48 | May 17, 2020 the sunday tiMes of Malta LIFE AND WELLBEING SCIENCE ALEXANDER FARRUGIA The square root of a number N is a number which, when squared (multiplied by itself), results in N. How did the ancient Greeks find square roots? Their method is nowadays called a ‘Greek ladder’, which uses whole numbers throughout. To illustrate Greek ladders, let us find the square root of five, so that N is five. We first find m, the largest number that, when squared, results in a number less than N. In our case, m is two, because two times two is four, which is less than five, but three times three is nine, which is larger than five. Then we can find q, the difference between N and the square of m. In this case, q is five minus the square of two, which is one. A sequence of numbers is then produced according to the following two rules: 1) The first two numbers in the sequence are zero and one. 2) Any other number in the sequence is the previous number times twice m, plus the number before that times q. For the square root of five, remembering that m is two and q is one, we obtain the sequence: 0, 1, 4, 17, 72, 305, 1292, 5473, … for example, the number 305 in the above sequence is 72, the previous number in the sequence, times twice m (four), plus 17 (the number before 72 in the sequence) times q (one). The Greeks stopped this sequence when the numbers became prohibitively large for them to be worked out efficiently. Once the se- quence is stopped, the square root of five is obtained by dividing the last two numbers in the sequence by each other, then subtracting m. This means that the square root of five is, roughly, 5473 divided by 1292, minus two, which is the fraction 2889 / 1292, which, in today’s notation, is 2.236068111…. Comparing this number with the decimal expansion of the square root of five, which is 2.236067977…, we notice that 2889 / 1292 is the square root of five correct to six decimal places! Moreover, to obtain an even better approximation, we simply go further in the above sequence of whole numbers. It is not known how the Greeks discovered this method. Since the above sequence is infi- nitely long, they might have concluded that there cannot exist an exact fraction for such square roots, possibly leading them to dis- cover irrational numbers. Nowadays, we have more sophisticated methods to approximate numbers such as square roots at our disposal. Scientists are ac- tively pursuing further improvement in effi- ciency of these methods. Dr Alexander Farrugia is the mathematics subject coordinator at the University of Malta Junior College. MYTH DEBUNKED Did Pythagoras discover Pythagoras’ theorem? Pythagoras of Samos (570BC-495BC) was a Greek philosopher whose teach- ings and philosophy not only influenced other Greek philosophers such as Plato (424BC-347BC), but also Western scientists such as Johannes Kepler (AD 1571-AD 1630) and Isaac Newton (AD 1642-AD 1727). He is most famous today for the theorem bearing his name, Pythagoras’ theo- rem. It states that, in a right-angled triangle, the square of the longest side is equal to the sum of the squares of the other two sides. But did Pythagoras actually discover this theo- rem himself? Many historians argue that Pythagoras did not dis- cover the theorem himself, but add that, perhaps, he was the one who introduced it to the Greeks, after learn- ing it from either Babylon- ian or Indian mathematics. Both the Babylonians and the Indians knew of Pythagoras’ theorem cen- turies before the Greeks. But did Pythagoras, at least, provide the first proof of the theorem? According to Walter Burkert, professor emeri- tus of classics, the answer is probably no, noting that no proof was ever attributed to Pythagoras. Moreover, Dan- ish historian Jens Høyrup remarks that the Babyloni- ans not only knew of Pythagorean triples (three numbers a,b,c satisfying a 2 +b 2 =c 2 ) but also how to apply them. This seems to suggest that they may have had a proof of Pythagoras’ theorem available. Whatever happened, it is certain that Pythagoras’ theorem is a hugely impor- tant result in geometry. This photo, taken on September 15, 2019 in Syracuse, Sicily, shows the author of this article, Alexander Farrugia, in front of the statue of the Greek mathematician Archimedes of Syracuse. Greek ladders and square roots PHOTO OF THE WEEK A 19th-century statue of the ancient Greek mathematician Euclid of Alexandria. PHOTO: WIKIPEDIA DID YOU KNOW? • The mathematics introduced to the world by ancient Greece is fundamental to today’s knowledge of