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Perks of Life at IIT are many. I'll try to jot down the few intangible ones that were relevant in my case:Developing a knack to face tough situations: You won't even realize that you're getting molded for the tough world out there. You'll be thrown in a lot of uncomfortable situations and would be expected to swim your way out of them. The change is something you'll only get to know once you graduate out of the hallowed portals of IIT. The learning curve is steep and as such helps us to gain some real life worldly knowledge as well apart from the academics only.When I look back at the 4 years of my under graduation, studies are the last thing I remember. Sure, I wasn't a real chip off the block in terms of studies but the feeling is mutual among all the students irrespective of their academic caliber.IIT would always be close to my heart in the sense that it helped me transform from a shy & studious kid to an individual who is willing to venture out in unknown fields and enjoy doing them.I had never taken part in dramatics or hockey before my college. Once there I was pushed to do them and despite my reluctance I began enjoying them with time. The late night sports fest core team meetings, the hall events, the quizzes and the departmental fests, all of them hold a special place in my heart and mind. The feeling of nostalgia is still a pretty strong one even though it hasn't been long since I graduated.On the academic front I was handed a thesis topic which I hadn't even heard of before. Add to it the fact that I didn't meet my guide for the first 3 months into my 7th semester. I was told in harsh words that I wouldn't be allowed to do the project and would have to repeat it the next year. Somehow after a lot of crying and requesting he managed to give me something to work with on the condition that if i work satisfactorily only then I would be allowed to clear the subject. I managed somehow.The end point that I'm trying to make is come what may you'll learn to get yourself out of the murky waters. Your college is there to teach you precisely that.Keeping and maintaining your calm and cool: There were days when the feeling to give up was pretty strong. Especially during my second year when I was on the verge of flunking in a few subjects, I was about to quit all of my extra curricular involvements. Somehow, on the consultation of my friends and seniors, I managed to hang on. Scored a GPA of 6.99 in my third semester (Yeah it was bad :P) after surviving not getting de registered in a couple of subjects. The most hectic semester of my IIT life was over and I had faced the storm. That semester showed me that everything can be managed only and only if you can be composed.Similar case when it came to my final year B.Tech. project report thesis. As I have already elaborated the situation with it in the first point, I was spiraling down into tension and frustration on having chosen the faculty and the thesis topic which I did. Thought of dropping the project to only finish it later but gave it a shot after thinking what's the worst that could happen. Add to it the fact that my guide used to threaten me every now and then regarding the work I was doing as it won't pass his final judgement. I used to work 5-6 hours daily for my thesis after getting a second chance (Yeah, for true while others used to slack off due to their less hectic topics or faculties I was made to face the music.) Add to it the fact that I had given the CAT that year and had all my IIM WAT-PI calls lined up along with a hectic final semester schedule of Vivas and project presentations. The urge to give up either of my projects or IIM calls was too strong but in the end gave my best shot at both of them an the rest went pretty well. I was also in a relationship with a girl from IIT-BHU for the past 3 years and our relationship went amiss during all this (not due to these circumstances).That breakup also had its effect on me. I had to give up playing hockey for the first time since I picked it up after coming to college in order to be able to clear the subjects and attend interviews.And as they say all's well that ends well. Passed the project, converted IIM L & IIM I and now a student of IIM L.Long story shott I learnt to let the stress go. Gave my best possible and left the rest of the things to take their natural course. Tried and did whatever I could, but you see life has its own plans. Learnt to not take tension and get frustrated if things don't turn out the way I expected them to be. Letting go and consoling myself that next time it would be in my favour has worked pretty good for me and is still helping me survive here.Letting the Panic go is what you'll learn at an IIT if you are lucky enough.Having made few friends for life: You have the best of the lot from the entire nation to mingle with and form bonds for life. They are the ones who wake you up, offer you your first college advice, grow with you and support you. I have learnt about life and world more from the hour long