A Mathematical Theory of Communication

I highly recommend working through Claude Shannon's Mathematical Theory of Communications [0]. It's originally a paper but was later restructured as a book, in either form it works quite well. The reason I recommend it is because it shows mathematical reasoning that is easy to follow and relevant to your daily life. It's real math, but very easy to read through and understand. If your unfamiliar this paper is where the very idea of "bits" comes from. One of the most important things in the paper for non-mathematicians to see is that the definition Information Entropy is derived simply from the mathematical properties Shannon desires it to have. This is important because I find that one of the biggest questions people ask about mathematical formula and idea is "What does this mean? Why is it this way?" without realizing that math is really not engineering nor physics. When deriving his definition of Information, Shannon simply states that information should have the following x,y... properties and then goes on to show that the now standard definition of information meets all these criteria.…

A few books at various levels with very high signal-to-noise ratio: Feynman's little popular science book _QED: The Strange Theory of Light & Matter_ is a gem. It's very short and non-mathematical, but still manages to get remarkably close to the truth. If you're more mathematically inclined (e.g., fluent in multivariate calculus) and know a little classical physics, the first 5 chapters of Wald's _General Relativity_ are an essentially complete introduction to the subject. Another classic is Shannon's _A Mathematical Theory of Communication_, which introduces the subject of information theory, and then solves basically all the subject's main problems. In economics, I think the record for signal to noise is probably held by Keynes _The Econonic Consequences of the Peace_, 118 pages, in which he basically forecasts the disastrous course of events from 1918 to 1939.