Thread: Books for improving math skills?

I get quite annoyed with being unable to understand the equations of things that interest me (i.e I understand the concepts of things like pascals constants, and general relativity, but have no idea how the maths 'prove' it)

are there any books I can get to learn/understand maths better? I am fairly competent at maths, I got a very good GCSE in it, but never pursued it beyond that.


also, a little niggling question. Can a law be 'wrong'? I frequently see things along the lines of "black holes break the laws of physics" surely if the law can be broken the law is, on some level at least, false? (I'm not trying to argue against top mathematicians, I'm just trying to understand if it's commonly accepted; either that things like BH's only appear to break laws, and we simply don't understand it well enough; or the Laws are not fully accurate?)

The subjects you are referring to (Pascal's constants and the theory of relativity) are physics, which operates exclusively by the scientific method. In other words, science does not attempt to prove anything, rather, the objective of science is to propose a theory of reality.

Theories become laws over time if they are duplicated repeatedly and continue to hold true. They may be incorrect in certain respects, but I do not know of any that one can simply say are wrong (for example, Newton's laws of motion fail at speeds approaching the speed of light, but hold true otherwise).

In mathematics, any proven theorem is a "law" that is simply correct.

Originally Posted by Booms

I get quite annoyed with being unable to understand the equations of things that interest me (i.e I understand the concepts of things like pascals constants, and general relativity, but have no idea how the maths 'prove' it)

are there any books I can get to learn/understand maths better? I am fairly competent at maths, I got a very good GCSE in it, but never pursued it beyond that.


also, a little niggling question. Can a law be 'wrong'? I frequently see things along the lines of "black holes break the laws of physics" surely if the law can be broken the law is, on some level at least, false? (I'm not trying to argue against top mathematicians, I'm just trying to understand if it's commonly accepted; either that things like BH's only appear to break laws, and we simply don't understand it well enough; or the Laws are not fully accurate?)

"Laws" is traditional but unfortunate terminology. A "law" is a well-established physical principle.

Physical theories advance as a series of progressively better approximations. The fundamental theories produce extremely accurate predictions of natural phenomena, within limited circumstances known as the domain of validity. Newtonian mechanics is valid for speeds that are small with respect to the speed of light and for masses that are large with respect to atoms and molecules. More accurate "laws" that include high speed and small scale are available with special relativity and quantum mechanics, but the models are more complicated. The best available theories are quantum field theories, the electroweak theory and quantum chromodynamics (together known as the Standard Model) which are relativistic quantum field theories describing the electromagnetic, weak and strong forces.

Newtonian gravity is a simple theory that is sufficient for almost all calculations involving the gravitational force. But it fails for cases requiring extreme accuracy or involving very large gravitational fields. The best available theory of gravitation is general relativity. Black holes, for instance, require general relativity. So does an understanding of cosmology which involves the big bang and an expanding universe.

Newtonian mechanics, Newtonian gravity, special relativity and elementary quantum mechanics involve relatively simple mathematics. Basic algebra, calculus, linear algebra, plus ordinary and partial differential equations are enough. Quantum field theories and general relativity are much more demanding. General relativity requires Riemannian geometry which itself requires a firm background in modern mathematics. Quantum field theories require advanced functional analysis, and even so is still an area needing more work to be put on a rigorous mathematical foundation.

There are hundreds of advanced mathematics books, but there is no single book that can give you all of the pre-requisites for the more advanced physical theories, starting from, say, college algebra or introductory calculus.

Where in world did you run across Pascal's constants ? That is a fairly esoteric subject in physical chemistry.

http://en.wikipedia.org/wiki/Pascal's_constants

http://pubs.acs.org/doi/abs/10.1021/ed085p532