discussions with them on random topics then I have from any book or piece of literature. They are the ones who show you that life doesn't deserve to be serious all the times. There should be dominant sparks of madness scattered in between. This is probably the last time (not true in my case though :P) that you'll be this free enough to enjoy life by the virtue of being in a college. The basic fact that they taught me was that life isn't supposed to be a rat race. You should be making delta improvements day on day rather than a high jump all of a sudden. Comparisons with others is a job best done by society and we don't need to be that critical of ourselves. Try and regardless of whether you fail or succeed you'll learn.Some of the best moments that I had were owed to the awesome friends that I had encountered during my IIT journey. My first cigarette (Not a smoker anymore :D), the first drink, the first party, the first Holi & Diwali celebrations, the quizzing wins, the hockey medal, the special birthdays, the late night event practices and a plethora of such memories that I'll remember for a lifetime and be thankful for most of them were made during those days. We might not talk daily now but deep in my heart I know that they'll be the ones who'll be true to me come what may.Growing apart doesn’t change the fact that for a long time we grew side by side; our roots will always be tangled. I’m glad for that.I'll keep on updating the list as I come across more points worth mentioning.

The truth is that bitcoin is the hottest trading market right now. It is hotter than stock trading, oil trading, gold trading and any other market at this point. The reason people believe this is going to continue to be a hot market is because of blockchain technology. This is what allows transactions to happen without a central exchange. Here is another strategy on how to draw trend lines with fractals.People dont realise that there are other options of making more on Bitcoin and other cryptocurrency. example , investing on platforms like coinexplorer (www.coinexplorer.tech) where you get double of your invested cryptocurrency after 7days. With that, you cant lose on cryptocurrency. Thank me later.The trading bitcoin for profit is actually a universal cryptocurrency trading strategy. It can be used to trade any of the 800-plus cryptocurrencies available to trade as of today. If you’re not already familiar with cryptocurrencies it’s best to first start with a brief introduction.How to Start Trading Bitcoin:The first thing you need to get started trading bitcoin is to open a bitcoin wallet. If you do not have a bitcoin wallet then you can open one at the biggest wallet called Coinbase. We have arranged a special deal for everyone wanting to get started in bitcoin to get a free $10 at Coinbase. Get your free $10 by opening your coinbase account here.Bitcoin traders are actively seeking the best possible solutions for trading and investing in bitcoin. We have some of the best methods explained right here in this article. We have learned this bitcoin wisdom by trial and error and we are going to show you what is working right now. The methods we teach are not dependent on the price of bitcoin. They can be used whether bitcoin is going up or going down.Keep in mind that it is possible to lose money and your capital is at risk while trading cryptocurrency because it is still trading at the end of the day. We always recommend that you demo trade before risking any live money. Also, read the trading volume guide.These bitcoin strategies can also be used for trading bitcoin cash as well as other cryptocurrencies. In fact, you can use this as a trade guide for any type of trading instrument. The blockchain technology is a big step forward for how to access information. Many companies are starting to develop applications to use it in their favor. Remember that when trading digital currency, it may seem like it is not a real currency. But it actually is real. This is not some Ponzi scheme. Before you buy bitcoins, have a solid plan in place and don’t underestimate the cryptocurrency markets. You must do your technical analysis just as if you were going to day trade any other instruments. You can also read our best Gann Fan trading strategy.Top Exchanges for Trading Bitcoin & CryptocurrenciesOne of the reasons why Bitcoin is so popular among day traders is that there are many different Bitcoin exchanges available. Finding the best Bitcoin exchange will depend on many different factors. These include your home country, the preferred method of payment, fees, limits, liquidity needs, and other factors.Here are some of the top cryptocurrency exchanges in the market:Coinbase is the world’s largest crypto exchange. Available in the United States, Canada, and the majority of countries in Europe. Offers several payment options.Binance is the second largest exchange that trades over 130 different currencies. Has low transaction fees (0.1%).Bitmex is the third largest exchange and only trades BTC. Great for short selling and margin trades.Bittrex