Tag: Mathematics

Numbers Too Large (and Small) to Count


At one point in the children’s novel “The Phantom Tollbooth,” its protagonist, Milo, sets out to reach infinity. When he abandons the quest as hopeless, he is advised “Infinity is a poor place.”

“Fantastic Numbers and Where to Find Them: A Cosmic Quest from Zero to Infinity,” by Antonio Padilla, holds a different view. As the author shows, infinity can be terrifying, but it is filled with an endless amount of numbers.

This is not a book just about numbers. It is about the relationship between numbers and physics, and how the world works at both the largest and the smallest scales. The numbers Padilla examines are fantastic in two senses. They are so extreme as to challenge belief and they are so extravagant as to seem fancy. And they define how the universe works.

This week on “The Learning Curve,” co-hosts Cara Candal and Gerard Robinson talk with Dr. Wilfried Schmid, Dwight Parker Robinson Emeritus Professor of Mathematics at Harvard University, who played a major role in drafting the 2000 Massachusetts Mathematics Curriculum Framework and served on the U.S. National Mathematics Advisory Panel (NMAP) in 2008. Dr. Schmid shares how he became interested in mathematics, and how it was taught and encouraged in the German schools he attended. He also talks about his academic career at Harvard, his teaching experiences there and at other elite universities, and the wide disparities in the level of academic math preparation between students from America and other countries. Professor Schmid explains how he became interested in K-12 mathematics, and his work with education expert Dr. Sandra Stotsky in drafting Massachusetts’ nation-leading math standards. They discuss the “math wars” that occurred across American K-12 education, and why, even after the landmark NMAP report, this country continues to struggle with teaching students basic mathematics.

Stories of the Week: In Texas, teachers who choose to resign during the school year are being stripped of their professional certification. US News & World Report’s annual ranking of top high schools is out, and the latest list features several from Massachusetts, including Boston’s exam and charter public schools, as well as wealthy suburban districts.

Visualizing Numbers Effectively


Most people have trouble with numbers. They easily visualize up to twelve. Once beyond 100 numbers kind of blur together. There is a difference between a one in 500 chance of something happening and a one in a million chance, but most people do not really understand it. Or the difference between a million and a billion.

“Making Numbers Count: The Art and Science of Communicating Numbers,” by Chip Heath and Karla Starr, offers a solution to that problem. It presents tools to understand numbers and effectively communicate the meaning of numbers to others. The authors provide a step-by-step process to give readers mastery of numbers using a few simple rules.

Take the difference between a million and a billion. Heath and Starr have readers visualize the difference this way: a million seconds is twelve days; a billion seconds is 32 years.  You suddenly appreciate the scale of the difference. Twelve days is two workweeks linked by a weekend. Thirty-two years? Depending on your age, it could be twice your lifespan, your lifespan, or half your lifespan. Regardless of your yardstick, you know it is a whole lot more than two workweeks linked by a weekend.

Quote of the Day – A Perfect Order


It is the function of science to discover the existence of a general reign of order in nature and to find the causes governing this order. And this refers in equal measure to the relations of man – social and political – and to the entire universe as a whole.

– Dmitri Mendeleev

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“The history of π is a quaint little mirror of the history of man.  It is the story of Archimedes of Syracuse, whose method of calculating π defied substantial improvement for some 1900 years, and it is also the story of a Cleveland businessman, who published a book in 1931 announcing the grand discovery that π was exactly […]

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Half-Elves, Third-Orcs, Gilgamesh, and Infinity


In one of the less-reputable areas of Ricochet – you know, the areas your mother warns you about – a conversation went something like this… (Spoiler alert, this post gets nerdy so fast, all the other posts will threaten to take your lunch money just for reading the next sentence.)

Person 1: What races do you usually play in Dungeons & Dragons? (See? Told ya. Cough up that money.)

Easy as ABC


S340LlNz4TmALRBxAAC2qVIrvC470So for years, like all of you, I’ve been idly wondering if it’s true that For every ε > 0, there are only finitely many triples of coprime positive integers a + b = c such that c > d1+ε, where d denotes the product of the distinct prime factors of abc. I figure that’s a question that occurs to everyone in the small hours of the morning, now and again. Once, when I was stuck in the métro during a wildcat strike, I reckoned I’d figured out a marvellous little proof. But I scribbled it down on the back of a pack of Gitanes, shoved it into the Pile of Papers, and never found it again.

Anyway, in 2012, Shinichi Mochizuki had one of those restless nights, then quietly posted 2,000 pages of scribblings on his website:

In them, Mochizuki claimed to have solved the abc conjecture, a 27-year-old problem in number theory that no other mathematician had even come close to solving. If his proof was correct, it would be one of the most astounding achievements of mathematics this century and would completely revolutionize the study of equations with whole numbers.

In Memorium: John Nash


John_Forbes_Nash I want to take note of John Nash’s death this weekend. Nash was a pivotal figure in several disciplines, and was the subject of the movie A Beautiful Mind. He was a man who suffered mental illness, and at times the suffering was severe.

I came across his work by studying game theory, in which his theory of “Nash equilibrium” is a basic building block. His equilibrium is what allows us to predict rational behavior (although it can’t predict whether the players are rational). The short version is that there are situations in which a person is rational to choose a particular strategy, regardless of what the other players do. That’s what makes that strategy predictable.

The genius, really, was in the simplicity of it. Some of the greatest geniuses became geniuses because they took some subject that had baffled people previously, but they were able to present it in such a way that everyone who followed said, “of course.” Nash was one of those guys.

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Here is an excerpt from a common core 9th grade mathematics textbook, from the chapter  Modeling and Using Exponential Functions: It can be amazing how many different historical events are connected in one way or another. For example, there are some environmentalists who claim that the increase in the world’s population has led to an […]

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Two or more years ago I stumbled upon the Numberphile channel on YouTube. While catching up on their videos, I discovered an interesting one on “How big is a billion?”. It got me to thinking about how Obama could use this to appear to be reducing the debt. The summary of the video is that there […]

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