Practice 7 4 Write An Equation Answer Key: Fill & Download for Free

Thanks for putting up this question .Mains is an important stage in the UPSC exam ,since it determines your selection for the next stage .For beginners ,just a clarification ,your prelims marks do not get added to mains marks and hence you just need to qualify the prelims (until you are aiming for Forest Services ,which was a distant dream for me atleast ).Writing practice has wreaked havoc in the lives of UPSC aspirants, especially when many toppers have constantly emphasised it's need . Almost every cocahing institute mentions in its brochure , advertisements that it teaches anwer writing but does none of them actually do it ?I still remember when I had joined a coaching(but left it within few months ) I expected that they will tell me how to begin but as the classes were coming to an end (after finishing the basic subjects ),all I was handed over was a pink book which contained some quotes and some content for common topics ,alongwith some techincal Actually, I was never told how to begin with ,so when mains was approaching ,I had to figure it out myself in the first attempt in 2015 (where I couldn't clear prelims ).It was like giving a quadratic equation to a class 5 kid .Thankfully in my second attempt, I had a mentor and one -two friends who were preparing so we could do it in a planned way .I was in the misbelief, that may be because I didn't complete the coaching that's why I was in this crisis but I was totally wrong .Now when I have started mentoring aspirants ,I find even those who have completed their coaching classes sincerely ,facing this issue ,in short we all sail in the same boat .Also those who have just started preparing are anxious about writing answers,I did the writing practice during the time between prelims and mains,tried to utilise maximum of the things I prepared in prelims.I really don't want to come at loggerheads with anyone who say that start answer writing from day one because it might have worked for them but it didn't for me .Hence believing that there are many (not so intelligent)aspirants, like me (who could try my way with some alterations suited to them )I have come up with a plan for easing this out (it is tried and tested because it was my strategy).There are some rules for answer writing (See Demystifying UPSC series for essay on telegram ):Rule 1:Don't start answer writing from DAY 1,it will lead you to nowhere .Rule 2:Complete the basic subjects first .Only after 5- 6 months(when you have built up on some basic content ) ,you are ready to hit the ground.Note:You can start ethics answer writing and case studies, once you complete the basic reading of ethics portion !Rule 3:The first or second reading should decide the content for mains(you can take help of your mentor ,this is an important segregation !See Demystifying UPSC series)Rule 4:The content can built by going through some extra sources (mentioned in Demystifying UPSC series )Rule 5:Start by writing 2 answers daily (from any site ,insights on india is what I preferred but you can chose any website )Rule 6:You need evaluation only after writing for about a month.Rule 7: Getting it evaluated properly is a crucial requirement . How(?): Mentioned in Demystifying UPSC series .Rule 8:Practice UPSC questions once you get in full swing.Rule 9:Try to be as multidimensional in writing your answers as much as possible along with good presentation (will elaborate upon in some other question)Rule 10:Revision is the key to write answers in the exam .I could write only those things that I had revised 3–4 times .In the exam you have 7 minutes to write an answer for a question which is totally new to you so unless you have fodder points and have them at your finger tips nothing could help you .Last and very important ,write only 2–3 test series but honestly .I wrote 2 tests for each GS paper in a gap of 1.5 months .SO THE NUMBER OF TEST SERIES DO NOT MATTER .Your attempt should be as your actual exam and evaluation should be proper .These were the rules which I made because I was also puzzled like you all . They helped me, but I always say ,different people have different strategies , stick to one person whom you feel is genuine and matches the most with your pattern of studying .Don't run around many people and end up getting super confused because it's the availability of so much resources that makes the exam difficult to crack .The journey is long , with its own way of siding you from the track . Go step by step ,with each move calibrated .Trying to jump even one step might cost you an entire year ! Planning and execution both are important in this journey !!Hope it helps !!You can follow my telegram channel or Civil Services chronicle FB page !Picture : Uttarkashi trek during IRS trainingEdit 1:Mrunal Sir's video for answer writing emphasising that starting answer writing from Day 1 of preparation is a bad idea .Best wishes!!