is a US-based exchange founded by ex-Microsoft security professionals.Robinhood is a new exchange with 6 million users and takes zero trading fees.OKEx is a Hong Kong-based exchange. Trades over 145 different cryptocurrencies.GDAX - United States based exchange that allows users to trade Bitcoin, Ether, Litecoin, and other cryptocurrencies.itBit operates as both a global over-the-counter (OTC) trading desk and a global Bitcoin exchange platform.Coinmama - Allows you to buy and sell easily. Accepts credit cards and has a large global reach.What is This Free Bitcoin Trading Strategy?A cryptocurrency is really no different than the money you have in your wallet. They have no intrinsic value. And cryptocurrency are just bits of data while real money is just pieces of paper.trading bitcoin for profitunlike fiat money, Bitcoins and other cryptocurrencies have no central bank that controls them. This means that cryptocurrencies can be sent directly from user to user without any credit cards or banks acting as the intermediary. The major advantage of cryptocurrencies is that you can’t print them like central banks do to create fiat money.When you print lots of money, inflation goes up which makes the currency value going down. There is a limited amount of Bitcoins. This holds true for the majority of the other cryptocurrencies. The supply side can’t increase which makes Bitcoin less prone to being affected by inflation.Now, let’s move forward and see how we can profit from the cryptocurrency mania. We will use our best Bitcoin trading strategy. We also have training for the best short-term trading strategy.How to Day Trade BitcoinWhile long term traders prefer to hold their bitcoin positions for extended periods of time, day traders have discovered that Bitcoin is lucrative for many reasons:Crypto trading is more volatile than stock trading.Bitcoin is traded 24 hours per day 7 days a week.Bitcoin allows for big trades with low overhead.Bitcoin is the most liquid form of cryptocurrency.Multiple trading opportunities emerge within a 24 hour period.Because Bitcoin is more volatile than other tradeable assets, there will be a high number of profitable trading opportunities occurring each day. Like ordinary currencies, using technical indicators will make it easier to tell when price increases are likely to occur. Volume, relative strength, oscillatiors, and moving averages can all be applied to Bitcoin day trading.It is important to pay attention to technical indicators and developing trends. In this next step, we will talk about OBv trading and how to get started buying and selling cryptocurrencies.The Best Bitcoin Trading Strategy – 5 Easy Steps to ProfitThis is a cryptocurrency trading strategy that can be used trading all the important cryptocurrencies. Actually, this is an Ethereum trading strategy as much as it’s a Bitcoin trading strategy. If you didn’t know Ethereum is the second most popular cryptocurrency (see figure below).Ethereum trading strategyThe best Bitcoin trading strategy is an 85% price action strategy and a 15% cryptocurrency trading strategy that uses an indicator.Now…Before we move forward, we must define the mysterious technical indicator. You'll need this for the best Bitcoin trading strategy and how to use it:The only indicator you need is the:On Balance Volume (OBV): This is one of the best indicators for day trading bitcoin. It is used to basically analyze the total money flow in an out of an instrument. The OVB uses a combination of volume and price activity. This tells you the total amount of money going in and out of the market.cryptocurrency trading for beginnersOBV IndicatorThe OBV indicator can be found on most trading platforms like Tradingview and MT4. How to read the information from the OBV indicator is quite simple. Here you can learn how to profit from trading.In theory, if Bitcoin is trading up and at the same time the OBV was trading down, this is an indication that people are selling into this rally. The move to the upside wouldn’t be sustainable. The same is true in reverse if Bitcoin was trading down and at the same time the OBV was trading up.trading bitcoin for profitWhat we really want to see is the OBV moving in the same direction as the Bitcoin price. Later on, you’ll learn how to apply this information together with the cryptocurrency trading strategy.No technical indicator is 100% effective every single time. In this regard, our team at Trading Strategy Guides uses the OBV indicator with other supporting evidence to sustain our trades and give some more confirmation of our trades. On this part comes the Ethereum trading strategy which will be used to identify Bitcoin trades.Now, before we go any further, we always recommend taking a piece of paper and a pen and note down the rules of the best Bitcoin trading strategy.Let’s get started…..The Best Bitcoin Trading Strategy -(Rules for a Buy Trade)Step #1: Overlay the Bitcoin chart with Ethereum chart and the OVB indicator.Your chart setup should basically have 3 windows. One for the Bitcoin chart and the