geometry. The idea of for- mal proof in mathematics also stemmed from ancient Greece. • Perhaps the most important work that ancient Greece gifted to the modern world is Euclid’s Elements. Written in 300BC, it is a collection of 13 books containing most, if not the entire knowl- edge, of the mathematics that was known to the ancient Greeks at that time. It is considered as the first mathematics book presenting its contents via definitions, from which theorems are deduced using logical arguments. • Another champion of ancient Greek mathematics is Archimedes of Syracuse (250BC). Among other achievements, Archimedes contributed a proof that pi is a number between the fractions 223/71 and 22/7, the latter of which is still used as an approxi- mation of pi to this day. For more trivia, see: www.um.edu.mt/think. SOUND BITES • Diophantus of Alexandria, a mathematician from ancient Greece, was the author of several books called Arithmetica, the majority of which are lost. In the paper ‘J. Christianidis & A. Megremia. Tracing the Early History of Algebra: Testimonies on Diophantus in the Greek-speaking World (4th-7th century CE), Historia Mathematica, 47 (2019), 16-38’, the authors challenge the common belief that Diophantus’s Arithmetica is an isolated phenomenon in the history of ancient Greek mathematics. This is done by investigating testimonies that suggest that Diophan- tus’s heritage was already present in the intellectual setting of the Greek-speaking world during the late antique times. • Conic sections constitute a major contribution to mathematics by the ancient Greeks. The paper ‘E. Rinner. Ancient Greek Sundials and the Theory of Conic Sections Reconsidered. In: John Steele, Mathieu Ossendrijver (eds.), Studies on the Ancient Exact Sciences in Honor of Lis Brack-Bernsen, Berlin Studies of the Ancient World, 44 (2018), 165-182’, argues that craftsmen who built sundi- als in the Roman period had basic knowledge of conic sections. For more soundbites, listen to Radio Mocha Malta https://www.fb.com/ RadioMochaMalta/.
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48 | May 17, 2020 the sunday tiMes of Malta
LIFEANDWELLBEING SCIENCE
ALEXANDER FARRUGIA
The square root of a number N is a numberwhich, when squared (multiplied by itself),results in N. How did the ancient Greeks findsquare roots? Their method is nowadayscalled a ‘Greek ladder’, which uses wholenumbers throughout.
To illustrate Greek ladders, let us find thesquare root of five, so that N is five. We firstfind m, the largest number that, whensquared, results in a number less than N.
In our case, m is two, because two times twois four, which is less than five, but three timesthree is nine, which is larger than five. Thenwe can find q, the difference between N andthe square of m. In this case, q is five minusthe square of two, which is one.
A sequence of numbers is then producedaccording to the following two rules:
1) The first two numbers in the sequenceare zero and one.
2) Any other number in the sequence is theprevious number times twice m, plus thenumber before that times q.
For the square root of five, rememberingthat m is two and q is one, we obtain the
sequence: 0, 1, 4, 17, 72, 305, 1292, 5473, … forexample, the number 305 in the above sequence is 72, the previous number in the sequence, times twice m (four), plus 17 (the number before 72 in the sequence) timesq (one).
The Greeks stopped this sequence when thenumbers became prohibitively large for themto be worked out efficiently. Once the se-quence is stopped, the square root of five isobtained by dividing the last two numbers inthe sequence by each other, then subtractingm. This means that the square root of five is,roughly, 5473 divided by 1292, minus two,which is the fraction 2889 / 1292, which, intoday’s notation, is 2.236068111….
Comparing this number with the decimalexpansion of the square root of five, which is2.236067977…, we notice that 2889 / 1292 isthe square root of five correct to six decimalplaces! Moreover, to obtain an even better approximation, we simply go further in theabove sequence of whole numbers.
It is not known how the Greeks discoveredthis method. Since the above sequence is infi-nitely long, they might have concluded thatthere cannot exist an exact fraction for suchsquare roots, possibly leading them to dis-cover irrational numbers.
Nowadays, we have more sophisticatedmethods to approximate numbers such assquare roots at our disposal. Scientists are ac-tively pursuing further improvement in effi-ciency of these methods.