Sometimes, simplicity is the best solution.Let me just share my story about how I arrived at the techniques that finally allowed me to study effectively for the first time in my life leading up to my IGCSE exams.Since I was about 11, I had been trying to find the answer to this question by devising and trying new wacky and complex ways to study effectively for the exams leading up to my Primary School leaving exam. However, I tripped up every time including for that crucial end of Primary education exam as well. Every method I had tried fell short and I constantly did poorly.However, with about 3 months left to spare before my IGCSEs (end of Secondary education exam) when I was 15, I realised I did not know enough to do even averagely well. The bad part was that I was still lazy and now desired studying in a very simple way just to save energy. I was fed up with weird and complex ways such as having millions of post it notes all over the place, everything in varying colours, fonts, shapes and sizes, odd mind maps, flow charts and diagrams bursting out of nowhere going to who-knows-where etc...you get the idea. Perhaps beauty and complexity was not related to effectiveness?Then, just out of sheer laziness, I started studying in a new way because it seemed simple to implement and follow. I did not think it would take me far or give me great results, I just expected it to get me by. However, it did quite the opposite. Perhaps laziness does have its perks.I used this method and managed to get 7A*s and 2As for what I did. I am not saying this to boast but rather just to show you that it worked for me (someone who had never done great in exams and desired a simple approach).For my study process, it just involved two simple parts where everything else hinged upon. Many people like to overthink different ways of studying for your exam but for me simplicity is what worked best, based on hard work. Also, try not to listen to music when studying as it tends to distract you and prevents information being absorbed, the same goes for multi-tasking.What are the two simple parts to how I studied?1. Note taking.Your brain is not going to remember every bit of information so don't even try to walk into the exam hall thinking you're an expert because 'you remember everything from class'. You may think you do, but you probably don't, unless you're a genius, which I by the way am not.When you are in class, Listen!I am not saying take down detailed notes in class or write down every word the teacher says...No. Just listen, listen, listen to what the teacher has to say and take down very brief summary notes of what is most important such as a key concept or an equation for example. Your exercise book should not be filled with more than a page and a half every lesson. Anymore than that, and you probably have not been absorbing or listening to what is going on as much as you should.When you are at home, Write!Once you get home, everyday, review your work and what exactly you have learnt in class, refer back to the textbook or whatever you did that day and refresh yourself on what you learnt. From that, and referring back to the syllabus (to see what you must learn and what topics you are unsure of), textbook and your school notebook every now and then to grab the key takeaways, write or type up a document of what you should know for that topic based on what you learnt that day.Do this everyday and it should not take more than half an hour. Leave it to the last minute, and you are going to be swamped with 101 topics from 101 subjects and its just not going to work at all. Do not carry on with your notes if there is something you are unsure about, only move on if you understand everything you write. If you are writing page 3 of your notes, you should understand Pages 1 & 2 inside out. If not, stop, re-read the chapter and seek help from your teacher.Regarding the notes, they absolutely do not have to be colourful, in your face mind maps or exquisitely designed flow charts (Depends on what works for you though).As for myself, I am not sorry to admit that mine was simply a word document built on sub-headings, bullet points, short explanations and diagrams. That's it. I used limited colour, a very small amount of shapes, and virtually no smart art or word art whatsoever. And yet, it worked well.I just filled my document with the core content of what needed to be known and put it into visible easy to read point form. This allowed me to easily refer back to it if I forgot or did not clearly understand something. The biggest problem I have with exquisite mind maps and notes is that it is difficult to pinpoint something and refer back to it quickly when I needed to, it also is extremely time consuming which for a set of revision notes, in my opinion, is unnecessary, but again, whatever works best for you. Remember, your notes is not a graphic design project.2. Practice PapersI am not an advocate of too many textbook questions, they are usually too simple and are nothing like what appears in the exam, so only do textbook questions to test your really fundamental understanding of the topic