Here’s Looking at Euclid: From Counting Ants to Games of Chance – An Awe-Inspiring Journey Through the World of Numbers by Alex Bellow.

4/5 rating. 291 pages.

Book #11 of 2022. Read February 9, 2022.

This book is incredible; I absolutely loved it! I gave it a 4, but the first 2/3rds or so of the book was an easy 5. It was enthralling and so interesting I just couldn’t help but talk about it with people. I think my roommate was getting annoyed with my constant references to what crazy thing I just learned or what conundrum I was thinking about because of this book.

Alex covers so many topics in math in this book, it’s insane. What’s even more interesting is that within our math department, just in conversation and fun exploration we had discussed a ton of these topics – yes, this is literally what math teachers talk about, we can’t help it!

Some of the ideas that I especially liked:

• We all are born thinking logarithmically; our “counting numbers” are cultural
• English-speakers are at a disadvantage remembering numbers because of the length of the words
• The reason we solve for “x” and not some other arbitrary letter is because the book printer was running out of letters and it is one of the less-common French letters
• “Calculus” is pebble in Latin
• In a room of 23 people, there is over a 50% chance that two people share a birthday

I would recommend this for anyone who loves math. If that’s not you, you should buy this book for your favorite math nerd in your life.

Quotes:

“We are all born conceiving of numbers [as logarithmic].”

“The kindergarten pupil, with no formal math education, maps numbers out logarithmically. By the first year at school, when the pupils are being introduced to number words and symbols, the graph is straightening. And by the second year at school, the numbers are at last evenly laid out along the line.”

“By contrast, understanding quantities in terms of exact numbers is not a universal intuition; it is a product of culture.”

“Faced with a group of spear-wielding adversaries, we needed to know instantly whether there were more of them than us. When we saw two trees we needed to know instantly which had more fruit hanging from it. In neither case was it necessary to enumerate every enemy or every fruit individually. The crucial thing was to be able to make quick estimates of the relative amounts.”

“Matsuzawa has increased the number of digits in further experiments, and Ayumu [a chimpanzee] can remember the positioning of eight digits made visible for only 0.21 second. Matsuzawa also reduced the time interval, and Ayumu can now remember the positioning of five digits visible for only 0.09 second, which is barely enough time for a human to register the numbers, let alone remember them.”

“In his book The Number Sense, the French cognitive scientist Stanislas Dehaene gives the example of a person who can discriminate 10 dots from 13 dots with an accuracy of 90 percent. If the first set is doubled to 20 doors, how many dots does the second set need to include so that this person still has 90 percent accuracy in discrimination? The answer is 26, exactly double the original number of the second set.”

“Researchers had previously assumed that the intuitive ability to discriminate amounts did not contribute much to how good a student is at a task like solving equations and drawing triangles. Yet this study found a strong correlation between a talent at reckoning and success in formal math. The better one’s approximate number sense, it seems, the higher one’s chance of getting good grades.”

“English speakers have a 50 percent chance of remembering the seven numbers correctly. By contrast, Mandarin Chinese speakers can memorize nine digits in this way. Dehaene says that this is because the number of digits we can hold in our heads at any one time is determined by how many we can say in a two-second loop. The Chinese words for one to nine are all concise single syllables: yi, er, san, si, wu, liu, qi, ba, jiu. They can be uttered in less than a quarter of a second each, so in a two-second span, a Chinese speaker can make it through nine of them.”

“The Latin for pebble is calculus, which explains the origin of the word ‘calculate.’”

“In other words, the square of the number n is the sum of the first n odd numbers.”

“This posthumous mythmaking has parallels with the story of another Mediterranean spiritual leader, and in fact, Pythagoras and Jesus were temporarily religious rivals. Aiming to stem the growth of Christianity in the second century C.E., the Roman empress Julia Domna encouraged her citizens to worship Apollonius of Tyna, who claimed he was Pythagoras reincarnated.”

“By taking more and more iterations you can make the surface area larger than any area you want, meaning that as the number of iterations approaches infinity, the surface area of the [Menger] sponge also approaches infinity. In the limit, the Menger sponge is an object with an infinitely large surface area that is also invisible.”

“When it comes to sums, nothing [here, meaning a zero] makes all the difference.”

“Due to its ease of use, the Indian method spread rapidly to the Middle East, where it was embraced by the Islamic world, which accounts for why the numerals have come to be known, erroneously, as Arabic. From there, the method was brought to Europe by an enterprising Italian, Leonardo Fibonacci, his last name meaning ‘son of Bonacci.’”

