Math Better Explained Pdf
Tea Rochlitz
Kalid Azad is the founder of Better Explained, one of the world's most popular maths websites that makes hard concepts easy to understand. After studying Computer Science at Princeton, Kalid spent a few years at Microsoft as a program manager, founded a Y Combinator startup, and currently works as a developer.
For many people, maths is the subject they used to hate most in school and they carry this fear of numbers into later life. But it doesn't have to be that way - and understanding basic mathematical principles can be both fun and useful.
Kalid uses an intuition first approach to explain difficult ideas in a way that anyone can understand and this makes him a great person to talk to about any subject or skill including his current profession of computer programming.
In this conversation we discuss a range of topics including:
- How to get better at maths and why some people find it so hard
- The secrets behind great explanations of tricky concepts
- How Kalid has learned skills from coding to snowboarding and weight training
So whether you're looking to get better with numbers, learn to code or pick up physical skills, this episode will give you the actionable principles and techniques needed to succeed.
In mechanics, our professor made the declaration that "all laws of physics" have been disproven. He mentioned several examples including the Law of Gravity, mentioning briefly that it is better explained by Einstein's theory of General Relativity. He also mentioned Bohr's model of the electron, and how it was better explained by other models, which I've since learned.
This is one of those comments that are true but misleading. To understand why you need to understand what physicists mean by a theory. A theory is just a mathematical model, that is set of equations, that allow us to predict what happens when we feed in a set of initial data. So we can take Newton's theory of gravity, feed in planet positions, spacecraft velocities, etc and calculate how to send spacecraft to Jupiter - and it works!
But all theories are approximate because they're based on simplifying assumptions. That means no theory is true, but all (good) theories are almost true in a limited range of conditions. Newton's laws work perfectly for sending spacecraft to Jupiter, but we know that they would break down if we attempted to use them to describe relativitic speeds or extremely high densities. To do that we need a more accurate theory, and that's general relativity.
So far general relativity has passed every experimental test we've done, but we expect that general relativity too will break down at very small distances when we expect quantum effects to become important. So general relativity is also only an approximation that we expect to break down when pushed beyond its limits.
So now you see what your professor was on about. There is no physical theory that isn't an approximation, so we expect that every physical theory will have limits beyond which it fails to describe reality. Whether this is a useful point to make outside of philosophy of science classes I'll leave you to decide.
Since you specifically asked about general relativity: in GR you feed in the distribution of masses and the theory tells you how spacetime is curved. Then you use an equation called the geodesic equation that tells you how masses move in curved space. If you do this for e.g. the Earth and the Moon then GR will indeed predict the Moon orbits the Earth (or more precisely they both orbit the barycentre). Whether GR explains why the Moon orbits the Earth depends on exactly what you mean by explain, but this is where we hand over to the philosophers.
You probably learned about Einstein's Theory of Special Relativity in your Modern Physics class. It is Einstein's Theory of General Relativity that provides a more accurate description of what we normally call gravity.
Basically, general relativity explains gravity not as an interaction between two bodies but as a warp in space-time in the presence of matter. Rather than thinking of gravity as a force, general relativity treats gravity as a change in space and time itself. It is more of a theory of geometry than of forces and attractions. But that is in general relativity, not special relativity.
The truth is that neither Newton nor Einstein actually explained gravitational attraction. They both gave us some equations, but neither gave the explanation concerning the true mechanism of the underlying force. (That's why physics is still so desperately trying to unite GR and QM, and why it cannot get rid of the concept of the graviton.)
Einstein took a shot at it, and he brought the distance much closer, but still failed to explain the very force. Because curvature of space(time) itself simply cannot make an apple move (toward another body or just toward a point). If curvature produces movement, it is only because there is gravitation underneath it. That's why an apple put on the ground rolls down the slope. It takes curvature (of the ground) and also it takes gravitation under (the ground) to do that. If you remove the force and leave only the curvature ... nothing happens to the apple; it will stay right where it was until it experiences real force. So curvature cannot replace gravitational pull, it cannot replace a real force. (Should somebody invoke the Einstein tensor her, please read through first, as I refer to this later on).
