history-page31 - x 3 − 2 x 2 10 x − 1 = 5 K υ 1 ς 10...

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Algebra Literal Algebra Algebra with letters. Kline: Use of symbolism far more signiFcant than technical advances. Stages Rhetorical: No abbreviations or symbols Syncopated: Abbreviations for some frequent operations Symbolic: Largely mathematical shorthand Babylonians had rhetorical algebra. Diophantus Syncopated algebra Introduced the symbol ς for the unknown, roughly the same as a sigma used at the end of a word. Powers ⋆ x 2 Δ υ from the Frst two letters of dýnamis, meaning power. ⋆ x 3 K υ from the Frst two letters of kýbos, meaning cube. ⋆ x 4 Δ υ Δ ⋆ x 5 Δ K υ ⋆ x 6 K υ K ⋆ x k was represented by appending a superscript χ to the symbol for x k (k > 0 ) ; e.g., x 2 Δ υχ . ⋆ x 0 M , from the Frst two letters of the Greek word for unity. (To balance units?) Others =∼ − ∼ conjectured to be a condensation of λι , the Frst two letters of the Greek root for “less”. Arabic numbers and Diophantine notation for
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Unformatted text preview: x 3 − 2 x 2 + 10 x − 1 = 5: K υ 1 ς 10 ⋔ Δ υ 2 ◦ M 1 iσ ◦ M 5 • Except for the Hindus, who syncopated their algebra, Diophantus’ innovation was largely neglected for many hundreds of years. • Al-Khowârizmî’s work was rhetorical. • In Western Europe, algebra was mainly rhetorical until the 15th century. Symbolic algebra was not prevalent there until 1650. ◦ Vi’‘ete (late 1500s) ⋆ The inventor of literal calculus. ⋆ Used symbols for parameters as well as unknowns. ⋆ Used vowels for unknowns, consonants for parameters. ⋆ Introduced square brackets and braces as grouping symbols. (Parentheses appeared in 1544.) ◦ Descartes (early 1600s) ⋆ Dispensed with homogeniety. ⋆ Used modern naming conventions. ⋆ Made systematic use of positive integer exponents. (±or n = 2, only occasionally.) ◦ Miscellaneous...
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This note was uploaded on 12/29/2011 for the course MATH 378 taught by Professor Wen during the Fall '10 term at SUNY Stony Brook.

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