Customer Reviews

Mathematics and the Imagination

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Originally published in 1940, the material in this book is beginning to show a little age. However, the quality of the writing renders those defects to near irrelevancy. Popular descriptions of mathematics are differentiated by the quality of the writing rather than the distinctiveness of the mathematics, and this one shines.
I like this book, starting with the title. It takes an enormous amount of imagination to do mathematics, something unappreciated by the public. It is easy to understand the use of linear segments to approximate the length of a curve. However, it requires an enormous leap of abstraction to believe that if they are made of zero length and then summed up, the result is the true length. Calculus students dutifully record and apply this, but in most cases don't appreciate the significance of the idea. In nearly all cases of major mathematical advancement, a fundamental change in thought processes was necessary. Those changes require imagination and the advances explained in this book are well documented and described.
Mathematicians are containers of some of the greatest concentrations of imagination that humans possess. Their leaps of abstraction often include descriptions of objects that cannot be visualized. Kasner and Newman capture this essential ingredient, serving it up in palatable portions.

Having had this book around the house for ages, I picked it up and to my surprise within a few minutes really understood (not just enough to use, but actually understood) what logarithms really are, where they come from. The chapter on e, pi, and i is another great one to get the story behind the story, as it were. For me the book could better have continued in this vein of explaining concepts we've seen before but never really grasped intuitively, and perhaps because I'm not terribly interested in mathematical games I found that segment less fun. But in fairness, they've done a good job getting away from textbook math and into some interesting themes. I don't know if it's all still valid, as it is so old--references to Fermat's last theorem are at least outdated!

My school teacher gave me this book to read when I was 13 years old, based on the interest I showed in Mathematics that went beyond the curriculum at school. In many ways it was way beyond my comprehension at the time, but little did I know that it would have such a lasting effect on me. Reading about concepts of infinity, that you could only describe to a fellow teenager as "different sizes of infinity", I realized that there really is a philosophy of mathematics that transcends all other subjects and that there is also an art to working with the subject. I can't recommend this book enough, and I never did give it back to my teacher!

This book came to me by chance.Instantly got my attention.It is written in such a way,that makes interesant to travel through different chapters.In each one you have the mathematical theme mixed with stories , mathematicians histories and puzzles.You learn about people with the greatest imagination.Their personality and a lot of other things,that make you enjoy the reading,no matter if you love mathematics or you have hated it all the time.I've enjoyed specially the chapter about Mathematical Analysis,with the story about rivalry between the egocentric Newton and the humble Leibnitz. This book is the opportunity to learn that mathematics are not the boring thing we have learned at school.

This book is for very few as it is not for the following group of readers:
a) only peripherally interested in math;
b) interested in math but wanting to learn more in a concise manner and readily bored with exhaustive math history and early 20th century verbosity;
c) mathematicians who already know that stuff;
d) interested in math but requiring well organized and structured material.

If you do not belong to any of the above four groups of readers, you probably will like this book. You may even love it. To improve the probability you will like it you should not read chapters sequentially. This will bore you. Instead jump in directly to the mathematics domains you are familiar with and you like best. For instance, chapter VII covers Probability and chapter IX covers Calculus. If you like Geometry, you are in luck as the authors dedicate two chapters to it (IV and VIII).

The chapter on calculus covers a lot of grounds including not only derivatives but also definite and indefinite integrals, calculus and the laws of motions, and trigonometry. Your head may spin after this broad and deep excursion into calculus. And, this may be one of the authors' most successful chapters.

The authors divulge a lot of fascinating information even though the latter is not always easily extractable. This includes very intuitive explanations of the discovery and calculations of Pi, e, i, and logarithms. They explain how mathematicians leveraged those discoveries. The numbers Pi and e have so many applications that they suggest our civilization in terms of mathematical understanding and its practical application would be truly primitive if not for them (Pi, e). Similarly, they show how the invention of logarithm essentially powered math and physics forward by rendering complicated calculations simple (replacing exponents by multiplication, multiplication by addition, etc...). They also make fascinating connections between algebra (equations) and geometry (visual representations of the same equations).

Their narrative on large numbers is pretty interesting. It taught me that even though Sergey Brin and Lawrence Page are very successful billionaire geniuses, they apparently do not know how to spell math terms. Indeed, Google refers to a very large number consisting of 1 followed by 100 zeros. But, it is spelled Googol! At least, Brin and Page got Googleplex right when naming the company headquarter (Googleplex equals 1 followed by a Googol of zeros). Those numbers are gigantic as The Harper Collins dictionary of Mathematics discloses there is only 10 to the 85th power number of atoms in the universe (less than a Googol that is 10 to the 100th power and obviously a lot less than a Googleplex).

In the appendix of the calculus chapter, the authors open the door on upcoming mathematical developments. Indeed, they introduce Pathological Curves which will become a foundation for Mandelbrot's Fractal Geometry as explained in The Fractal Geometry of Nature. In turn, the latter will leverage this concept to develop a new theory of finance in The (MIS)Behaviour of Markets: A Fractal View of Risk, Ruin, and Reward. And, Taleb will turn the same concept into a bestseller in The Black Swan: The Impact of the Highly Improbable. In the same appendix, the authors also introduce the Criscross Curve. Stephen Wolfram will leverage this concept and develop cellular automata as explained in A New Kind of Science. As you can see, ultimately everything is related.

However, if you want to learn math basics in a more effective and efficient manner, I recommend instead the series: Forgotten Statistics: A Refresher Course with Applications to Economics and Business, Forgotten Algebra and Forgotten Calculus. Those will give you a pretty good foundation on which to build upon. If you want to deepen your knowledge of statistics I also recommend the excellent Intuitive Biostatistics.

this book is more than 50 years old, but it still provides valuable insight for any math-oriented high school student and especially so for those students who plan for careers in mathematics or any of the sciences.

The book provides a logical and entertaining explanation of the principals of mathematics and their development.

My Granddad was a great engineer, and loved puzzles and games. After his funeral I was looking through his books and found this one, the original edition! I read through it and loved it. My Granddad had highlighted several passages. This book and those highlighted passages has been a great insight into the mind of my granddad. He was a true technologist before there was such a word, and he loved this book. Now I love this book as well. I have used my granddad as a role model in my engineering and mathematical studies, and it has been a precious gift to understand who his role models were. While undoubtedly this book will not hold to same emotional connection for you as it does for me, it is nonetheless entertaining, well written, and thought provoking. Read it!

I am thoroughly enjoying the challenges to my brain that I am finding in this book. it was well worth the price I paid

Wonderful Service! Your customer service is very much appreciated ... from the description of the condition to the packaging of the product ... all very top shelf!