at the beginning, but bear in mind, simply acing textbook questions is not going to get you very far in the exam.In terms of practice papers, try not to wait till the last minute and cram 3 per day. That is just unsustainable and bloody tiring. Same as with your notes, try and do your practice papers over a pro-longed period of time, 2-3 papers a week of alternating subjects worked well for me, however I increased that frequency to 3-4 per week in the final month of the exam just to make sure I left no stone unturned.You may ask how are there enough practice papers to keep this going? Simple. Find the exam papers and answer keys online then print them all out starting from a few years back such as 2008.When you do your practice papers, have a copy of the answer sheet too. The point of this is because what you may begin to notice is that the questions they ask tend to repeat and be pretty similar across the years, therefore, their answers will also tend to be very similar too, the answer sheet will expose first hand to these 'model' answers that will come up and again and again and will also show you exactly what answers the exam board wants to specific questions, hence, even if you go into the exam and have a familiar question but you don't fully understand the concept, you should know the model answer and therefore could just put it down and have a high probability of getting it right simply because you've done your background work. This also puts you at an advantage above the people who may understand the concept but lack the ability to articulate it in the way the exam board wants you to do so.After finishing every paper and learning from your mistakes, file it away somewhere so when you refer back to it or refresh yourself before the exam, it will be easily accessible.Once you have completed refreshing yourself with the work you have done, do no work at all up to the day of the exam. Enjoy yourself and do what you want, you have worked hard. There is nothing to fear.The above content was mostly for content and word-based subjects such as Biology, Chemistry, Literature etc... rather than a number based topic such as Maths. After all, you cannot really take good quality notes for a subject like that.Therefore, I came up with a logical step by step study method that I'll share with you meant just for Maths which somehow got me an A* and an A even though I have never in my life been great at it:Get a copy of the syllabus to get a clear view of the exact topics and their components that you will be tested on. Make a little mark beside the topics you do not think you are very confident in.Get your textbook again and focus on those topics you are not too sure about. Do about 10-15 beginner/intermediate questions ensuring your foundation is there. Then move on to complete all the hard questions in that topic to ensure you can handle the tougher stuff. You will probably run into difficulties along the way and that is when you seek help from your teacher.Once you have finished the topics you are not confident in and have become comfortable with them, move onto all the other topics you feel neutral and confident in. Do about 5-7 of the beginner/intermediate questions in those topics just to make sure your foundation is still very much intact. Once again, move on to do all the 'hard' questions even if you think they are easy. This will ensure you are comfortable with those topics and can handle the harder questions there too. Do not be complacent and skip topics as that is where you will lose easy marks.Find and print out all the past year exam papers from all the different time zones along with their respective answer keys.Start work on the past year exam papers under timed conditions. Do all the questions that you know how to do. Leave out the questions you do not know how to do. Once the paper is done use the mark scheme to mark yourself and to fill in the questions that you did not know how to do and learn from and understand your mistakes. If unsure, seek the help of a teacher. Never move onto another paper without understanding every single question and every single method used in the previous paper. This part is very important as it gives you an idea of the types of questions the exam board likes to ask and also shows you how the questions have evolved over the years. The answer key shows you the answers and methods the markers like to see and also gives you an idea of how the answers required have evolved over the years. Knowledge of these will give you a big advantage on the day of the exam.Once all the papers are completed, go through each of the papers to review what you have done and to ensure you remember and understand why you have done what you have done for each question. Just make a little mark on the question if you do not understand why you did something and then seek help from a teacher. Only move onto the next question when you fully understand why you have done what you have done for the previous one.Spend the remaining time before the Maths exam; relaxing