history-page31 - x 3 2 x 2 + 10 x 1 = 5: K 1 10 2 M 1 i M 5...

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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 1 10 2 M 1 i M 5 Except for the Hindus, who syncopated their algebra, Diophantus innovation was largely neglected for many hundreds of years. Al-Khowrizms work was rhetorical. In Western Europe, algebra was mainly rhetorical until the 15th century. Symbolic algebra was not prevalent there until 1650. Viete (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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