This might be objected to with arguments that the movement is produced by the energy expressed by the Einstein tensor resulting from the stress-energy tensor. Well, I tried to address this issue here: At which point of the universe $R_\mu \nu=0$ if there is a source of gravitation (point mass). From Muphrid's answer and the following discussion, you can see that even Einstein's field equations are admitted to be "incomplete", and actually misleading. Textbooks say the equation shows the curvature of space(time) through the Ricci tensor, metric tensor and cosmological constant. If you inquire further, however, you will be told the equation does not actually refer to the space at all, but to matter only ... You will be told that the full space curvature is described by the Riemann tensor consisting of the Ricci tensor (null for the vacuum) and the Weyl tensor. But if you then decide to dig even more and ask the quite obvious question, whether the Weyl tensor is the one responsible for the space curvature that makes massive bodies come together, the answer is ... silence ...
As a follow-up, I think you should see my answer here: Why Newton's law of universal gravitation is a valid law? What causes any two bodies in the universe attract each other with a force? and watch Feynman's video linked.
I discovered that I am able to understand concepts much quicker and better when they are explained/taught in a visual way. For example the epsilon-delta definition of a limit of a sequence/function only became really clear to me when explained with the help of a graph similar to this one:
Proofs without words - Exercises in visual thinking:Volumes I and II by Roger B. Nelson are a good way to work through relatively simple problems. But since they are presented with very nice and sometimes enlightening graphics, it is fun to work through them.
Beautiful Evidence by Edward Tufte is a wonderful book about how to present data and statistics. I deeply appreciate all his books around this theme and from my point of view it's a must have for statisticians.
The Ashley Book of Knots: Although this book is not written for mathematicians, I recommend it to all who like topology. The theme of the last chapter from the Topological Picture Book is knot theory. And if you are a visual learner with a faible for knots you will appreciate this book . It is a guide containing thousands of wonderfully drawn knots most of them are masterpieces of art. You can delve into an incredible world of different knots and after that you will look at Topology with different eyes.
Analysis by Its History by E. Hairer and G. Wanner is an approach to Analysis following the chronological order of the subject. It's a valuable supplement to ordinary textbooks in calculus providing also a wealth of highly instructive graphics. When looking at the many graphics it's obvious, that hundreds of hours had been invested with great sensitivity by the authors to serve the visual needs of the students.
Calculus by Michael Spivak is a well known (modern) classic which also contains a lot of good drawings. The contents is great of course, the drawings are numerous and instructive, but they do not play in the same league as those by E. Hairer and G. Wanner.
For some proofs with good visual input are (I think these are the books that most agree with the question asked, in fact some demonstrations have almost only drawings.) : Proofs without Words I, II and III, R. B. Nelsen.
For those who have an interest in graph theory, the combination with algorithms and drawing the graph can be interesting. Graph Drawing: Algorithms for the Visualization of Graphs, I. G. Tollis, G. Di Battista, P. Eades, R. Tamassia (Authors).
Something I think should be mentioned is Robert Ghrist's "Elementary Applied Topology". Ghrist's illustrations in this book are some of the best I've seen; in fact, he allegedly had wanted to develop a career in graphic design, but eventually chose math instead!
Mathematics is the language of science. It is the art of problem-solving. And if we care to look closely enough at the world around us, we will be amazed at the mathematics at play. It is my hope as your professor that you will take from this class a newfound appreciation of how mathematics fits into your world. Whether you are interested in the microscopic cells and chemical reactions occuring in your body, the local geography of the beautiful Upper Peninsula of Michigan, the global changes in our climate, the large-scale behavior of our universe, or maybe you are just interested in playing video games and listening to music - all of these things can be better understood, better explained, better appreciated with mathematics.
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