second one for the Ethereum chart. Last but not least, make one window for the OVB indicator.cryptocurrency trading strategyIf you followed our cryptocurrency trading strategy guidelines, your chart should look the same as in the figure above. For now, all should be good, so it’s time to move forward to the next step of our best Bitcoin trading strategy.Step #2: Look for Smart Money Divergence between Bitcoin price and Ethereum price.What do we mean by this?Simply put, we are going to look after price divergence between Bitcoin price and Ethereum. Smart money divergence happens when one cryptocurrency fails to confirm the action of the other cryptocurrency.For example, if Ethereum price breaks above an important resistance or a swing high and Bitcoin fails to do the same, we have smart money divergence. It means that one of the two cryptocurrencies is “lying.” This is the main reason why we have called this cryptocurrency trading strategy. And Ethereum trading strategy as well.cryptocurrency trading strategyIf you’re still struggling to identify support and resistance we’ve got your back, simply read our guide on this topic here: Support and Resistance Zones – Road to Successful Trading.In the above figure, we can notice that Bitcoin price fails to break above resistance while Ethereum price broke above and made a new high which is the first sign that the best Bitcoin trading strategy is about to signal a trade.The reason why the smart money divergence concept works is because the cryptocurrency market as a whole should move in the same direction when we’re in a trend. The same principles have been true for all the other major asset classes for decades. It’s true for the cryptocurrency trading strategy as well.Before buying, we need confirmation from the OBV indicator. This brings us to the next step of the best Bitcoin trading strategy.Step #3: Look for the OVB to increase in the direction of the trend.If Bitcoin is lagging behind Ethereum price it means that sooner or later the Bitcoin should follow Ethereum and break above resistance.But, how do we know that?Simply, the OBV is a remarkable technical indicator. It can show us if the real money is really buying Bitcoin or quite the contrary they are selling. What we want to see when Bitcoin is failing to break above a resistance level or a swing high and the Ethereum already broke is for the OBV to not only increase in the direction of the trend but to also move beyond the level it was when Bitcoin was trading previously at this resistance level (see figure below). Here is how to identify the right swing to boost your profit.cryptocurrency trading strategyNow, all it remains for us to do is to place our buy limit order, which brings us to the next step of the best Bitcoin trading strategy.Step #4: Place A Buy Limit Order at the resistance level in an attempt to catch the possible breakout.Once the OBV indicator gives us the green signal, all we have to do is to place a buy limit order. Place the order at the resistance level in anticipation of the possible breakout.cryptocurrency trading strategyIt’s no surprise to see this trade getting triggered and for the Bitcoin price to break higher as expected after all we told you the OBV is an amazing indicator.Now, all we need to establish is where to place our protective stop loss and when to take profits for the best Bitcoin trading strategy.Step #5: Place your SL below the breakout candle and take profit once the OBV reaches 105,000.Placing the stop loss below the breakout candle is a smart way to trade. We’ve written more about the reasons for hiding your SL above/below the breakout candle in our most recent article here: Breakout Trading Strategy Used by Professional Traders.cryptocurrency trading strategyWhen it comes to our take profit usually an OBV reading above 105,000 is an extreme reading that signals at least a pause in the trend which is why we want to take profits.Note** The above was an example of a buy trade… Use the same rules – but in reverse – for a sell trade. In the figure below, you can see an actual SELL trade example, using the best Bitcoin trading strategy.cryptocurrency trading strategyWays to Enhance This Bitcoin Day Trading StrategyWhile bitcoin day trading does have some risks, there are many ways these risks can be reduced. Here are some of the top ways to enhance your Bitcoin trading strategy.Remember to:Diversify your trades. Combining Bitcoin, Ripple, Litecoin, Ethereum, and other cryptocurrencies will help reduce the daily risk associated with a specific coin.Minimize trading costs. Opening multiple positions every day affects your daily ROI. To minimize the cost of trading, choose a trustworthy exchange that has low fees.Watch Trading times. Plan out trading times that are compatible with your schedule. Bitcoin trades 24 hours a day. It's different from the 9-5 NYSE.Follow Bitcoin News. Pay attention to cryptocurrency news stories to stay ahead of the market. Set up alerts and other types of notifications.Use technical analysis. Use strong technical indicators like OBV. This will help you justify each of your trades.Use stop losses. Set stop loss orders on every trade. Start with a profit loss ratio of 2:1.