and having a good time. Do no work at all. No more maths. No more school material. Behave like you are on holiday. Go out with friends, exercise, start a business, get a job, do whatever makes you happy. Most importantly, get tonnes of sleep as there is no better preparation for a Maths exam than to be well rested and have a clear, quick mind. This part is absolutely fine to do and actually good because you have spent the past time working hard and understanding everything, therefore if you were truly diligent about it, everything should make sense to you and there is no point going over stuff you have already completed hence, you are prepared for the exam.On the day of the exam, remain calm but focused and alert. Understand and have the confidence that you have put in the hard work and that the results you aimed for will follow as a result of that. After you finish the paper, check your answers thoroughly. There is nothing worse than coming to terms with the reality of careless mistakes undoing what you have spent the past time practicing.Finally,For any exam, try to enjoy the journey! I struggle with this immensely too, possibly because I do not see myself as an academic guy, but I try to find a source of enjoyment somewhere inside the world of exams and grades. Make sure you exercise and have a good time in between as well as it will help to clear your mind and study more efficiently. Try to do the same, it definitely helps.Hope the answer has been helpful and all the best for whatever you are studying for!

For reference, the full quote of Grothendieck is reproduced here along with its source.The introduction of the cipher 0 or the group concept was general nonsense too, and mathematics was more or less stagnating for thousands of years because nobody was around to take such childish steps …– Alexander Grothendieck in a letter to Ronnie Brown, 1982[1][1][1][1]The year was 1945, and World War II was coming to an end. It was at this time that two bright, young mathematicians—Saunders Mac Lane and Samuel Eilenberg—published one of the great papers in all of mathematics. Titled General Theory of Natural Equivalences, Eilenberg and Mac Lane’s joint work birthed the entire field of category theory (cf. [1]), a field that has been dubbed “general abstract nonsense.” In most cases, the moniker is not intended to be derogatory. Mac Lane was actually the first person who used the term in print!Really, “general abstract nonsense” is an apt term for categorical algebra: It describes how category theory can be seen as a mathematical theory of mathematics, a sort of framework or language, and how its intrinsically abstract nature can often feel like nonsense, not clearly linked to anything concrete. For many more descriptions of category theory, see this Twitter thread, which includes perspectives from bonafide category theorists such as Emily Riehl, whose recommendation of Barry Mazur’s When is one thing equal to some other thing? is enthusiastically repeated here.The definition of a category makes clear its utility:A category is a collection of objects with arrows between them. Arrows can be composed, i.e., one can follow an arrow [math]g[/math] after the preceding arrow [math]f[/math] (we write this [math]g \circ f[/math]), and this law of composition is associative, meaning taking [math]h[/math] followed by [math]f[/math] after [math]g[/math] is the same as taking [math]g[/math] after [math]h[/math] followed by [math]f[/math]. We write this symbolically as [math](f \circ g) \circ h = f \circ (g \circ h)[/math]. Finally, for each pair [math]x[/math] and [math]y[/math] of objects, if [math]f[/math] is an arrow between them, then [math]1_y \circ f = f = f \circ 1_x[/math], where [math]1_x[/math] and [math]1_y[/math] are the identity arrows on [math]x[/math] and [math]y[/math], respectively.In less technical terms, a category is a bunch of objects with arrows between them such that we can combine arrows and where every object has an arrow pointing from itself into itself. This is a very general and powerful sort structure, whose study easily captures most algebraic mathematics and can be used as a foundation[2][2][2][2] of all mathematics [2].Seeing exactly how category theory is of practical use to mathematicians requires some time and effort, too much to go on about here. The author should have a course coming out on the subject on The Educational Blog soon, and he is likely teaching a course covering somewhat similar material this fall at the University of Maryland. One of these may be of interest to the reader who wants to know more.For an easier to understand example of the sort of "general nonsense," the "childish steps," of which Grothendieck wrote: groups [3].As with most mathematical objects, there are many ways to think of groups. We describe them only as algebraic structures, which is, admittedly, perhaps the worst possible way to present them to a general audience. We include it only so those with a bit of background come out with some idea of what groups are in formal