main issue with math is that children are given problems and not much real world applications to it. I had the same issue in high school but once I started taking physics everything started making sense to me. And once I started calculus in college I went in a bit confused from precalculus but I felt that calculus just made everything easier to understand.
Additionally in some cases mobius strips are used in belts for drive systems. This is becasue they have one side and can provide uniform wear (material removal of a surface due to friction).
what about the math necessary for quantum computing?
Abstract algebra is the foundation of functional programming which has significantly increased in market use in the past few years which means it can definitely be useful for computer science.