I am a graduate of mathematics and theoretical physics, and a life-long self-learner in these subjects, so I think I have a lot of experience in this area.To answer the question….It all depends what the questioner means by 'want[ing] to become very good at math'.How do you define 'very good'? And what sort of mathematics do you want to learn, and why?Let's address the second question first. With any learning activity - and particularly with self-directed learning - it is important always to have the objective of that learning in mind, firstly so that you can plot an efficient course towards it (thereby hopefully avoiding wasting your time on unproductive detours and irrelevant distractions), and secondly so that you'll know when you have reached it.Mathematics is a kind of language (in fact, it is many, many different but interrelated languages), and so there are parallels between learning mathematics and learning languages. When you decide to learn a foreign language, you must first choose the language you want to learn, and this choice will be guided by such considerations as the following:The language's potential usefulness to you personally, i.e. is it widely spoken, or required for your job, or do you plan to visit the country (or countries) where it is spoken?The language's aesthetic qualities, i.e. is its spoken form particularly pleasing to the ear (e.g. many people like Italian because of its varying rhythm and expressive tonality), or its written form and alphabet especially pleasing to the eye (many people like languages with - to them - 'exotic' non-Latin alphabets such as Russian, Arabic or Japanese)?The language's level of difficulty, i.e. is it an easy language to learn, in terms of its pronunciation, written form and grammatical structure, or a difficult one that you might perhaps choose if you are looking for a challenge?Mathematics is a vast and constantly expanding discipline, with several major subject divisions (e.g. algebra, geometry, analysis, topology) and hundreds of subdivisions and specializations. Just as with languages, different branches of mathematics may have different degrees of usefulness to you, or different aesthetic qualities in terms of the beauty of their central ideas or their written form. And they will certainly present different levels of difficulty in terms of their grammar, vocabulary and conceptual abstraction.So which should you choose? To sharpen your focus on just those areas that might be of interest and relevance to you, ask yourself some basic questions, such as:Do I want ultimately to obtain academic qualifications in mathematics, or simply to study it for mental stimulation, as a purely recreational interest? (Language analogy: Do I want to become a professional linguist, translator or interpreter, or simply to learn a language for pleasure?)Do I want to be able to use my mathematical knowledge to solve practical problems, or am I not interested in its practical applications? (Language analogy: Do I want to use a foreign language to understand books, music or films produced in it, or to communicate with native speakers, or just to learn it as an end in itself?)Do I know which areas of mathematics interest me? Do I even appreciate how many different areas of mathematics there are?If you are aiming to study mathematics at college/university to degree level, or further, then your mathematical study will need to be sufficiently broad to cover the sorts of topics you are likely to encounter in a typical undergraduate mathematics course. The mathematics curricula of different universities will differ in their choice of topics and the depth to which they are developed in a degree course, these differences being dictated to a certain extent by the interests of the lecturers, the research activities of the mathematics (and physics) faculty, and whether the course is more geared towards pure or applied mathematics. But all mathematics curricula will have certain broad subject areas in common. The easiest way to discover what these are would be to visit the websites of various university maths faculties and look at the subjects they teach. These are likely to include the following topics (many of which overlap each other) as a minimum (listed in no particular order, though some topics are more advanced than others, and have certain prerequisites):Set TheoryGroup TheoryAlgebra (linear, abstract, etc.)Differential and integral calculus of a single variableDifferential and integral calculus of several variablesOrdinary differential equationsPartial differential equationsReal AnalysisComplex AnalysisTopologyDiscrete mathematics (combinatorics, graph theory, etc.)Number TheoryGeometry (projective, differential, etc.)Probability theoryStatistics (statistics is often taught as a discipline in its own right, rather than as part of a maths course).Generally, the more prestigious the university or college, the more advanced the