mathematics. Coming back to pedagogically sound exposition, we now examine an example: the cyclic groups [math]C_n \cong \mathbb{Z} [/math][math][/math][math]/ n \mathbb{Z}[/math] [4].Clocks are good physical models of one of the cyclic groups (there is exactly one for each natural number [math]n \geq 1[/math] [5]) Consider the everyday problem of telling time. If it is 6:00 p.m. now, then what time will it be in 12 hours? The answer, of course, is 6:00 a.m. Ignoring the Latin initialisms for ante meridiem (“before midday”) and post meridiem (“after midday”), the time is the same. Similarly, the time will be the same (6:00) in 24 hours. In this sense, we have something that looks like [math]6+12=6[/math] and [math]6+12+12=6[/math] and [math]6+12+12+12=6[/math], repeat ad infinitum. How strange!It turns out this general situation where applying an operation can “wrap around” is of great use in mathematics. Whenever we have some objects with some notion of addition (or multiplication or subtraction … any operation that takes in two objects and spits out another from the original collection) that is associative, has a distinguished member called the identity, and has inverses, we have a cyclic group. These groups are especially important for group theory in general because they are very simple (e.g., they are abelian). Furthermore, they provide a complete characterization of the finite abelian groups [6], that is those groups for which the operation is also commutative, meaning the order in which it is applied does not matter.More formally, a group is an algebraic structure that consists of a set along with a binary operation that is associative and for which identities and inverses exist.Why would one choose to appeal to such generality? Pure mathematicians often argue group theory is of intrinsic interest and value. The author agrees; however, describing the intrinsic beauty of groups is no small task. We choose the easy route in describing an application of groups, one which is half mathematical and half practical, and which will appear again later in the answer.The first application is at least mathematical. It will also come back in a moment. Suppose we have some polynomial equation, that is an equation such as [math]2x^2+3x-7=9[/math] where the terms are all of the form [math]ax^n[/math] for [math]a[/math] some number and [math]n[/math] a natural number and where the terms are combined only by addition and subtraction.Experimental evidence suggests polynomials equations such as the above can serve as good physical models of projectile motion such as when one throws a baseball through the air. Solving such an equation in this context might correspond to the time [math]t[/math] at which the flight of the ball concludes, or, in a more pressing example, perhaps the time it will take for a rocket to return to Earth [7].An equation one might actually see in this context is[math]0=v_0 t \sin\theta - \frac{1}{2}(9.81)t^2, [/math][math][/math][math] \tag*{}[/math]where [math]\sin[/math] is the trigonometric sine function, [math]\theta[/math] is the angle of launch, and [math]v_0[/math] is the initial velocity[3][3][3][3].Thankfully, there are methods, both approximate and exact, for solving such equations. Without such mathematical tools, much of engineering would be rendered impossible thus leaving modern society nonexistent. The best known and perhaps nicest way to solve such an equation is with the quadratic formula, which is reproduced below:[math]x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}. \tag*{}[/math]The two example equations above are both quadratic equations, meaning the highest power on [math]x[/math] is [math]2[/math]. What if that power is larger? Can we still solve these equations for any power? Is there a nice formula like the one above? (And, yes, polynomials of higher order do have practical use.)For the cases of [math]n=3[/math] and [math]n=4[/math], we have known exact formulas since the sixteenth century. Amazingly, however, there is no such formula for the [math]n=5[/math] case. Indeed, for all [math]n>5[/math], there is no algebraic solution to general polynomial equations with arbitrary coefficients [8]. This is known as the Abel-Ruffini theorem, named for Paolo Ruffini who provided a partial proof in 1799 and Niels Henrik Abel who proved the result in full in 1824[4][4][4][4].That proof, or conclusive demonstration of the formula, works, but it is not particularly nice. It was not long before it was superseded by the brilliant work of the young French mathematician Évariste Galois whose work not only appealed to the aesthetic sensibilities of mathematicians (both then and now) but also did much more than solve this one particular problem. Galois invented an entire field, one named for him: Galois theory. Fruitful work on this subject continues today.Care to take a guess what else Galois laid the foundations for in his work? That is right, Galois also provided some of the initial development of group theory! As odd as it may seem, the