Diophantine Algebra
Diophantus was a Hellenistic mathematician who lived c. 250 CE, but the uncertainty of this date is so great that it may be off by more than a century. He is known for having written Arithmetica, a treatise that was originally thirteen books but of which only the first six have survived. Arithmetica has very little in common with traditional Greek mathematics since it is divorced from geometric methods, and it is different from Babylonian mathematics in that Diophantus is concerned primarily with exact solutions, both determinate and indeterminate, instead of simple approximations.
In Arithmetica, Diophantus is the first to use symbols for unknown numbers as well as abbreviations for powers of numbers, relationships, and operations; thus he used what is now known as syncopated algebra. The main difference between Diophantine syncopated algebra and modern algebraic notation is that the former lacked special symbols for operations, relations, and exponentials. So, for example, what we would write as
Diophantus would have written this as
- ΚΥ α̅ς ι̅ ⫛ ΔΥ β̅ Μ α̅ ἴσ Μ ε̅
where the symbols represent the following:
| Symbol | Representation |
|---|---|
| α̅ | represents 1 |
| β̅ | represents 2 |
| ε̅ | represents 5 |
| ι̅ | represents 10 |
| ς | represents the unknown quantity (i.e. the variable) |
| ἴσ | (short for ἴσος) represents "equals" |
| ⫛ | represents the subtraction of everything that follows it up to ἴσ |
| Μ | represents the zeroth power of the variable (i.e. a constant term) |
| ΔΥ | represents the second power of the variable, from Greek δύναμις, meaning strength or power |
| ΚΥ | represents the third power of the variable, from Greek κύβος, meaning a cube |
| ΔΥΔ | represents the fourth power of the variable |
| ΔΚΥ | represents the fifth power of the variable |
| ΚΥΚ | represents the sixth power of the variable |
Note that the coefficients come after the variables and that addition is represented by the juxtaposition of terms. A literal symbol-for-symbol translation of Diophantus's syncopated equation into a modern symbolic equation would be the following:
and, to clarify, if the modern parentheses and plus are used then the above equation can be rewritten as:
Arithmetica is a collection of some 150 solved problems with specific numbers and there is no postulational development nor is a general method explicitly explained, although generality of method may have been intended and there is no attempt to find all of the solutions to the equations. Arithmetica does contain solved problems involving several unknown quantities, which are solved, if possible, by expressing the unknown quantities in terms of only one of them. Arithmetica also makes use of the identities:
Read more about this topic: History Of Elementary Algebra
Famous quotes containing the word algebra:
“Poetry has become the higher algebra of metaphors.”
—José Ortega Y Gasset (1883–1955)