Visual maths book

Question

Background

Even when complex maths get abstract, usually there are helpful visual models. John Wheeler said he couldn't understand some functions without a picture of them. It is in this sense I use the word visual, but not only restricted to functions.

What I mean by concrete

This video is a very good example of what I mean by visual maths.

Typing visual mathematics book google returns a list but I don't know if any would be good.

Also Euler's elements is being rather pleasant.

Question

Is there any book with this kind of approach to maths?

Well, there's a book called Concrete Mathematics, by Graham, Knuth, and Patashnik, https://www.csie.ntu.edu.tw/~r97002/temp/Concrete%20Mathematics%202e.pdf — Gerry Myerson, Jun 18 '18 at 06:51
Don't be too discouraged; what you describe is actually pretty close to the standard way of thinking about areas and volumes of weird shapes. The Riemann integral starts getting to the bottom of how we define these concepts, and that essentially involves a "limiting" approximation where we fill up the shapes with rectangles / rectangular prisms and sum their areas / volumes, computing each in the standard way (product of the side lengths). — Kaj Hansen, Jun 18 '18 at 07:14
The only problem is, of course, that you can't fill, say, a circle with a finite number of squares, but you can get arbitrarily close approximations with a finite number of them (and then "taking the limit"—calculus formalizes this—you can get the exact, theoretical value). There are a number of ways of doing this, but think something along the lines of this. — Kaj Hansen, Jun 18 '18 at 07:17
It would help if you explain what you mean by concrete. Do you mean math that is applied? Do you mean math that requires little prior mathematical background but might be conceptually at a high level? Do you mean math that requires little prior mathematical background and is not conceptually at a high level? Do you mean math that is applied in everyday stuff, and not just applied math (which might be applied only to physics or engineering)? — Dave L. Renfro, Jun 18 '18 at 14:40
Possible suggestions based on my trying to guess the answers to what others are asking: What is Mathematics? and the W. W. Sawyer books Mathematician's Delight and Prelude to Mathematics. — Dave L. Renfro, Jun 18 '18 at 14:44
Does this answer your question? Textbooks for visual learners — user1147844, Feb 26 '23 at 00:15

Gerry Myerson · Answer 1 · 2023-02-26T05:11:38.167

You might have a look at Hutz, Am Experimental Introduction to Number Theory. It says,

"This book presents material suitable for an undergraduate course in elementary number theory from a computational perspective. It seeks to not only introduce students to the standard topics in elementary number theory, such as prime factorization and modular arithmetic, but also to develop their ability to formulate and test precise conjectures from experimental data. Each topic is motivated by a question to be answered, followed by some experimental data, and, finally, the statement and proof of a theorem. There are numerous opportunities throughout the chapters and exercises for the students to engage in (guided) open-ended exploration. At the end of a course using this book, the students will understand how mathematics is developed from asking questions to gathering data to formulating and proving theorems.

"The mathematical prerequisites for this book are few. Early chapters contain topics such as integer divisibility, modular arithmetic, and applications to cryptography, while later chapters contain more specialized topics, such as Diophantine approximation, number theory of dynamical systems, and number theory with polynomials. Students of all levels will be drawn in by the patterns and relationships of number theory uncovered through data driven exploration."

Also, Weissman, An Illustrated Theory of Numbers. It says,

"An Illustrated Theory of Numbers gives a comprehensive introduction to number theory, with complete proofs, worked examples, and exercises. Its exposition reflects the most recent scholarship in mathematics and its history.

"Almost 500 sharp illustrations accompany elegant proofs, from prime decomposition through quadratic reciprocity. Geometric and dynamical arguments provide new insights, and allow for a rigorous approach with less algebraic manipulation. The final chapters contain an extended treatment of binary quadratic forms, using Conway's topograph to solve quadratic Diophantine equations (e.g., Pell's equation) and to study reduction and the finiteness of class numbers.

"Data visualizations introduce the reader to open questions and cutting-edge results in analytic number theory such as the Riemann hypothesis, boundedness of prime gaps, and the class number 1 problem. Accompanying each chapter, historical notes curate primary sources and secondary scholarship to trace the development of number theory within and outside the Western tradition.

"Requiring only high school algebra and geometry, this text is recommended for a first course in elementary number theory. It is also suitable for mathematicians seeking a fresh perspective on an ancient subject."