abstract groups we discussed before are actually needed for Galois’ proof—actually, they are used extensively. The sorts of groups most important for the proof are now known as Galois groups after their discoverer who sadly died due to injuries sustained in a duel (yes, seriously[5][5][5][5]) soon after completing his work.It is hopefully clear now why passing to abstraction is powerful. If not, the author still has one example up this sleeve: the humble number [math]0[/math].For those not familiar with the history of mathematics, it may come as a surprise that the advent of [math]0[/math] as a widespread concept is relatively recent [9]. It was not immediately easy that having a symbol to denote nothingness was of any use. Perhaps the reader is unsure why we even have it. Here are two incredibly useful applications of the concept.Writing numbers without a positional notation can be a pain for humans. To see this fact, try doing some basic arithmetic using only Roman numerals. Of course, the skill can be taught nonetheless[6].Often in the course of solving polynomial equations, one comes to something like [math](A)(B)=0[/math] (where [math]A[/math] and [math]B[/math] can be basically arbitrarily complicated expressions), at which point one can conclude [math]A=0[/math] or [math]B=0[/math] [10]. Now, how could we do this without zero?And the fun does not end because the second example above shows the utility of the group concept as well. Algebraic structures such as groups allow us to say exactly when the implication [math]ab=0 [/math][math][/math][math] \implies a=0[/math] or [math]b=0[/math] holds! (This implication actually goes both ways.) As it turns out, the answer is this works in any integral domain; it is this way by definition[7][7][7][7].In Grothendieck’s letter, he is actually alluding to schemes, abstract objects in algebraic geometry that Grothendieck introduced in the 1960s in his masterpiece Éléments de géométrie algébrique (EGA). Like the group concept or the use of [math]0[/math], Grothendieck was considering something very simple and abstract, something that could seem at first to be useless.Schemes are a generalization of algebraic varieties, historically the primary objects of study in algebraic geometry. The simplest definition of a variety is as no more than the solution set of a system of polynomial equations, which comes with a natural geometry[8][8][8][8]. One famous example of a variety is an elliptic curve, which is a “nice” curve defined by a polynomial in two variables [11][math]y^2=x^3+ax+b. \tag*{}[/math]These curves have an aesthetic geometry, happen to form a group, and are of key importance to not just algebraic geometry but also number theory and even computer security [9][9][9][9][12] (despite Grothendieck et al. obviously not knowing about the latter at the time of developing scheme theory). Below is a picture of two elliptic curves.(Source: Elliptic curve - Wikipedia (https://en.wikipedia.org/wiki/Elliptic_curve#/media/File:ECClines-3.svg))Hopefully, mathematicians’ interest in curves such as the above is immediately clear. How exactly schemes generalize varieties such as elliptic curves is unfortunately beyond the scope of this answer (see Vakil for an introduction, arguably the best one), but there are two important components: i.) Scheme theory requires a non-trivial amount of very abstract algebra, including category theory. ii.) The development of scheme theory involves a seemingly endless string of abstract and confusing, but relatively easy to prove, results.The first point is simply a matter of fact. Vakil starts his text (oops, I mean notes) with a chapter on categories for a reason. Similarly, most graduate scheme theory courses require at least a year of graduate algebra, in the course of which some category theory is taught.The second point is perhaps more interest, more revealing of the “childish steps” Grothendieck mentions. Rather than describe what Grothendieck’s style of mathematics is like, let us allow him to do the honors.I can illustrate the ... approach with the ... image of a nut to be opened. The first analogy that came to my mind is of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better, and otherwise you let time pass. The shell becomes more flexible through weeks and months — when the time is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado! . . .A different image came to me a few weeks ago. The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration ... the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it ... yet finally it surrounds the resistant substance.