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]]>Based on the name Abstract algebra, you might think it’s similar to the algebra course most people take in high school, just a little more abstract. But if you open a book on Abstract Algebra, you’ll be in for quite a shock. It looks nothing like the algebra most people know about. So, to help you understand the subject, let’s go back in time .

The year is 1800, and for some time now, people have known how to solve linear equations, quadratic equations, cubic equations and even quartic equations. But what about equations of higher degree? Degrees five, six, seven and beyond? A young teenager named Evariste Galois answered this question. And to do so, he used a tool that he called a “group”. Around this time, Carl Friedrich Gauss was busy making discoveries of his own. He ironed out a new technique called modular arithmetic which helped him solve many problems in number theory.

Modular arithmetic shared many similarities to the groups used by Galois. The 1800’s also saw a revolution in geometry. Euclid dominated the scene for more than 2,000 years with his book “The Elements”. Mathematicians began to realise that there are other geometries beyond the one devised by the ancient Greeks. It didn’t take long before groups were found to be a useful tool in studying these new geometries.

It soon became clear that groups were a powerful tool that could be used in many different ways. So, it made sense to “abstract” out the common features of this tool used by Galois, Gauss, and others into a general tool, and to then learn everything about it. Thus, group theory was born. And if “groups” were so useful, it’s natural to ask: would this approach work elsewhere? Soon, new abstract objects began to take shape: rings, fields, vector spaces, modules… This didn’t happen overnight. It took years of hard work to find the right definitions. Too specific, and they wouldn’t be very useful. Too general, and they would be kind of boring and NOT very useful. Altogether, they form the subject we now call Abstract Algebra.

At first glance Abstract Algebra may not seem very applicable to the world around us. But it’s a young subject, and its usefulness continues to grow. We can see the uses found in the real world. Not just in mathematics: Physics, chemistry, computer science and other areas are discovering just how useful abstract algebra can be. Quick note – “abstract algebra” is sometimes called “modern algebra.” And if you’re ever at a cocktail party with mathematicians, they’ll simply call it “algebra.” So, when are you ready to begin learning Abstract Algebra? First, you really need to know the more familiar algebra, but the most important requirements are these: mathematical experience and mental maturity. Have you seen many mathematical proofs before? Are you able to think VERY abstractly? If so, then get ready, Abstract Algebra will challenge you like never before.

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]]>It’s complex, and unfortunately a lot of people adopt the belief that they are just not maths people. They even do not know how to solve a simple problem. Which is patently untrue because maths is a skill that can be learned just like any other. But since you clicked on this article hopefully you are not one of those people. Hopefully you have at least some degree of belief that you can become better at maths. And if you do, the obvious questions are **how do you get better at maths, to study maths on your own?**. How to study maths in college? and how to study maths? So, here I will provide you the best tips to study maths.

Well, fortunately, these questions have a pretty simple answer. If you want to get better at maths, you must do lots and lots of maths. This is the best way to study maths. Practice, practice and practice. And the tougher the problems are, the better. Because tough problems will stretch your understanding and lead you to new break through. Once you know how to solve it then you are a master. But, working through these tough problems you are eventually going to come to problems that you get completely stuck on.

When you get to these points, it’s important to know how to solve these problems. Because these are the ones that are really going to stretch and build your skill set. So that is what I want to focus on in this article. I want to give you practical techniques for working through, and eventually solving those problems that seem insurmountable at first.

To start, I want to focus on a piece of advice. The Hungarian mathematician George Polya shared in his 1945 book “How to Solve It”. This is the important technique to understand and put into practice when you are trying to solve maths problems. Because maths builds upon itself. More complex concepts are built upon simpler concepts. And if you don’t have grasp on the fundamental principles, then a complex problem will be hard. So, if you come across a problem that you can’t solve, Try to solve it. First, identify the components or the operations that it wants you to carry out.

Of course, sometimes you have too shaky of an understanding of the concepts and operations themselves for you to work with them and solve that problem. And in that case, it’s time to go do some learning. Go dig into your book, look through your notes, or find example problems online that you can follow along with step-by-step so you can see how people are getting to the solutions, using these concepts. And, if you need to, you can get a step-by-step solution to the exact problem you are working on as well.

There are several tools out there that you can use to do this. The two that I want to focus on in this article which are the best ones I have been able to find are Wolfram Alpha and **Symbolab** . Both websites will allow you to type in an equation and get an answer and also gypha and **Symbolab** that you can follow along with. The difference between the two is that **WolframAlpha** , while being much more power and capable, does require you to be part of their paid plan if you want to get those step-by-step solutions. By contrast, while I found that typing in equations into **Symbolab** was a little bit slower and less intuitive than it is with **WolframAlpha** their step-by-step solutions are free.

