SafekipediaDiophantus
Adapted from Wikipedia · Discoverer experience
Diophantus of Alexandria was a Greek mathematician who lived around 250 CE. He wrote a famous book called Arithmetica, which has thirteen parts, and ten of them still exist today. In this book, Diophantus solved problems using algebraic equations, a method that later became very important in mathematics.
Many great mathematicians, like Joseph-Louis Lagrange, praised Diophantus for his work, calling him "the inventor of algebra." His ideas influenced math for many years, especially after his book was translated into Arabic. One interesting story is that Pierre de Fermat, a famous mathematician, wrote his challenging "Last Theorem" in the margins of a copy of Diophantus's book.
Today, we still use Diophantus's name in many areas of math. Equations where we look for whole number solutions are called Diophantine equations. There are also areas of math called Diophantine geometry and Diophantine approximations, all named after his important contributions. Problems from his book have even inspired modern research in abstract algebra and number theory.
Biography
We do not know many details about Diophantus' life. He was a Greek mathematician who lived a long time ago, possibly between 170 BCE and 350 CE. Some guesses place him in the 3rd century CE.
One fun puzzle from a long time later tries to tell us how old Diophantus was when he died. It says he lived to be 84 years old, but we cannot be sure if this is true. The puzzle describes different parts of his life, like his childhood, youth, marriage, and having a son.
Arithmetica
Arithmetica is the most important work by Diophantus and a key book in the history of algebra. It has 290 problems where readers solve equations to find numbers that work. Originally, there were thirteen books, but today only six survive in Greek and four in Arabic.
Diophantus used early algebra methods to solve arithmetic problems. He introduced special symbols to make solving easier, though his symbols were different from the ones we use today. His work includes solving equations that we now call Diophantine equations. Many of these problems lead to what we know as quadratic equations.
| Symbol | What it represents |
|---|---|
| α ¯ {\displaystyle {\overline {\alpha }}} | 1 (Alpha is the 1st letter of the Greek alphabet) |
| β ¯ {\displaystyle {\overline {\beta }}} | 2 (Beta is the 2nd letter of the Greek alphabet) |
| ε ¯ {\displaystyle {\overline {\varepsilon }}} | 5 (Epsilon is the 5th letter of the Greek alphabet) |
| ι ¯ {\displaystyle {\overline {\iota }}} | 10 (Iota is the 9th letter of the modern Greek alphabet but it was the 10th letter of an ancient archaic Greek alphabet that had the letter digamma (uppercase: Ϝ, lowercase: ϝ) in the 6th position between epsilon ε and zeta ζ.) |
| ἴσ | "equals" (short for ἴσος) |
| ⋔ {\displaystyle \pitchfork } | represents the subtraction of everything that follows ⋔ {\displaystyle \pitchfork } up to ἴσ |
| M {\displaystyle \mathrm {M} } | the zeroth power (that is, a constant term) |
| ζ {\displaystyle \zeta } | the unknown quantity (because a number x {\displaystyle x} raised to the first power is just x , {\displaystyle x,} this may be thought of as "the first power") |
| Δ υ {\displaystyle \Delta ^{\upsilon }} | the second power, from Greek δύναμις, meaning strength or power |
| K υ {\displaystyle \mathrm {K} ^{\upsilon }} | the third power, from Greek κύβος, meaning a cube |
| Δ υ Δ {\displaystyle \Delta ^{\upsilon }\Delta } | the fourth power |
| Δ K υ {\displaystyle \Delta \mathrm {K} ^{\upsilon }} | the fifth power |
| K υ K {\displaystyle \mathrm {K} ^{\upsilon }\mathrm {K} } | the sixth power |
Other works
Diophantus wrote several works besides his famous Arithmetica. One of these, On Polygonal Numbers, survives in an incomplete form and discusses numbers that can be arranged into shapes like triangles and squares. Two other works, Porisms and On Parts, are lost but mentioned in other writings.
Recent research suggests that a book called Preliminaries to the Geometric Elements, usually attributed to Hero of Alexandria, might actually be by Diophantus. The Porisms included mathematical lemmas, or helper statements, that Diophantus used in his other work. On Parts seems to have explored fractions, or parts of whole numbers, based on a single reference in another ancient text.
Influence
Diophantus' work has had a large influence in history. Although Joseph-Louis Lagrange called Diophantus "the inventor of algebra", his work Arithmetica created a foundation for algebra and much of advanced mathematics. Diophantus and his works influenced mathematics in the medieval Islamic world, and editions of Arithmetica helped shape the development of algebra in Europe from the late sixteenth through the eighteenth centuries.
The Latin translation of Arithmetica by Bachet in 1621 became widely available. Pierre de Fermat studied it and wrote notes in the margins, including his famous "Last Theorem". This theorem remained unsolved for centuries until Andrew Wiles found a proof in 1994. Diophantus was among the first to use positive rational numbers as numbers, allowing fractions in his solutions.
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