Editing Diophantus (section)

===Notation===
Diophantus introduced an algebraic symbolism that used an abridged notation for frequently occurring operations, and an abbreviation for the unknown and for the powers of the unknown. 

Similar to medieval Arabic algebra, Diophantus uses three stages to solution of a problem by algebra:{{sfn|Christianidis|Oaks|2023|pp=53–66}}
# An unknown is named and an equation is set up
# An equation is simplified to a standard form (''al-jabr'' and ''al-muqābala'' in Arabic)
# Simplified equation is solved

Diophantus does not give classification of equations in six types like [[Al-Khwarizmi]] in extant parts of ''Arithmetica''. He does says that he would give solution to three terms equations later, so this part of work is possibly just lost.{{sfn|Christianidis|Oaks|2013|pp=158-160}}

The main difference between Diophantine notation and modern algebraic notation is that the former lacked special symbols for operations, relations, and exponentials.{{sfn|Cooke|1997|pp=167-168}}{{sfn|Derbyshire|2006|pp=35-36}} So for example, what would be written in modern notation as
<math display="block">x^3 - 2x^2 + 10x -1 = 5,</math>
which can be rewritten as
<math display=block>\left({x^3}1 + {x}10\right) - \left({x^2}2 + {x^0}1\right) = {x^0}5,</math>
would be written in Diophantus's notation as

:<math>\Kappa^{\upsilon} \overline{\alpha} \; \zeta \overline{\iota} \;\, \pitchfork \;\, \Delta^{\upsilon} \overline{\beta} \; \Mu \overline{\alpha} \,\;</math>{{lang|grc|ἴ}}<math>\sigma\;\, \Mu \overline{\varepsilon}</math>

{| class="wikitable"
|-
! Symbol
! What it represents
|-
| &nbsp;<math>\overline{\alpha}</math>
| 1 ([[Alpha]] is the 1st letter of the [[Greek alphabet]])
|-
| &nbsp;<math>\overline{\beta}</math>
| 2 ([[Beta]] is the 2nd letter of the Greek alphabet)
|-
| &nbsp;<math>\overline{\varepsilon}</math>
| 5 ([[Epsilon]] is the 5th letter of the Greek alphabet)
|-
| &nbsp;<math>\overline{\iota}</math>
| 10 ([[Iota]] is the 9th letter of the [[History of the Greek alphabet|{{em|modern}} Greek alphabet]] but it was the 10th letter of an [[Archaic Greek alphabets|ancient archaic Greek alphabet]] that had the letter [[digamma]] (uppercase: Ϝ, lowercase: ϝ) in the 6th position between [[epsilon]] ε and [[zeta]] ζ.)
|-
| {{lang|grc|ἴσ}}
| "equals" (short for {{lang|grc|[[wiktionary:ἴσος|ἴσος]]}})
|-
| <math>\pitchfork</math>
| represents the subtraction of everything that follows <math>\pitchfork</math> up to {{lang|grc|ἴσ}}
|-
| <math>\Mu</math>
| the zeroth power (that is, a constant term)
|-
| <math>\zeta</math>
| the unknown quantity (because a number <math>x</math> raised to the first power is just <math>x,</math> this may be thought of as "the first power")
|-
| <math>\Delta^{\upsilon}</math>
| the second power, from Greek {{lang|grc|δύναμις}}, meaning strength or power
|-
| <math>\Kappa^{\upsilon}</math>
| the third power, from Greek {{lang|grc|κύβος}}, meaning a cube
|-
| <math>\Delta^{\upsilon}\Delta</math>
| the fourth power
|-
| <math>\Delta\Kappa^{\upsilon}</math>
| the fifth power
|-
| <math>\Kappa^{\upsilon}\Kappa</math>
| the sixth power
|-
|}

Unlike in modern notation, the coefficients come after the variables and addition is represented by the juxtaposition of terms. A literal symbol-for-symbol translation of Diophantus's equation into a modern equation would be the following:{{sfn|Derbyshire|2006|pp=35-36}}
<math display=block>{x^3}1 {x}10 - {x^2}2 {x^0}1 = {x^0}5</math>
where to clarify, if the modern parentheses and plus are used then the above equation can be rewritten as:{{sfn|Derbyshire|2006|pp=35-36}}
<math display=block>\left({x^3}1 + {x}10\right) - \left({x^2}2 + {x^0}1\right) = {x^0}5</math>