By delving into older, purely mathematical Babylonian texts written between 1800 B.C.E. and 1600 B.C.E., which also described computations with a trapezoid, he realized that the astronomers who made the tablets had gone a step further. To compute the time at which Jupiter would have moved halfway along its ecliptic path, the astronomers divided the 60-day trapezoid into two smaller ones of equal area. The vertical line dividing the two trapezoids marked the halfway time; because of the different shapes of the trapezoids, it indicated not 30 days but slightly fewer.
The Babylonians had developed “abstract mathematical, geometrical ideas about the connection between motion, position and time that are so common to any modern physicist or mathematician,” Ossendrijver says.
Indeed, compared with the complex geometry embraced by the ancient Greeks a few centuries later, with its cycles and epicycles, the inscriptions reflect “a more abstract and profound conception of a geometrical object in which one dimension represents time,” says historian Alexander Jones of New York University in New York City. “Such concepts have not been found earlier than in 14th century European texts on moving bodies,” he adds. “Their presence … testifies to the revolutionary brilliance of the unknown Mesopotamian scholars who constructed Babylonian mathematical astronomy.”
A review of When Computers Were Human - the period of time when numerical problems too large for one person to solve were broken into parts and solved by a group of people. It sounds like an interesting bit of history.
Human computers were usually women - probably the most famous groups was "Pickering's Harem" at the Harvard Observatory. It was created in a time where the amount of data was beginning to surpass the individual processing ability of astronomers at the observatory and was female as women were paid about half men's wages, so processing power per dollar was twice as high. The Harem produced a few remarkable astronomers including Annie Jump Cannon. She was central to the development of astrophysics and worth reading about in her own right.
Not mentioned in the review was Lewis Richardson - a polymath of the early to mid 20th century who developed numerical techniques for forecasting weather. In the early 1920s he estimated 64,000 people could be used as a massively parallel human computer to solve numerically solve a set of differential equations and forecast the world's weather.
The error detection scheme in the IBM 1401 - it operated on decimal digits (not binary) using binary coded decimal where a trip to qui-binary was made to an adder .. a bit Rube Goldberg-ish, but it made the digital computer robust enough for regular business use.
from Ken Shirriff’s excellent history of technology blog
This article has discussed how the 1401 adds or subtracts a single digit. The complete addition/subtraction process in the 1401 is even more complex because the 1401 handles numbers of arbitrary length; the hardware loops over each digit to process the entire numbers.
Studying old computers such as the IBM 1401 is interesting because they use unusual, forgotten techniques such as qui-binary arithmetic. While qui-binary arithmetic seems strange at first, its error-detection properties made it useful for the IBM 1401. Old computers are also worth studying because their circuitry can be thoroughly understood. After careful examination, you can see how arithmetic, for instance, works, down to the function of individual transistors.
Tiling a plane means you can cover a flat surface completely with patterns of the same shape and size. There won't be any gaps or overlaps... All triangles tile a plane as do all quadrilaterals. Things break with pentagons - a regular pentagon with equal angles and side lengths - doesn't, but 14 pentagons that allow different angles and side lengths have been shown to do the job. That is 14 until recently. A 15th was added about a month ago.
Perhaps you can have a tile maker create something for the bathroom floor, but more interesting looking are tessellations. Now the plane can be tiled with more than one geometric shape. There are repeating and non-repeating patterns. M.C. Escher was an artistic master,
Math is usually seen as a tool in the US rather than part of curiosity and even excitement. There have been exceptions. For years Martin Gardner's Mathematical Games column ran in Scientific American inspiring many to play with math often without needing a deep background.
a tip of the hat to Barb
Scientific American also ran C.L. Strong's Amateur Scientist column for years - effective a gateway drug to professional science for thousands as well as the basis for basement science.
The Canadian mathematician made a multimillion-dollar fortune by writing calculus textbooks for universities and high schools. Last year alone he sold 500,000 books, accounting for about $26.6m (£17.5m) in sales, according to his estate.
Stewart was also an unlikely architectural trailblazer. He devoted many years of his life, and much of his income, to building his dream home in an upmarket Toronto neighbourhood. Integral House – named after the “integral”, a concept in calculus – is a shrine to calculus, the mathematics of flowing change.
Stewart died last December, aged 73, and Integral House is now for sale at £11.4m .
“The house is a piece of art,” said Paul Maranger, of Sotheby’s International Realty. “When buyers go into the house the first reaction is a sense of awe.”
Parents who are uneasy about their own math skills often worry about how best to teach the subject to their kids.
Well ... there's an app for that. Tons of them, in fact. And a study published today in the journal Science suggests that at least one of them works pretty well for elementary school children and math-anxious parents.
A team from the University of Chicago used a demographically diverse group of first-graders and their parents — nearly 600 in all — across a wide swath of Chicago. One group got to use an iPad app called Bedtime Math, built by a nonprofit with the same name. (The app is also available for Android, but we're told most used the iPad version) The no-frills app uses stories and sound effects to present kids with math problems that they can solve with their parents.
Talia Berkowitz*, Marjorie W. Schaeffer*, Erin A. Maloney, Lori Peterson, Courtney Gregor, Susan C. Levine†, Sian L. Beilock†
University of Chicago, Department of Psychology, Chicago, IL 60637, USA.
With a randomized field experiment of 587 first-graders, we tested an educational intervention designed to promote interactions between children and parents relating to math. We predicted that increasing math activities at home would increase children’s math achievement at school. We tested this prediction by having children engage in math story time with their parents. The intervention, short numerical story problems delivered through an iPad app, significantly increased children’s math achievement across the school year compared to a reading (control) group, especially for children whose parents are habitually anxious about math. Brief, high-quality parent-child interactions about math at home help break the intergenerational cycle of low math achievement.