Editing Expression (mathematics) (section)

=== Syncopated stage ===
The "syncopated" stage of mathematics introduced symbolic abbreviations for commonly used operations and quantities, marking a shift from purely [[Geometry|geometric]] reasoning. [[Ancient Greek mathematics]], largely geometric in nature, drew on [[Egyptian numerals|Egyptian numerical systems]] (especially [[Attic numerals]]),<ref>Mathematics and Measurement By Oswald Ashton Wentworth Dilk. Pg [https://books.google.com/books?id=AKJZvXOS7n4C&pg=PA14 14]</ref> with little interest in algebraic symbols, until the arrival of [[Diophantus]] of [[History of Alexandria|Alexandria]],<ref>[http://www.ms.uky.edu/~carl/ma330/projects/diophanfin1.html Diophantine Equations]. Submitted by: Aaron Zerhusen, Chris Rakes, & Shasta Meece. MA 330-002. Dr. Carl Eberhart. 16 February 1999.</ref> who pioneered a form of [[syncopated algebra]] in his ''[[Arithmetica]],'' which introduced symbolic manipulation of expressions.<ref>Boyer (1991). "Revival and Decline of Greek Mathematics". pp. 180-182. "In this respect it can be compared with the great classics of the earlier Alexandrian Age; yet it has practically nothing in common with these or, in fact, with any traditional Greek mathematics. It represents essentially a new branch and makes use of a different approach. Being divorced from geometric methods, it resembles Babylonian algebra to a large extent. But whereas Babylonian mathematicians had been concerned primarily with approximate solutions of determinate equations as far as the third degree, the Arithmetica of Diophantus (such as we have it) is almost entirely devoted to the exact solution of equations, both determinate and indeterminate. [...] Throughout the six surviving books of Arithmetica there is a systematic use of abbreviations for powers of numbers and for relationships and operations. An unknown number is represented by a symbol resembling the Greek letter ζ {\displaystyle \zeta } (perhaps for the last letter of arithmos). [...] It is instead a collection of some 150 problems, all worked out in terms of specific numerical examples, although perhaps generality of method was intended. There is no postulation development, nor is an effort made to find all possible solutions. In the case of quadratic equations with two positive roots, only the larger is give, and negative roots are not recognized. No clear-cut distinction is made between determinate and indeterminate problems, and even for the latter for which the number of solutions generally is unlimited, only a single answer is given. Diophantus solved problems involving several unknown numbers by skillfully expressing all unknown quantities, where possible, in terms of only one of them."</ref> His notation represented unknowns and powers symbolically, but without modern symbols for [[Relation (mathematics)|relations]] (such as [[Equality (mathematics)|equality]] or [[Inequality (mathematics)|inequality]]) or [[Exponentiation|exponents]].<ref name="Boyer">Boyer (1991). "Revival and Decline of Greek Mathematics". p. 178. "The chief difference between Diophantine syncopation and the modern algebraic notation is the lack of special symbols for operations and relations, as well as of the exponential notation."</ref> An unknown number was called <math>\zeta</math>.<ref>A History of Greek Mathematics: From Aristarchus to Diophantus.  By Sir Thomas Little Heath. Pg [[iarchive:bub gb 7DDQAAAAMAAJ/page/n472|456]]</ref> The square of <math>\zeta</math> was <math>\Delta^v</math>; the cube was <math>K^v</math>; the fourth power was <math>\Delta^v\Delta</math>; the fifth power was <math>\Delta K^v</math>; and <math>\pitchfork</math> meant to subtract everything on the right from the left.<ref>A History of Greek Mathematics: From Aristarchus to Diophantus.  By Sir Thomas Little Heath. Pg [[iarchive:bub gb 7DDQAAAAMAAJ/page/n474|458]]</ref> So for example, what would be written in modern notation as:
<math display="block">x^3 - 2x^2 + 10x -1,</math>
Would be written in Diophantus's syncopated notation as:
: <math>\Kappa^{\upsilon} \overline{\alpha} \; \zeta \overline{\iota} \;\, \pitchfork \;\, \Delta^{\upsilon} \overline{\beta} \; \Mu \overline{\alpha} \,\;</math>
In the 7th century, [[Brahmagupta]] used different colours to represent the unknowns in algebraic equations in the ''[[Brāhmasphuṭasiddhānta]]''. Greek and other ancient mathematical advances, were often trapped in cycles of bursts of creativity, followed by long periods of stagnation, but this began to change as knowledge spread in the [[early modern period]].