(Roughly) Daily

“No part of mathematics is ever, in the long run, ‘useless’.”*…

The number 1 can be written as a sum of distinct unit fractions, such as 1/2 + 1/3 + 1/12 + 1/18 + 1/36…

Number theorists are always looking for hidden structure. And when confronted by a numerical pattern that seems unavoidable, they test its mettle, trying hard — and often failing — to devise situations in which a given pattern cannot appear.

One of the latest results to demonstrate the resilience of such patterns, by Thomas Bloom of the University of Oxford, answers a question with roots that extend all the way back to ancient Egypt.

“It might be the oldest problem ever,” said Carl Pomerance of Dartmouth College.

The question involves fractions that feature a 1 in their numerator, like 1/2, 1/7 or 1/122. These “unit fractions” were especially important to the ancient Egyptians because they were the only types of fractions their number system contained: With the exception of a single symbol for 23, they could only express more complicated fractions (like 3/4) as sums of unit fractions (1/2 + 1/4).

The modern-day interest in such sums got a boost in the 1970s, when Paul Erdős and Ronald Graham asked how hard it might be to engineer sets of whole numbers that don’t contain a subset whose reciprocals add to 1. For instance, the set {2, 3, 6, 9, 13} fails this test: It contains the subset {2, 3, 6}, whose reciprocals are the unit fractions 1/2, 1/3 and 1/6 — which sum to 1.

More exactly, Erdős and Graham conjectured that any set that samples some sufficiently large, positive proportion of the whole numbers — it could be 20% or 1% or 0.001% — must contain a subset whose reciprocals add to 1. If the initial set satisfies that simple condition of sampling enough whole numbers (known as having “positive density”), then even if its members were deliberately chosen to make it difficult to find that subset, the subset would nonetheless have to exist.

“I just thought this was an impossible question that no one in their right mind could possibly ever do,” said Andrew Granville of the University of Montreal. “I didn’t see any obvious tool that could attack it.”…

Bloom, building on work by Ernie Croot, found that tool. The ubiquity of ways to represent whole numbers as sums of fractions: “Math’s ‘Oldest Problem Ever’ Gets a New Answer,” by Jordana Cepelewicz (@jordanacep) in @QuantaMagazine.

* “No part of mathematics is ever, in the long run, ‘useless.’ Most of number theory has very few ‘practical’ applications. That does not reduce its importance, and if anything it enhances its fascination. No one can predict when what seems to be a most obscure theorem may suddenly be called upon to play some vital and hitherto unsuspected role.” – C. Stanley Ogilvy, Excursions in Number Theory

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As we recombine, we might send carefully-calculated birthday greetings to Ulugh Beg (or, officially, Mīrzā Muhammad Tāraghay bin Shāhrukh); he was born on this date in 1394. A Timurid sultan with a hearty interest in science and the arts, he is better remembered as an astronomer and mathematician.

The most important observational astronomer of the 15th century, he built the great Ulugh Beg Observatory in Samarkand between 1424 and 1429– considered by scholars to have been one of the finest observatories in the Islamic world at the time and the largest in Central Asia. In his observations he discovered a number of errors in the computations of the 2nd-century Alexandrian astronomer Ptolemy, whose figures were still being used. His star map of 994 stars was the first new one since Hipparchus. Among his contributions to mathematics were trigonometric tables of sine and tangent values correct to at least eight decimal places.

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March 22, 2022 at 1:00 am

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Cubed…

Nearly 40 years ago, a Hungarian architecture professor, Emo Rubik, created a puzzle to use with his design students- a puzzle with 43 quintillion possible combinations and one solution.  Within five years, it had been played by over 20% of the world’s population, and has so far sold over 350 million units (not counting “unofficial” versions).

Read an interview with Rubik at CNN, and find other confounding facts (like the ones above) here.

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As we twist and turn, we might spare a thought for Ulugh Beg; he died on this date in 1449.  Probably Mongolia’s greatest scientist, Beg was a Timurid ruler and sultan, a mathematician, and the greatest astronomer of his time.  In his observatory in at Samarkand he discovered a number of errors in the computations of the 2nd-century Alexandrian astronomer Ptolemy, whose figures were still being used; his star map (of 994 stars) was the first since Hipparchus’.

Forensic facial reconstruction

Written by (Roughly) Daily

October 27, 2012 at 1:01 am

Posted in Uncategorized