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Posts Tagged ‘Numbers

“Those who wish to know the art of calculating, its subtleties and ingenuities, must know computing with hand figures”*…

Scholars at an Abbasid library. Maqamat of al-Hariri Illustration by Yahyá al-Wasiti, 1237 [source]

The House of Wisdom sounds a bit like make believe: no trace remains of this ancient library, destroyed in the 13th Century, so we cannot be sure exactly where it was located or what it looked like.

But this prestigious academy was in fact a major intellectual powerhouse in Baghdad during the Islamic Golden Age, and the birthplace of mathematical concepts as transformative as the common zero and our modern-day “Arabic” numerals.

Founded as a private collection for caliph Harun Al-Rashid in the late 8th Century then converted to a public academy some 30 years later, the House of Wisdom appears to have pulled scientists from all over the world towards Baghdad, drawn as they were by the city’s vibrant intellectual curiosity and freedom of expression (Muslim, Jewish and Christian scholars were all allowed to study there).

An archive as formidable in size as the present-day British Library in London or the Bibliothèque Nationale of Paris, the House of Wisdom eventually became an unrivalled centre for the study of humanities and sciences, including mathematics, astronomy, medicine, chemistry, geography, philosophy, literature and the arts – as well as some more dubious subjects such as alchemy and astrology.

To conjure this great monument thus requires a leap of imagination (think the Citadel in Westeros, or the library at Hogwarts), but one thing is certain: the academy ushered in a cultural Renaissance that would entirely alter the course of mathematics.

The House of Wisdom was destroyed in the Mongol Siege of Baghdad in 1258 (according to legend, so many manuscripts were tossed into the River Tigris that its waters turned black from ink), but the discoveries made there introduced a powerful, abstract mathematical language that would later be adopted by the Islamic empire, Europe, and ultimately, the entire world.

Tracing the House of Wisdom’s mathematical legacy involves a bit of time travel back to the future, as it were. For hundreds of years until the ebb of the Italian Renaissance, one name was synonymous with mathematics in Europe: Leonardo da Pisa, known posthumously as Fibonacci. Born in Pisa in 1170, the Italian mathematician received his primary instruction in Bugia, a trading enclave located on the Barbary coast of Africa (coastal North Africa). In his early 20s, Fibonacci traveled to the Middle East, captivated by ideas that had come west from India through Persia. When he returned to Italy, Fibonacci published Liber Abbaci, one of the first Western works to describe the Hindu-Arabic numeric system.

When Liber Abbaci first appeared in 1202, Hindu-Arabic numerals were known to only a few intellectuals; European tradesmen and scholars were still clinging to Roman numerals, which made multiplication and division extremely cumbersome (try multiplying MXCI by LVII!). Fibonacci’s book demonstrated numerals’ use in arithmetic operations – techniques which could be applied to practical problems like profit margin, money changing, weight conversion, barter and interest…

Fibonacci’s great genius was not just his creativity as a mathematician, however, but his keen understanding of the advantages known to Muslim scientists for centuries: their calculating formulas, their decimal place system, their algebra. In fact, Liber Abbaci relied almost exclusively on the algorithms of 9th-Century mathematician Al-Khwarizmi. His revolutionary treatise presented, for the first time, a systematic way of solving quadratic equations. Because of his discoveries in the field, Al-Khwarizmi is often referred to as the father of algebra – a word we owe to him, from the Arabic al-jabr, “the restoring of broken parts”—and in 821 he was appointed astronomer and head librarian of the House of Wisdom…

Centuries ago, a prestigious Islamic library (tragically burned in the the Siege of Baghdad) brought Arabic numerals to the world; its mathematical revolution changed our world: “How modern mathematics emerged from a lost Islamic library.”

For more on The House of Wisdom– and the sad stories of other libraries and archives that have been destroyed through the ages– see Richard Ovenden‘s remarkable new Burning the Books- a History of the Deliberate Destruction of Knowledge.

* Leonardo da Pisa, known posthumously as Fibonacci [see here]

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As we count our blessings, we might spare a thought for John Pell; he died on this date in 1685.  An English mathematician, he is perhaps best remembered for having introduced the “division sign”– the “obelus,” a short line with dots above and below– into use in English.  It was first used in German by Johann Rahn in 1659 in Teutsche Algebra; Pell’s translation brought the symbol to English-speaking mathematicians.  But Pell was an important influence on Rahn, and edited his book– so may well have been, many scholars believe, the originator of the symbol for this use.  (In any case the symbol wasn’t new to them:  the obelus [derived from the word for “roasting spit” in Greek] had already been used to mark passages in writings that were considered dubious, corrupt or spurious…. a use that surely seems only too appropriate to legions of second and third grade math students.)

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“He told me that in 1886 he had invented an original system of numbering”*…

A visualization of the 3-adic numbers

The rational numbers are the most familiar numbers: 1, -5, ½, and every other value that can be written as a ratio of positive or negative whole numbers. But they can still be hard to work with.

The problem is they contain holes. If you zoom in on a sequence of rational numbers, you might approach a number that itself is not rational. This short-circuits a lot of basic mathematical tools, like most of calculus.

