image” for Math. What does the book contain ? It contains 12 chapters on various aspects of math with history and examples blended in such a way that a reader will love the pace of the book.

0. Head for Numbers

The book starts off with questioning the concept of NUMBER itself. Is it a cultural artefact or Is it something that we have an innate ability to comprehend ? Alex cites an interesting piece of research which reaffirms our tendency to think in logarithmic scale. Innately, we compare numbers by ratios and not by their distance between them. So, the distance between 10 and 100 appears same as 100 and 1000, though our schooling in math says they are different.  The point made by the author is that we are innately kind of wired with log scale BUT we encounter the linear scale in our math education. So, the suggestion is to marry both scales somehow and lessen the math phobia amongst some of the kids at the school level. If understanding quantities approximately in terms of estimating ratios is a universal human intuition, then the question remains ,”Why should Kids learn about times tables , instead of sharpening their estimation skills ?” 

I have a similar feeling about the distance formula taught at a school level. Every high school kid , when given two Cartesian coordinates  (x1, y1) and (x2, y2) can write the Euclidean distance in a jiffy. But ask the same kid , “ what happens if points on the x axis have a systematic error in measurement with average e1 and points on y axis have a systematic error in measurement with average e2 ? “. He would search for a formula in his textbook and give up. Basically , he is not taught that Euclidean distance is just one of MANY measures of distance between two points and in real world nobody hands him perfect coordinates of anything. Real world data is by default , uncertain. Shouldn’t estimation skills be taught , ground up , starting from as simple as a concept as Euclidean distance ? What should be done to kids to explore beyond the rudimentary knowledge ? May be better teachers ! !

**1. The Counter Culture:

**Author then moves on to talk about the counting system that had evolved over the centuries. He traces counting system and evolution of various bases like  20, 33, 8 , 12 , 60 and finally binary systems. The chapter also talks about the “way” in which we say numbers effects the learning speed.  There is also a mention about the amazing Japanese Kids who do mind blowing arithmetic calculations using Soraban . In fact years ago I saw a YouTube video(below)  about it and was truly fascinated by what I saw. BTW, there is an elderly couple in Goregaon, Mumbai who have been teaching Soroban to kids / teachers in India. According to them, they have seen kids doing extremely fast mental calculations in a flash of a second , though none of them have reached a state of manipulating beads in their mind with out the actual device. I guess it will take some time , but soon some talented Indian kid will amaze us all like this kid in the video below.

[youtube=http://www.youtube.com/watch?v=EueFhYZ4HxI]

2. Behold!

In this chapter, the author talks about geometry, origami , Euclid’s contribution etc.  You will never look at origami the same way again after reading the the math mentioned in this chapter.

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3. Something about Nothing:

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There is an account of the author going down to India to meet Sankaracharya, a person who is given demi-god status in India. He is also responsible for propagating Vedic mathematics / sutras , which have supposedly been culled out of Vedas, the sacred text of Hindus. The first person to have  inferred the sutras / principles from Vedas was Bharti Krishna Tirthaji who in the role of Sankarcharya toured the country to give lectures on Vedic math.  There is also a mention of shunya / zero in the Indian math context which lead to quicker calculations and more importantly to the evolution of place-holder system in arithmetic.  The concept of zero made a lot of calculations easier and thus lead to Arabic Numerals which stamped their supremacy on the other number systems such as Greek , Romans etc. By their very nature, the other systems were cumbersome for large number calculations. Not having place holder system in place, long calculations were painful. An offshoot of that system is that people did not think of very large numbers. In contrast, Buddha is supposed to have thought of very very small numbers and very very large number to the order of 10^421 . Arabic Numerals with zephyr (0) was a breakthrough.  The author ends the chapter reflecting on the concept of nothingness which is the underlying principle of Indian Spirituality and muses that zero was a manifestation of India’s spirituality in to the math world.

 4. Life of Pi

Pi is one of the fascinating numbers in the math world and it is no surprise that the author spends some time talking about it. Pi has a very colorful history that entire books have been devoted to the number. The author condenses all that in to a couple of pages to give the vibrant nature of the number whose digits have enthralled mathletes, scientists, random number generating algos , criminals, psychopaths etc. Why should someone try to find the decimals in pi is a mystery ? It has captivated a lot of people and out of all this craze, there has been one good practical application, i.e test the performance of computing / compare the performance of computing between devices .  The rate of convergence of any math formula , machine can be characterized in the rate of convergence of digits in Pi and thus is a good measure to compare stuff. Pi is known up to 2.7 trillion places after the decimal but that is not stopping people from exploring.  The race to find whether Pi is a “normal number” is on since ages