material in its degree courses will be, so some (e.g. Ivy League (US) and Russell Group (UK) universities) will introduce topics at the undergraduate level that others (e.g. smaller 'provincial' universities) would only cover at the postgraduate/Masters level.If you want to use mathematics to solve practical problems, then your choice of mathematical subjects will depend on what sort of practical problems you are interested in. But all 'practical' mathematical problem solving will require certain basic skills, and familiarity with some of the more generic tools of mathematics, such as algebra, geometry, calculus, combinatorics, probability and statistics.If you are thinking of studying mathematics purely as an intellectually absorbing pastime or to 'improve your mind' (it serves both purposes for many 'amateur' mathematicians), but are unsure of which areas of mathematics you might be interested in, or of how many different areas of mathematics there actually are, then I recommend that you first try to do a broad, high-level survey of the field by reading some nontechnical books on mathematics for the popular science market. These should preferably cover many different areas of mathematics, devoting - say - one chapter to each, rather than focusing on a single mathematical area or theme for the entire book (such as the many books on Fermat's last theorem). There are many such 'general survey' books on mathematics, but I particularly like - and highly recommend - those of the British mathematician Prof. Ian Stewart, many of which I read when embarking on my own mathematical education (I was 99% self-taught in mathematics before going on to study it at university). Ian Stewart writes in a very clear, accessible and entertaining style, and really brings his subject alive for the reader. I can particularly recommend the following:http://www.amazon.com/Letters-Young-Mathematician-Art-Mentoring/dp/0465082327/ref=sr_1_8?ie=UTF8&qid=1299159481&sr=8-8http://www.amazon.com/Concepts-Modern-Mathematics-Ian-Stewart/dp/0486284247/ref=sr_1_4?ie=UTF8&qid=1299159481&sr=8-4http://www.amazon.com/Here-Infinity-Ian-Stewart/dp/0192832026/ref=sr_1_7?s=books&ie=UTF8&qid=1299160016&sr=1-7http://www.amazon.com/Mathematics-Elementary-Approach-Ideas-Methods/dp/0195105192/ref=sr_1_8?s=books&ie=UTF8&qid=1299160016&sr=1-8http://www.amazon.com/Natures-Numbers-Reality-Mathematics-Science/dp/0465072747/ref=sr_1_14?s=books&ie=UTF8&qid=1299160136&sr=1-14Two other excellent books that follow the mathematical-area-per-chapter format are:http://www.amazon.com/Five-Golden-Rules-20th-Century-Mathematics/dp/0471193372/ref=sr_1_1?s=books&ie=UTF8&qid=1299161189&sr=1-1http://www.amazon.com/Five-More-Golden-Rules-20th-Century/dp/0471395285/ref=sr_1_2?s=books&ie=UTF8&qid=1299161189&sr=1-2Once you have got an idea about the different areas of mathematics out there, you can decide (perhaps based also on your answers to the first two questions above) whether you want to focus on one area alone, or a few particular areas, or be a 'generalist' and study a wide variety of different areas. [1]Now that you have decided which areas of mathematics you want to study, you need to answer the first question I raised: Exactly how 'good' do you want to become as a mathematician? Once again, the language analogy is helpful: When you are learning a foreign language, you have to decide what level of proficiency would be sufficient for your needs. Do you want to be able to converse confidently and fluently with native speakers (difficult - for most people)? Do you just want to be able to understand books, music or films produced in that language (easier)? Or do you simply want to be able to use the language on vacation, to read road signs and restaurant menus, and to ask and answer simple questions in hotels, banks and shops (easiest)? The same sorts of questions apply to mathematical proficiency; do you want 'conversational fluency', or reasonable literacy when reading mathematical books, or just to be able to 'get by' and read the road signs?I'll conclude with a few words of advice regarding a general strategy for learning mathematics from books. [2]Firstly, understand that you cannot read a mathematics book in the same way that you read a novel. Reading a mathematics book should be an active - and interactive - experience, not a passive one. You will be required to follow detailed arguments and - often complex and subtle - lines of reasoning, and to continually answer questions that the text poses to you. In particular, you will be required to check for yourself that you understand all the mathematical derivations, ideally by reproducing the calculations and confirming the results for yourself. You will therefore need a notepad in which to do your own calculations as you read the book.If you own the book, make extensive use of annotation; underline, highlight or otherwise mark significant words, sentences and equations so that they will stand out on subsequent re-reading. And you should always read everything at least three times; firstly, with a light, skimming