Mathematicians usually solve this problem by arranging the rationals in a line and filling the gaps with irrational numbers to create a complete number system that we call the real numbers.

But there are other ways of organizing the rationals and filling the gaps: the p-adic numbers. They are an infinite collection of alternative number systems, each associated with a unique prime number: the 2-adics, 3-adics, 5-adics and so on.

The p-adics can seem deeply alien. In the 3-adics, for instance, 82 is much closer to 1 than to 81. But the strangeness is largely superficial: At a structural level, the p-adics follow all the rules mathematicians want in a well-behaved number system…

“We’re all on Earth and we work with the reals, but if you went [anywhere] else, you’d work with the p-adics,” [University of Washington mathematician Bianca] Viray explained. “It’s the reals that are the outliers.”

The p-adics form an infinite collection of number systems based on prime numbers. They’re at the heart of modern number theory… which is itself at the heart of computer science, numerical analysis, and cryptography: “An Infinite Universe of Number Systems.”

* Jorge Luis Borges, Labyrinths

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As we dwell on digits, we might send carefully-calculated birthday greetings to Klaus Friedrich Roth; he was born on this date in 1925. After escaping with his family from Nazi Germany, he was educated at Cambridge, then taught mathematics first at University College London, then at Imperial College London. He made a number of important contribution to Number Theory, for which he won the De Morgan Medal and the Sylvester Medal, and election to Fellowship of the Royal Society. In 1958 he was awarded mathematics’ highest honor, the Fields Medal, for proving Roth’s theorem on the Diophantine approximation of algebraic numbers.

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“It’s exact and indefinite. It’s like pi– you can keep figuring it out and always be right and never be done”*…

 

piPie

 

It’s Pi Day!  What better way to “prove” 3.14 than with that most perfect of pies– pizza!

Via the ever-illiminating Boing Boing.

See also: “Pi Day: How One Irrational Number Made Us Modern.”

* Peter Schjeldahl, quoting the painter John Currin

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As we celebrate the irrational, we might recall that it was on this date in 1958 that “Tequila” hit the top of the pop charts (sales and radio plays, both pop and R&B).

 

 

Written by (Roughly) Daily

March 14, 2020 at 1:01 am

“When you have mastered numbers, you will in fact no longer be reading numbers, any more than you read words when reading books. You will be reading meanings.”*…

Errors of judgment about large numbers can have a big impact on the way you view policies and government decisions. The rationale goes like this: The National Science Foundation received $7.463 billion for fiscal year 2016 through the Consolidated Appropriations Act. The total United States budget outlay for 2016 was $3.54 trillion. If you’re someone who perceives the difference between a billion and a trillion as relatively small, you’d think the US is spending a lot of money on the National Science Foundation—in fact, depending on your politics, you might applaud the federal government’s investment or even think it wasteful. But, if you understand that a billion is a thousand times less than a trillion, you can calculate that the Foundation got a paltry 0.2 percent of the budget outlay last year. (It may be more straightforward to think of the budget as roughly one-half to one-third of reported costs for the proposed US-Mexico border wall, and let your values guide you from there.)…

On the significance of scale: “How to Understand Extreme Numbers.

[The image above is, of course, from the ever-wonderful xkcd.]

* W.E.B. Du Bois

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As we nudge ourselves toward numeracy, we might spare a thought for Sewall Wright; he died on this date in 1988.  A geneticist, he was known for his influential work on evolutionary theory and also for his work on path analysis. He was a founder (with Ronald Fisher and J.B.S. Haldane) of population genetics– a major step in the development of the modern evolutionary synthesis combining genetics with evolution.   He is perhaps best remembered for his concept of genetic drift (called the Sewall Wright effect): when small populations of a species are isolated, the few individuals who carry certain relatively rare genes may fail, out of pure chance, to transmit them. The genes may therefore disappear and their loss may lead to the emergence of new species– although natural selection has played no part in the process.

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Written by (Roughly) Daily

March 3, 2017 at 1:01 am

“God made the integers; all the rest is the work of Man”*…

 

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From Alex Bellos: the results of his global online poll to find the world’s favorite number…

The winner?  Seven—  and it wasn’t even close…

Leopold Kronecker

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As we settle for anything but snake eyes, we might send symbolic birthday greetings to John Pell; he was born on this date in 1611.  An English mathematician of accomplishment, he is perhaps best remembered for having introduced the “division sign”– the “obelus”: a short line with dots above and below– into use in English.  It was first used in German by Johann Rahn in 1659 in Teutsche Algebra; Pell’s translation brought the symbol to English-speaking mathematicians.  But Pell was an important influence on Rahn, and edited his book– so may well have been, many scholars believe, the originator of the symbol for this use.  (In any case the symbol wasn’t new to them:  the obelus [derived from the word for “roasting spit” in Greek] had already been used to mark passages in writings that were considered dubious, corrupt or spurious…. a use that surely seems only too appropriate to legions of second and third grade math students.)

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Written by (Roughly) Daily

March 1, 2015 at 1:01 am

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