Regardless of the online maths tool that you choose to use here the underlying point is that, sometimes it can be useful to see a step-by-step solution for a problem. But, there are two very important caveats here. First and foremost, before you go running off to find a solution, ask yourself Honestly, have I pushed my brain to the limit trying to solve this problem first? Expending the mental effort required to solve the problem yourself is going to stretch your capabilities. It’s going to make you a better mathematician in a way that just looking through solutions won’t.

Now, if you do need to look up a solution, that’s fine. Look it up, follow the steps and make sure that you understand how the answer was arrived at. But, once you have done that, challenge yourself to go back and rework the problem without looking at that reference. It is important to stay vigilant about this. Because if you want to get better at maths the whole point is to master the concepts that you are working with.

The danger that comes with looking up solutions is that, with maths it’s easy to follow along with a step-by-step solution and comprehend what’s going on. But that is very different than being able to do it on your own. And that brings me to my final tip for you. And this is especially important for anybody in a maths class working through assigned homework. Don’t rush when you work through maths problems. I know it’s really tempting to try to work through homework as fast as you can and heck, I even made an article about it recently.

With maths and science and any sort of complex subject especially rushing is only going to hurt you down the road. Because when you rush, you don’t master the concepts. You just brute force your way to answers or you look things up, or you otherwise kind of cheaty-face your way to a completed homework assignment. And later, when you are sitting in a testing room, or you have to apply what you have learned in the real world you are going to get a harsh lesson about exactly what it is you don’t know. So, let’s recap here. If you want to get better at maths and you want to improve your ability to solve those tough problems first, identify the combination of concepts or operations being used in a problem and then isolate them. Work simpler problems that use just one and then master each concept.

You can also simplify the problem by leaving the combination of concepts intact but swapping in smaller, easier to handle numbers. If you need help with the concepts themselves go to your book or an explained article online look up sample problems, or use a tool like WolframAlpha or Symbolab to get step-by-step solutions to the problem you are working on. And finally, don’t rush through your homework assignments. Make sure that you are focusing intently on mastering the concepts, not just finishing.

Hopefully these tips will give you the confidence to tackle some tough maths problems. To expand your maths skill set, I want to leave you with a quote from the great physicist, Richard Feynman.

“The bottom line is this: Ultimately, your ability to get good at maths and anything else for that matter starts with having the confidence to approach it. And as you solve problems and make mental breakthroughs your confidence is going to naturally increase. It becomes a self-sustaining cycle.”

**Here is my another article about the history of maths**

** http://www.candlemind.com/map-mathematics-history/**.

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]]>The mathematics we learn in school doesn’t quite do the field of mathematics justice. We only get a glimpse at one corner of it, but the mathematics is a huge and wonderfully diverse subject. My aim with this article is to show you all that amazing stuff. We’ll start back at the very beginning.

The origin of mathematics lies in counting. In fact, counting is not just a human trait, other animals are able to count as well and evidence for human counting goes back to prehistoric times with check marks made in bones. There were several innovations over the years with the **Egyptians having the first equation, the ancient Greeks made strides in many areas like geometry and numerology, and negative numbers were invented in China.** And zero as a number was first used in India.

Watch this video: https://youtu.be/OmJ-4B-mS-Y

In the Golden Age of Islam Persian mathematicians made further strides and the first book on algebra was written. Then mathematics boomed in the renaissance along with the sciences. Now there is a lot more to the history of mathematics then what I have just said, but I’m going to jump to the modern age and mathematics as we know it now. Modern mathematics can be broadly being broken down into two areas, pure maths: the study of mathematics for its own sake, and applied maths: when you develop mathematics to help solve some real-world problem. But there is a lot of crossover.

In fact, many times in history someone’s gone off into the mathematical wilderness motivated purely by curiosity and kind of guided by a sense of aesthetics. And then they have created a whole bunch of new mathematics which was nice and interesting but doesn’t really do anything useful. But then, say a hundred years later, someone will be working on some problem at the cutting edge of physics or computer science and they’ll discover that this old theory in pure maths is exactly what they need to solve their real-world problems! Which is amazing, I think! And this kind of thing has happened so many times over the last few centuries.

It is interesting how often something so abstract ends up being useful. But I should also mention, pure mathematics on its own is still a very valuable thing to do because it can be fascinating and on its own can have a real beauty and elegance that almost becomes like art.

Pure maths is made of several sections. The study of numbers starts with the natural numbers and what you can do with them with arithmetic operations. And then it looks at other kinds of numbers like integers, which contain negative numbers, rational numbers like fractions, real numbers which include numbers like pi which go off to infinite decimal points, and then complex numbers and a whole bunch of others. Some numbers have interesting properties like Prime Numbers, or pi or the exponential. There are also properties of these number systems, for example, even though there is an infinite amount of both integers and real numbers, there are more real numbers than integers. So, some infinities are bigger than others. The study of structures is where you start taking numbers and putting them into equations in the form of variables.