– Alexander Grothendieck in his autobiographical work Récoltes et semailles, as quoted in Vakil [10][10][10][10](p. 9) [13]Finally, lest one think we have finally ventured into the realm of abstract nonsense proper, we should note that even scheme theory has its uses. Besides being used extensively in arithmetic geometry and other subjects, schemes were used by Pierre Deligne to prove the Weil conjectures in 1974[11][11][11][11]. These conjectures, due to celebrated number theorist André Weil, were modeled directly on the Riemann hypothesis, one of the most famous unsolved problems in mathematics and one of the Millenium Prize Problems of the Clay Mathematics Institute, worth $1,000,000[12][12][12][12].Again, describing the Riemann hypothesis in any detail is too much to do here, but the interested reader can see Daniel McLaury’s answer to What is the Riemann hypothesis?. For the sake of showing schemes’ significance, it suffices to know the Riemann hypothesis tells us something about the prime numbers (i.e., a number greater than [math]1[/math] whose only factors are [math]1[/math] and itself), which, by the fundamental theorem of arithmetic, one of the most fundamental results in all of mathematics, are the building blocks of all integers in the sense that any integer is a unique product of primes.Like elliptic curves, the Riemann hypothesis has some relation to cryptography, although claimed practical consequences of a proof are, almost as a rule, overstated[13][13][13][13].Notes[1] While the 1945 paper is indeed the first work in the mathematical literature devoted to category theory, the notion of a category was officially introduced in 1942 in Natural Isomorphisms in Group Theory[14][14][14][14]. It is here that the term “general abstract nonsense” is first used to describe the subject. This is despite the abstract nonsense being invented to do more concrete work at the intersection of Eilenberg’s topology and Mac Lane’s algebra. One of those uses was formalizing the notion of a construction or map being “natural,” hence the title of category theory’s seminal work.[2] This is somewhat surprising given the following two facts: i.) The usual definition of a category requires sets and classes. This suggest category theory ought to require set theory, which is already the de facto foundation for mathematics[15][15][15][15]. ii.) Category theory can be seen, somewhat perversely, as a subfield of graph theory! [16][16][16][16][3] Incidentally, the study of groups led quite directly to the invention of category theory. The historical line can be traced back almost arbitrarily far, but Emmy Noether’s emphasis on structure-preserving maps (homomorphisms or, more generally, morphisms) and her first isomorphism theorem are especially noteworthy.[4] The notation [math]\mathbb{Z}[/math] is used for the integers; the composite notation [math]\mathbb{Z}/n\mathbb{Z}[/math] denotes the quotient group of the integers by their normal subgroups, [math]n\mathbb{Z}[/math], which can be thought of as collapsing each of the numbers that differ by a multiple of [math]n[/math] into one entity. The symbol [math]\cong[/math] means the two are isomorphic, which is a sort of weak notion of equivalence. See the author’s exposition on these matters at his blog.[17][17][17][17][5] This is readily seen by proving the cyclic groups are exactly the quotient groups [math]\mathbb{Z}/n\mathbb{Z}[/math], up to isomorphism.[6] The structure theorem for finite abelian groups states that all finite abelian groups are isomorphic to direct sums of cyclic groups.[18][18][18][18][7] Physics is the natural science concerned precisely with modeling such situations via mathematics[19][19][19][19]. We owe this science much, for without it the electricity, telephones, and the Internet would not exist. In this example, we use exceptionally simple models, the likes of which one might see in a first course on the subject in high school or college.[8] There is some nuance here. We actually know formulas for these equations, but they are not particularly nice, not just in the sense that they may be long (that is true even in the case of the cubic and quartic), but in that they involve “non-standard” operations. There are also many methods of solving special cases of these equations, and we can find approximate solutions with relative ease.[9] From so many perspectives, the story of [math]0[/math] is a fascinating one. See, for example, the popular book by Seife for more details.[10] This is why we carry out of the procedure of setting both parts equal to zero when solving polynomial equations. Sadly, this is rarely explained.[11] Here is a small lie. Elliptic curves can be somewhat more complicated, but provided a few conditions hold, which they often do, we can write elliptic curves in this form, called the Weierstrass form after nineteenth century German mathematician Karl Weierstrass, one of the founders of modern analysis, known to students everywhere due to the Bolzano-Weierstrass theorem.[12] The book in the adjacent footnote is written by my (informal) advisor. No doubt, I am biased, but I was familiar with Larry’s exceptional exposition—primarily through his monograph Introduction to Cyclotomic Fields, one of the works that made his name and still one of the standard references on the subject—before even coming to UMD and meeting him. Moreover, the opinion that his book is one of the cleanest and most accessible works for pure and applied mathematicians alike is far from unique to me.