![]() He noted that, after each monthly generation, the number of pairs of rabbits increased from 1 to 2 to 3 to 5 to 8 to 13, etc, and identified how the sequence progressed by adding the previous two terms (in mathematical terms, F n = F n-1 + F n-2), a sequence which could in theory extend indefinitely. He discovered the sequence – the first recursive number sequence known in Europe – while considering a practical problem in the “Liber Abaci” involving the growth of a hypothetical population of rabbits based on idealized assumptions. The discovery of the famous Fibonacci sequenceįibonacci is best known, though, for his introduction into Europe of a particular number sequence, which has since become known as Fibonacci Numbers or the Fibonacci Sequence. The horizontal bar notation for fractions was also first used in this work (although following the Arabic practice of placing the fraction to the left of the integer). However, common sense eventually prevailed and the new system was adopted throughout Europe by the 15th century, making the Roman system obsolete. ![]() Despite its obvious advantages, uptake of the system in Europe was slow (this was after all during the time of the Crusades against Islam, a time in which anything Arabic was viewed with great suspicion), and Arabic numerals were even banned in the city of Florence in 1299 on the pretext that they were easier to falsify than Roman numerals. In particular, in 1202, he wrote a hugely influential book called “Liber Abaci” (“Book of Calculation”), in which he promoted the use of the Hindu-Arabic numeral system, describing its many benefits for merchants and mathematicians alike over the clumsy system of Roman numerals then in use in Europe. On his return to Italy, he helped to disseminate this knowledge throughout Europe, thus setting in motion a rejuvenation in European mathematics, which had lain largely dormant for centuries during the Dark Ages. Little is known of his life except that he was the son of a customs offical and, as a child, he travelled around North Africa with his father, where he learned about Arabic mathematics. The 13th Century Italian Leonardo of Pisa, better known by his nickname Fibonacci, was perhaps the most talented Western mathematician of the Middle Ages. Here, instead of only adding the next number, we alternate between subtraction and addition.LEONARDO FIBONACCI – ITALIAN MATHEMATICIAN (WROTE LEBER ABACI) One of the most well-known divergent series in music is the Harmonic series, where the denominator gets bigger with each step: The Harmonic Series, or the Overtone Series Let’s see a few divergent series, and then we’ll discuss about a few convergent examples. They are opposite to convergent series, the terms of which get closer and closer to a certain number. This means that, with infinite iterations of the series, said term will reach ∞ or -∞. Divergent Seriesĭivergent series are sequences of numbers in which a term diverges from its starting point. Today is sort of bridge between the two semesters, as what we’ll see can be applied to rhythm as well as harmony and melody: number sequences. I’ve exhausted the rhythm topics I wanted to address, and I wanted to talk more about verticality. ![]() The first semester of Advanced Mathematics has now ended.
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