Algebra contains the rules of how you then manipulate these equations. Here you will also find vectors and matrices which are multi-dimensional numbers, and the rules of how they relate to each other are captured in linear algebra. Number theory studies the features of everything in the last section on numbers like the properties of prime numbers. Combinatorics looks at the properties of certain structures like trees, graphs, and other things that are made of discreet chunks that you can count. Group theory looks at objects that are related to each other in, well, groups. A familiar example is a Rubik’s cube which is an example of a permutation group. And order theory investigates how to arrange objects following certain rules like, how something is a larger quantity than something else.

The natural numbers are an example of an ordered set of objects, but anything with any two-way relationship can be ordered. Another part of pure mathematics looks at shapes and how they behave in spaces. The origin is in geometry which includes Pythagoras, and is close to trigonometry, which we are all familiar with form school. Also, there are fun things like fractal geometry which are mathematical patterns which are scale invariant, which means you can zoom into them forever and the always look kind of the same.

Topology looks at different properties of spaces where you can continuously deform them but not tear or glue them. For example, a Möbius strip has only one surface and one edge whatever you do to it. And coffee cups and donuts are the same thing – topologically speaking. Measure theory is a way to assign values to spaces or sets tying together numbers and spaces. And finally, differential geometry looks the properties of shapes on curved surfaces, for example triangles have got different angles on a curved surface, and brings us to the next section, which is changes. The study of changes contains calculus which involves integrals and differentials which looks at area spanned out by functions or the behavior of gradients of functions. And vector calculus looks at the same things for vectors.

Here we also find a bunch of other areas like dynamical systems which looks at systems that evolve in time from one state to another, like fluid flows or things with feedback loops like ecosystems. And chaos theory which studies dynamical systems that are very sensitive to initial conditions. Finally, complex analysis looks at the properties of functions with complex numbers. This brings us to applied mathematics. At this point it is worth mentioning that everything here is a lot more interrelated than I have drawn. This map should look like more of a web tying together all the different subjects but you can only do so much on a two-dimensional plane so I have laid them out as best I can. Okay we’ll start with physics, which uses just about everything on the left-hand side to some degree.

Mathematical and theoretical physics has a very close relationship with pure maths. Mathematics is also used in the other natural sciences with mathematical chemistry and biomathematics . Mathematics is also used extensively in engineering, building things has taken a lot of maths since Egyptian and Babylonian times. Very complex electrical systems like aeroplanes or the power grid use methods in dynamical systems called control theory. Numerical analysis is a mathematical tool commonly used in places where the mathematics becomes too complex to solve completely. So instead you use lots of simple approximations and combine them all together to get good approximate answers. For example, if you put a circle inside a square, throw darts at it, and then compare the number of darts in the circle and square portions, you can approximate the value of pi. But in the real world numerical analysis is done on huge computers.

Game theory looks at what the best choices are given a set of rules and rational player sand. It’s used in economics. when the players can be intelligent, but not always, and other areas like psychology, and biology. Probability is the study of random events like coin tosses or dice or humans, and statistics is the study of large collections of random processes and analysis of data. This is obviously related to mathematical finance, where you want model financial systems and get an edge to win all those fat stacks. Related to optimization, where you are trying to calculate the best choice amongst set of many different options or constraints. You can normally visualize as trying to find the highest or lowest point of a function.

Optimization problems are second nature to us humans. We do them all the time. Trying to get the best value for money, or trying to maximize our happiness in some way. Another area that is very deeply related to pure mathematics is computer science. The rules of computer science were derived in pure maths and is another example that was worked out way before programmable computers were built. Machine learning, the creation of intelligent computer systems uses many areas in mathematics like linear algebra, optimization, dynamical systems and probability. Finally the theory of cryptography is very important to computation and uses a lot of pure maths like combinatorics and number theory. So, that covers the main sections of pure and applied mathematics.

This area tries to work out at the properties of mathematics itself, and asks what the basis of all the rules of mathematics is. Is there a complete set of fundamental rules, called axioms, which all of mathematics comes from? And can we prove that it is all consistent with itself? Mathematical logic, set theory and category theory try to answer this and a famous result in mathematical logic are Gödel’s incompleteness theorems.

Most people, means that Mathematics does not have a complete and consistent set of axioms. It means that it is all kind of made up by us humans. Which is weird seeing as mathematics explains so much stuff in the Universe so well. Why would a thing made up by humans be able to do that? That is a deep mystery right there. Also, we have the theory of computation which looks at different models of computing. How efficiently they can solve problems and contains complexity theory which looks at what is and isn’t computable?, how much memory and time you would need, which, for most interesting problems, is an insane amount. Ending So that is the map of mathematics.

Now the thing I have loved most about learning maths is that feeling you get. Something that seemed so confusing finally clicks in your brain and everything makes sense. Like an epiphany moment, kind of like seeing through the matrix. It’s great, I love it. Ending Making a map of mathematics was the most popular request I got. I was happy about because I love maths and it’s great to see so much interest in it. So, I hope you enjoyed it.

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