“To those who asked whether the method was math or magic, he had a set reply: ‘It is both. It is magic until you understand it; and it is mathematics thereafter.’”

“Mathematics is a creative subject. Once you have a variety of methods children realize you can invent your own and they become inventive too. Math is a really fun, playful subject and [Vedic math] brings out a way to teach it that way.” – Kenneth Williams

“The ability to calculate rapidly has no great correlation with mathematical insight or creativity. Only a few great mathematicians have demonstrated lightning calculator skills, and many mathematicians are surprisingly poor at arithmetic.”

“There is no practical reason to know pi to 72 digits, or to 35 digits for that matter. Four decimal places are enough for the engineers of precision instruments. Ten decimal places are sufficient to calculate the circumference of the Earth to within a fraction of a centimeter. With 39 decimal places, it is possible to compute the circumference of a circle surrounding the known universe to within an accuracy of a radius of a hydrogen atom.”

“In medieval Spain, barbershops displayed signs saying Algebrista y Sangrador. The phrase means Bonesetter and Bloodletter, two trades that used to be part of a barber’s repertoire. (This is why a barber’s pole has red and white stripes – the red symbolized blood, and the white symbolizes the bandage.)”

“The root of algebrista is the Arabic al-jabr, which, in addition to referring to crude surgical techniques, also means restoration or reunion.”

“Al-Khwarizmi’s algebra book, together with another one he wrote on the Indian decimal system, became so widespread in Europe that his name was immortalized as a scientific term: al-Khwarizmi became Alchoarismi, Algorismi and, eventually, algorithm.”

“It was Descartes’s decision to use lowercase letters from the beginning of the alphabet for known quantities, and lowercase letters from the end of the alphabet for the unknowns. When the book was being printed, however, the printer started to run out of letters. He inquired if it mattered if x, y or z was used. Descartes replied not, so the printer chose to concentrate on x since it is used less frequently in French than y or z. As a result, x became fixed in math – and the wider culture – as the symbol for the unknown quantity.”

“The marvel of algebra is that once a problem is restated in symbolic terms often it is almost solved.”

“Yet often pilots flying the most modern jets prefer to use their [circular slide rules called] whiz wheels. Having a slide rule at hand means you can work out estimates very quickly, and also have a more visual understanding of the numerical parameters of the flight. Flying is safer because of pilots’ dexterity with an early-seventeenth-century calculating machine.”

“One of the most exciting characteristics of math is how seemingly different topics are interrelated, and how this in itself leads to vibrant new discoveries.”

“Two weeks later the [Sudoku] puzzle first appeared, and the days after that the Daily Mail introduced its own version. In January 2005, the Daily Telegraph joined the game, and not long afterwards, every British newspaper had to have a daily puzzle to keep up with the competition. That same year, the Independent reported a 700 percent rise in U.K. pencil sales, and attributed it to the craze.”

“The puzzle is math by stealth. Though Sudoku contains no arithmetic, it does require abstract thought, pattern recognition, logical deduction and the generation of algorithms. The puzzle also encourages an aggressive attitude toward problem solving and fosters an appreciation of mathematical elegance.”

“Math is not sums, calculations and formulae. It is pulling things apart to understand how things work.”

“I asked him if math gave him that same wonder. He replied, ‘Absolutely, yes.’”

“In the 1970s, a fascinating (if brutal) experiment examined how important a sense of control was for elderly patients in a nursing home. Some patients were allowed to choose how their rooms were arranged and allowed to choose a plant to look after. The others were told how their rooms would be and had a plant chosen and tended for them. The result after 18 months was striking. The patients who had control over their rooms had a 15 percent death rate, but for those who had no control the rate was 30 percent. Feeling in control can keep us alive.”

“While at first this seems like an amazing series of coincidences, the list is mathematically unsurprising because whenever you have a randomly selected group of 23 people, it is more likely than not that two people in it will share the same birthday.”

“With four people it is 70 percent likely that two will have birthdays within the same month. You need only seven people for it to be likely that two of them were born in the same week, and in a group of 14 it is as likely as it is not that two people were born within a day of each other. As group size gets bigger, the probability rises surprisingly fast. In a group of 35 people, the chance of a shared birthday is 85 percent, and with a group of 60 the chance is more than 99 percent.”

“I recently bought an electronic kitchen scale. It has a glass platform and an Easy to Read Blue Backlit Display. My purchase was not symptomatic of a desire to make elaborate desserts. Nor was I intending my apartment to become a haunt for local drug gangs. I was just interested in weighing stuff.”

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