The article, which should be required reading for all advanced high school students, creates a tantalizing picture of how much easier certain fundamental concepts of trigonometry could be in an alternate universe where we use tau. For example, with pi-based thinking, if you want to designate a point one third of the way around the circle, you say it has gone two thirds pi radians. Three quarters around the same circle has gone one and a half pi radians. Everything is distorted by a confusing factor of two. By contrast, a third of a circle is a third of tau. Three quarters of a circle is three quarters tau. As a result of pi, Palais says, “the opportunity to impress students with a beautiful and natural simplification is turned into an absurd exercise in memorization and dogma.”
— Read on www.scientificamerican.com/article/let-s-use-tau-it-s-easier-than-pi/
An eminent mathematician reveals that his advances in the study of millennia-old mathematical questions owe to concepts derived from physics.
— Read on www.quantamagazine.org/secret-link-uncovered-between-pure-math-and-physics-20171201/
The rational numbers include the integers and any number that can be expressed as a ratio of two integers, such as 1, –4 and 99/100. Mathematicians are particularly interested in rational numbers that solve what are called “Diophantine equations” — polynomial equations with integer coefficients, like x2 + y2 = 1. These equations are named after Diophantus, who studied them in Alexandria in the third century A.D.
by Roger » Mon Feb 09, 2009 6:17 pm
Heres a puzzle you may have heard before which you can build as a simple electric circuit. First, the puzzle: a farmer is traveling to market with his cat, a chicken and some corn. He has to cross a river, and the only way to cross is in a small boat which can hold the farmer and just one of the three items he has with him. The problem is, he has to be very careful about what he chooses to leave behind at any time. If the cat and chicken are left alone, the cat will eat the chicken. If the chicken and the corn are left alone, the chicken will eat the corn. To solve the puzzle, you must show how the farmer can get himself and his three items across the river without losing any of them. The goal of this project is to design a simple electrical circuit that follows the puzzle. Youll need a 6 V battery, a flashlight bulb, a bulb holder, some connecting wire, and four toggle switches: 3 SPDT single-pole, double throw and 1 DPDT double-pole, double throw. Each switch represents one of the items: the farmer, the cat, the chicken and the corn you have to figure out which need to be SPDT switches and which one needs to be a DPDT switch. The switches are mounted on a small panel, in a horizontal row representing the river, which you can draw in. Each switch is labeled “Farmer”, “Cat”, “Chicken”, “Corn”. The circuit is to be designed so that if either of the problematic pairs cat-chicken, or chicken-corn are left alone on the same side of the river, the light bulb lights up, indicating an incorrect solution you can add a 6 V buzzer, too, if you like. Since the boat can hold only two items, players can use only two switches per “move”. Irwin Maths book, Wires and Watts: Understanding and Using Electricity has the solution Math, 1981, 67–70, but see if you can figure this one out on your own. The puzzle Ive got figured out but I am stuck on the wiring and cannot find an Irwin Math book.
Karnaugh Maps are used for many small design problems. It’s true that many larger designs are done using computer implementations of different algorithms. However designs with a small number of variables occur frequently in interface problems and that makes learning Karnaugh Maps worthwhile. In addition, if you study Karnaugh Maps you will gain a great deal of insight into digital logic circuits.
In this section we’ll examine some Karnaugh Maps for three and four variables. As we use them be particularly tuned in to how they are really being used to simplify Boolean functions.
The goals for this lesson include the following.
Given a Boolean function described by a truth table or logic function,
Draw the Karnaugh Mapfor the function.
Use the information from a Karnaugh Map to determine the smallest sum-of-products function.
A Karnaugh Map is a grid-like representation of a truth table. It is really just another way of presenting a truth table, but the mode of presentation gives more insight. A Karnaugh map has zero and one entries at different positions. Each position in a grid corresponds to a truth table entry. Here’s an example taken from the voting circuit presented in the lesson on Minterms. The truth table is shown first. The Karnaugh Map for this truth table is shown after the truth table.
Even a rudimentary look at probability can give new insights about how to interpret data. Simple thought experiments an can give new insight into the different ways misunderstanding of statistics can distort the way we perceive the world.
We’ve selected five classic problems solved in unconventional ways that can help one get a new way to understand the way that data can be misleading and the story on the surface can take people in the wrong direction.
August 26, 2006
UCLA’s Terrence Tao recently won the Fields Medal, the highest award in mathematics. Their article on him has some interesting things to say about working on math problems.
Tao offered some insight. “I don’t have any magical ability,” he said. “I look at a problem, and it looks something like one I’ve done before; I think maybe the idea that worked before will work here. Nothing’s working out; then you think of a small trick that makes it a little better but still is not quite right. I play with the problem, and after a while, I figure out what’s going on.
“Most people, faced with a math problem, will try to solve the problem directly,” he said. “Even if they get it, they might not understand exactly what they did. Before I work out any details, I work on the strategy. Once you have a strategy, a very complicated problem can split up into a lot of mini-problems. I’ve never really been satisfied with just solving the problem. I want to see what happens if I make some changes; will it still work? If you experiment enough, you get a deeper understanding. After a while, when something similar comes along, you get an idea of what works and what doesn’t work.
“It’s not about being smart or even fast,” Tao added. “It’s like climbing a cliff: If you’re very strong and quick and have a lot of rope, it helps, but you need to devise a good route to get up there. Doing calculations quickly and knowing a lot of facts are like a rock climber with strength, quickness and good tools. You still need a plan — that’s the hard part — and you have to see the bigger picture.”
And on the relevance that all this problem-solving has on life, he says:
“Mathematicians often work on pure problems that do not have any applications for 20 years, and then a physicist or computer scientist or engineer has a real-life problem that requires the solution of a mathematical problem and finds that someone already solved it 20 years ago,” Tao said. “When Einstein developed his theory of relativity, he needed a theory of curved space. Einstein found that a mathematician devised exactly the theory he needed more than 30 years earlier.”
Then there’s the device itself: clearly there’s a lot of thoughtfulness and smarts that went into the design. But there’s also a palpable contempt for the owner. I believe — really believe — in the stirring words of the Maker Manifesto: if you can’t open it, you don’t own it. Screws not glue. The original Apple ][+ came with schematics for the circuit boards, and birthed a generation of hardware and software hackers who upended the world for the better. If you wanted your kid to grow up to be a confident, entrepreneurial, and firmly in the camp that believes that you should forever be rearranging the world to make it better, you bought her an Apple ][+.
But with the iPad, it seems like Apple’s model customer is that same stupid stereotype of a technophobic, timid, scatterbrained mother as appears in a billion renditions of “that’s too complicated for my mom” listen to the pundits extol the virtues of the iPad and time how long it takes for them to explain that here, finally, is something that isn’t too complicated for their poor old mothers.
I showed Dan’s video to my freshmen study hall. The first thing it showed me is how much they still need to read. They had a hard time following Dan’s vocabulary, which, while he is well spoken, he isn’t exactly Mary Shelley or anything.
Second, they totally agreed with him. I have a really great mix of high and low achievers in this study hall. All of them responded the same way. They said, why can’t we just ask simple questions that need math? One kid said he hates doing math, because he does every homework problem and repeats the same thing that he either understands or doesn’t. Notice that he said he can still do his homework whether he understands the concept or not. Yikes.
Test for math PROBLEM-SOLVING homework–if kids can do by dumb rote–is it really helping?
Dan Meyer: Math class needs a makeover: Patient Problem Solving