This is a terrific idea -- a math book that avoids the pitfalls of most math books aimed at nonmathematicians.
Such a book could cover many important mathematical ideas without necessarily lapsing into equations and overly technical explanations. For example, it should be possible to describe how compound interest works without falling into an obscure technical explanation, and understanding compound interest is very important in modern life.
Another example might explain why the stopping distance of s car is proportional to the square of the speed -- this is not well-known, and it's important for drivers to know, young ones especially.
Yet another example would explain why each member of the running sum of odd numbers is a perfect square. Expressed in words, it's not obvious that it's true or why it's true, but a picture conveys the reason immediately and intuitively: http://arachnoid.com/example/index.html#Math_Example
Just a few examples. I'm sure one could fill such a book with useful examples that would convey useful information, and make math sound like fun, or both, without being preachy or too technical.
> For example, it should be possible to describe how compound interest works without falling into an obscure technical explanation
I personally find these obscure technical (and highly abstract) explanations often far more easy to understand. I often found "easy" explanations highly illogical - not before I got the rather abstract explanations I found these explanations acceptable (and even this was not always the case - almost always the explanation for this phenomenon was that the definitions given in foundation courses could be abstracted a lot).
How can this be explained? The reason is simple: in highly abstract definitions anything that is not necessary is omitted - so there is less to think about. Additionally in this kind of definitions there is a lot more "internal logic". What does this mean? This is a little bit difficult to explain for non-mathematicians, but you can be sure that anything in the definition has a deep meaning. If this meaning seems strange to you, you can be sure that what remains to be understood often carries a deep meaning. On the other hand: when using "simple" definitions, you always have to worry whether, if something sounds strange, it is because you haven't understood it or if the "simple" explanation was simply bad.
Disclaimer: I'm a mathematician (as may be imagined). But I'm a computer scientist, too. :-)
> I personally find these obscure technical (and highly abstract) explanations often far more easy to understand.
I do, too, but I also know that nontechnical, nonmathematical people are turned off by a quick immersion in mathematical reasoning. I have a theory (not just mine by any means) that if the beauty of mathematics could be presented before the required discipline and attention to detail, we might lose fewer possible future mathematicians. As things stand, the public level of innumeracy is depressing.
> On the other hand: when using "simple" definitions, you always have to worry whether, if something sounds strange, it is because you haven't understood it or if the "simple" explanation was simply bad.
Yes, very true, one must be very careful to get it right while making it simple. I personally think a persuasive layman's explanation of something mathematical can go wrong in so many ways, and the more persuasive, the more room for error. Consider all the crazy "explanations" of quantum theory out there -- the more popular ones have no connection to reality.
I was actually thinking something just a little more technical aimed at coders only. But maybe that's the real reason it doesn't exist: no one can agree on what it is. :)
Such a book could cover many important mathematical ideas without necessarily lapsing into equations and overly technical explanations. For example, it should be possible to describe how compound interest works without falling into an obscure technical explanation, and understanding compound interest is very important in modern life.
Another example might explain why the stopping distance of s car is proportional to the square of the speed -- this is not well-known, and it's important for drivers to know, young ones especially.
Yet another example would explain why each member of the running sum of odd numbers is a perfect square. Expressed in words, it's not obvious that it's true or why it's true, but a picture conveys the reason immediately and intuitively: http://arachnoid.com/example/index.html#Math_Example
Just a few examples. I'm sure one could fill such a book with useful examples that would convey useful information, and make math sound like fun, or both, without being preachy or too technical.