5. The x-factor

Mathematical notation plays an important role in formulation of any problem. In fact the right notation is equivalent to problem half solved. The author explores the prevalence of x in most of the algebraic equations and gives a quick historical narrative of quadratics, cubic’s, quadratics.  Also the evolution of notations gave rise to new concepts like logarithms. History of slide rules, curta, HP-35 calculator is given for the reader to get a sense of the importance of measurement tools for the advancement of science and technology. The chapter ends with the narration of a breakthrough in math, i.e connecting algebra and geometry using Cartesian coordinates. Descartes was the man behind it and thus started a love affair between algebra and geometry. In today’s world we hardly notice the dichotomy when we solve a system of equations using vector spaces. However one must pause once in a while to appreciate the fact that some one took the effort to make the connection because of which we are able to use concepts from geometry and algebra together to solve problems. Imagine statistics with out any geometry. Statistics is one of those fields where the marriage between algebra(equations) and geometry is so powerful an alliance , that it helps in understanding the data driven world a lot better.

6. Playtime

An interesting note on the history of the popular puzzles can be found here. Magic Squares , Latin Squares, Sudoku, 15 Puzzle, Tangrams , Rubik’s cube, Tetris, etc are all mentioned here and it leaves a reader with the feeling that puzzles might seem practically useless things at the outset. But when looked closely, some of the most interesting developments in math have come from people pondering over puzzles. Graph Theory, Combinatorics , Developments in probability theory are all offshoots of solutions to puzzles.

7. Secrets of Succession

This chapter explores sequences and more importantly the role of primes in the number theory. Like pi, the prime numbers have been the fascination for many years and it continues to be.  Application wise, primes are useful in encryption stuff, random number generation algos, etc. 

8. Gold finger

A book on math with out the mention of golden ratio is unthinkable as it is is a ratio which is evident in a wide variety of natural phenomenon. One trivia I learnt from this book is that iPod is designed in such a way that it has golden ratio in its design. Nature evokes beauty and it is no wonder that Apple guys learnt things from the nature.

9. Chance is a fine thing

Girolamo Cardano, a 16th century gambler was the first to conceive of the idea of frequentist world, probabilities are proportional to the frequency of occurrence of the events. The important stuff about probability was kick started by interesting  questions from Chevalier de Mere. Question on the probability of obtaining double sixes on throwing 2 dice and “problem of points” was posed to Blaise Pascal who corresponded with Fermat to provide mathematical notions to chance events. Gambling was an area where people could easily see the probability in play and the payoff from the bets. The book then goes in to various gambling machines / games explaining that the law of large numbers makes sure that casinos never go bankrupt. Shannon – Thorpe deadly duo’s story is given where they discover a way to beat the casino. Thorpe goes further to master blackjack and other games where the gambler might have > 100 % payback percentage. His book “ Beat the Dealer”, spawned a ton on math related to gambling. Thorpe went on to create a hedge fund which to this date is revered in the finance world for its awesome CAGR returns. For interested readers, a journalistic account of the math used by Thorpe’s  is given by a book titled ,”Fortune’s Formula”,which also mentions Kelly Criterion ( a betting strategy) that has become an indispensable risk management criterion in various types of activities in the financial domain. Few interesting trivia from the book are

  • Did you know that John Venn, popularly known for his Venn diagram had developed a powerful graphic to show randomness?

  • Perso Diaconis and Frederick Mosteller -  Law of very large numbers : With a large enough sample , any outrageous thing can happen

10. Situation Normal

Gaussian Distribution is present in many phenomenon in the world. The author introduces this fascinating discovery and the people behind it. Carl Friedrich Gauss, Poincare, Adolphe Quetelet, Francis Galton are the characters that make up a great story and story leads up to concepts such as correlation, regression towards mean etc. William Sealy Gosset, is also mentioned  in the context of his superb discovery of t distribution which makes the underlying distribution of random variables irrelevant as the statistic is based on averages, a fact which has lead to major offshoots in estimation procedures and hypothesis testing.

11. The end of the line

The last chapter contains narrative about non-Euclidean geometry , one of its main postulates is that “on a hyperbolic surface , there are infinite number of parallel lines through a point”. The chapter ends with Hilbert spaces  and types of infinities. The book ends with a nice thought that math has come a long way, from a time when there were too many things and not enough numbers to count TO George Cantor who provided with so many different kinds of infinities that there are no longer enough things to count :)

image Takeaway :

It gives you an entertaining and a lively account of all the math developments that have happened over centuries. Definitely worth reading.