[13] Notice that Vakil’s text is named for the rising sea analogy Grothendieck here employs. These analogies are well known among algebraic geometers and even other mathematicians besides. The philosophy is core to algebraic geometry, and, in the author’s opinion, Grothendieck’s gifting it to us as he did is an underrated contribution.Footnotes[1] https://groupoids.org.uk/pdffiles/eureka-meth1.pdf[1] https://groupoids.org.uk/pdffiles/eureka-meth1.pdf[1] https://groupoids.org.uk/pdffiles/eureka-meth1.pdf[1] https://groupoids.org.uk/pdffiles/eureka-meth1.pdf[2] foundation of mathematics in nLab[2] foundation of mathematics in nLab[2] foundation of mathematics in nLab[2] foundation of mathematics in nLab[3] Projectile motion - Wikipedia[3] Projectile motion - Wikipedia[3] Projectile motion - Wikipedia[3] Projectile motion - Wikipedia[4] http://www.abelprisen.no/nedlastning/verker/oeuvres_1839/oeuvres_completes_de_abel_1_kap02_opt.pdf[4] http://www.abelprisen.no/nedlastning/verker/oeuvres_1839/oeuvres_completes_de_abel_1_kap02_opt.pdf[4] http://www.abelprisen.no/nedlastning/verker/oeuvres_1839/oeuvres_completes_de_abel_1_kap02_opt.pdf[4] http://www.abelprisen.no/nedlastning/verker/oeuvres_1839/oeuvres_completes_de_abel_1_kap02_opt.pdf[5] http://discovermagazine.com/2006/jan/mathematician[5] http://discovermagazine.com/2006/jan/mathematician[5] http://discovermagazine.com/2006/jan/mathematician[5] http://discovermagazine.com/2006/jan/mathematician[6] Roman Numeral Arithmetic[7] How do I prove that ab=0 if a=0 or b=0?[7] How do I prove that ab=0 if a=0 or b=0?[7] How do I prove that ab=0 if a=0 or b=0?[7] How do I prove that ab=0 if a=0 or b=0?[8] http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf[8] http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf[8] http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf[8] http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf[9] Elliptic Curves: Number Theory and Cryptography, Second Edition (Discrete Mathematics and Its Applications): Lawrence C. Washington: 9781420071467: Amazon.com: Books[9] Elliptic Curves: Number Theory and Cryptography, Second Edition (Discrete Mathematics and Its Applications): Lawrence C. Washington: 9781420071467: Amazon.com: Books[9] Elliptic Curves: Number Theory and Cryptography, Second Edition (Discrete Mathematics and Its Applications): Lawrence C. Washington: 9781420071467: Amazon.com: Books[9] Elliptic Curves: Number Theory and Cryptography, Second Edition (Discrete Mathematics and Its Applications): Lawrence C. Washington: 9781420071467: Amazon.com: Books[10] http://math.stanford.edu/~vakil/216blog/FOAGnov1817public.pdf[10] http://math.stanford.edu/~vakil/216blog/FOAGnov1817public.pdf[10] http://math.stanford.edu/~vakil/216blog/FOAGnov1817public.pdf[10] http://math.stanford.edu/~vakil/216blog/FOAGnov1817public.pdf[11] La conjecture de Weil : I[11] La conjecture de Weil : I[11] La conjecture de Weil : I[11] La conjecture de Weil : I[12] Clay Mathematics Institute[12] Clay Mathematics Institute[12] Clay Mathematics Institute[12] Clay Mathematics Institute[13] Proving the Riemann Hypothesis and Impact on Cryptography[13] Proving the Riemann Hypothesis and Impact on Cryptography[13] Proving the Riemann Hypothesis and Impact on Cryptography[13] Proving the Riemann Hypothesis and Impact on Cryptography[14] https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1078535/pdf/pnas01647-0031.pdf[14] https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1078535/pdf/pnas01647-0031.pdf[14] https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1078535/pdf/pnas01647-0031.pdf[14] https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1078535/pdf/pnas01647-0031.pdf[15] Joseph Heavner's answer to How does group theory relate to set theory?[15] Joseph Heavner's answer to How does group theory relate to set theory?[15] Joseph Heavner's answer to How does group theory relate to set theory?[15] Joseph Heavner's answer to How does group theory relate to set theory?[16] https://ocw.mit.edu/courses/mathematics/18-s996-category-theory-for-scientists-spring-2013/textbook/MIT18_S996S13_chapter4.pdf[16] https://ocw.mit.edu/courses/mathematics/18-s996-category-theory-for-scientists-spring-2013/textbook/MIT18_S996S13_chapter4.pdf[16] https://ocw.mit.edu/courses/mathematics/18-s996-category-theory-for-scientists-spring-2013/textbook/MIT18_S996S13_chapter4.pdf[16] https://ocw.mit.edu/courses/mathematics/18-s996-category-theory-for-scientists-spring-2013/textbook/MIT18_S996S13_chapter4.pdf[17] Quotients & Homomorphisms for Beginners: Part 1, Motivation[17] Quotients & Homomorphisms for Beginners: Part 1, Motivation[17] Quotients & Homomorphisms for Beginners: Part 1, Motivation[17] Quotients & Homomorphisms for Beginners: Part 1, Motivation[18] Abelian group - Wikipedia[18] Abelian group - Wikipedia[18] Abelian group - Wikipedia[18] Abelian group - Wikipedia[19] Physics - Wikipedia[19] Physics - Wikipedia[19] Physics - Wikipedia[19] Physics - Wikipedia