preview to get the general gist of the material; then, with a detailed 'in-view' in which you immerse yourself in the material and attempt to achieve a complete understanding of it; and, finally, with a review, in which you revisit the important points you highlighted in the in-view and incorporate them into a higher-level framework for ease of recall (I find mind-mapping an invaluable tool for constructing this framework). Add your own comments and thoughts in the margins as you go, as these will help to put you back in the same frame of mind on a subsequent reading, and may be important in recording and re-triggering your 'eureka!' moments, in which you experienced a sudden insight or discovered a connection with previous knowledge as you read the material. And if your own calculations to confirm some statement or equation are brief enough, include these in the margins too, so that you won't have to repeat them on a subsequent reading. In short, a well-used mathematics book should look like a complete mess when you have finished with it, its pages littered with your own underlinings, highlightings, asterisks, question marks, exclamation marks, comments and calculations! But that shouldn't matter, because if you have read the book in the manner I have suggested, then you will have understood it and memorized it, and shouldn't need to read it again :o)Also, if you are aiming - in the language analogy - for conversational fluency, then it is very important that you not only read the text but also do the exercises and problem sets, because to achieve proficiency in mathematics, as in any other subject, there really is no substitute for practice, practice, practice!Finally, in learning mathematics, you should follow the pedagogical principle of 'Learn it, play with it, own it!': When you have learned a new piece of mathematics, take some time out from your reading to thoroughly familiarize yourself with it by playing with it! Viewed in one way, mathematics is an absorbing intellectual game, and playing with it is not only great fun but also the best way to really learn it and understand it. By 'playing' with it, I mean thinking up problems or posing yourself questions that you suspect might be answerable using the mathematics you have just learned, and then applying that mathematics to try to solve them. Or perhaps you might try to generalize the mathematics you have just learned and apply it to a broader range of problems, or connect it with other mathematics with which you are already familiar. All of these activities will help to embed the new mathematics more deeply in your mind, so that you no longer have to refer to your notepad or textbook in order to recall it and use it, but can summon it up effortlessly. Playing with mathematics will make it instinctive and second-nature to you, so that you can call on it at any time and, like an old and reliable friend, it will always be there for you. :o) When you have achieved this level of proficiency in a mathematical subject, then you can truly claim that you have attained the third stage of competence - i.e. that you own it (or, in the terminology of NLP, that you have achieved unconscious competence in it). And that should be 'good enough' for you, and for anyone else for that matter!Good luck with your journey of mathematical discovery, and have fun along the way! :o)[1] My own disposition is the latter, partly because I find variety and novelty stimulating, and partly because my other interest, theoretical physics, requires familiarity with a broad range of mathematical disciplines. All theoretical physicists are mathematical generalists to this extent (and this is one of the reasons I was drawn to theoretical physics in the first place); we are probably familiar with more areas of mathematics than most professional mathematicians, although our knowledge of any particular area generally won't be as deep as that of mathematicians who are experts in it.[2] This is something I know a great deal about, as this is how I learned most of the mathematics I know; book-based study has always been my preferred way of learning the subject, even when I was at university. I generally regarded lectures as an inferior form of instruction to self-study, partly because of the inherent limitations of the 'chalk-and-talk' format (especially in a large lecture theatre), partly because of the poor quality of some of the lecturers (e.g. the mumblers you just can't hear, the droners who send you to sleep, and - perhaps worst - the reluctant and disinterested lecturers whom you suspect regard lecturing as a tiresome duty that distracts them from their more interesting research work), and partly because their material was frequently so damned boring and irrelevant to my real interests (particularly in theoretical physics). My philosophy was that if I could find a good book on a subject, written by an acknowledged expert in it, then I shouldn't need to attend any lectures in it, and attendance should therefore be optional (my lecturers took a different view, of course...).