Dr Alexander Farrugia is the mathematics
subject coordinator at the University of Malta
Junior College.
MYTH DEBUNKED
Did Pythagorasdiscover Pythagoras’ theorem?Pythagoras of Samos(570BC-495BC) was a Greekphilosopher whose teach-ings and philosophy notonly influenced other Greekphilosophers such as Plato (424BC-347BC), butalso Western scientists suchas Johannes Kepler (AD1571-AD 1630) and IsaacNewton (AD 1642-AD 1727).
He is most famous todayfor the theorem bearing hisname, Pythagoras’ theo-rem. It states that, in aright-angled triangle, thesquare of the longest side isequal to the sum of thesquares of the other twosides. But did Pythagorasactually discover this theo-rem himself?
Many historians arguethat Pythagoras did not dis-cover the theorem himself,but add that, perhaps, hewas the one who introducedit to the Greeks, after learn-ing it from either Babylon-ian or Indian mathematics.Both the Babylonians and the Indians knew ofPythagoras’ theorem cen-turies before the Greeks.
But did Pythagoras, atleast, provide the first proofof the theorem?
According to WalterBurkert, professor emeri-tus of classics, the answer isprobably no, noting that noproof was ever attributed toPythagoras. Moreover, Dan-ish historian Jens Høyrupremarks that the Babyloni-ans not only knew ofPythagorean triples (threenumbers a,b,c satisfyinga2+b2=c2) but also how toapply them. This seems tosuggest that they may havehad a proof of Pythagoras’theorem available.
Whatever happened, it iscertain that Pythagoras’theorem is a hugely impor-tant result in geometry.
This photo, taken on September 15, 2019 in Syracuse,Sicily, shows the author of this article, Alexander Farrugia,in front of the statue of the Greek mathematicianArchimedes of Syracuse.
Greek ladders andsquare roots
PHOTO OF THE WEEK
A 19th-century statue of the ancientGreek mathematician Euclid ofAlexandria. PHOTO: WIKIPEDIA
DID YOU KNOW?• The mathematics introduced to the world by ancient Greece is
fundamental to today’s knowledge of geometry. The idea of for-mal proof in mathematics also stemmed from ancient Greece.
• Perhaps the most important work that ancient Greece gifted tothe modern world is Euclid’s Elements. Written in 300BC, it is acollection of 13 books containing most, if not the entire knowl-edge, of the mathematics that was known to the ancient Greeksat that time. It is considered as the first mathematics book presenting its contents via definitions, from which theorems arededuced using logical arguments.
• Another champion of ancient Greek mathematics is Archimedesof Syracuse (250BC). Among other achievements, Archimedescontributed a proof that pi is a number between the fractions223/71 and 22/7, the latter of which is still used as an approxi-mation of pi to this day.
For more trivia, see: www.um.edu.mt/think.
SOUND BITES• Diophantus of Alexandria, a mathematician from ancient
Greece, was the author of several books called Arithmetica, themajority of which are lost. In the paper ‘J. Christianidis & A.Megremia. Tracing the Early History of Algebra: Testimonies onDiophantus in the Greek-speaking World (4th-7th century CE),Historia Mathematica, 47 (2019), 16-38’, the authors challengethe common belief that Diophantus’s Arithmetica is an isolatedphenomenon in the history of ancient Greek mathematics. Thisis done by investigating testimonies that suggest that Diophan-tus’s heritage was already present in the intellectual setting ofthe Greek-speaking world during the late antique times.
• Conic sections constitute a major contribution to mathematics bythe ancient Greeks. The paper ‘E. Rinner. Ancient Greek Sundialsand the Theory of Conic Sections Reconsidered. In: John Steele,Mathieu Ossendrijver (eds.), Studies on the Ancient Exact Sciencesin Honor of Lis Brack-Bernsen, Berlin Studies of the AncientWorld, 44 (2018), 165-182’, argues that craftsmen who built sundi-als in the Roman period had basic knowledge of conic sections.
For more soundbites, listen to Radio Mocha Malta https://www.fb.com/RadioMochaMalta/.
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