Diophantus

Diophantus of Alexandria was a Greek mathematician who lived around 250 CE. He wrote a famous book called Arithmetica. This book has thirteen parts, and ten of them still exist today.

In his book, Diophantus solved problems using algebraic equations. This method became very important in mathematics.

Many great mathematicians praised Diophantus for his work. They called him "the inventor of algebra." His ideas influenced math for many years. His book was translated into Arabic.

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, named after his contributions. Problems from his book have 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 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.

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, still exists but is incomplete. It talks about 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 thought to be by Hero of Alexandria, might actually be by Diophantus. The Porisms included helpful math ideas that Diophantus used in his other work. On Parts seems to have looked at fractions, or parts of whole numbers, based on a single reference in another ancient text.

Influence

Diophantus' work has been very important in history. His book Arithmetica helped create the basis for algebra and advanced mathematics. Diophantus and his work influenced mathematics in the medieval Islamic world. Editions of Arithmetica helped shape algebra in Europe from the late sixteenth through the eighteenth centuries.

The Latin translation of Arithmetica by Bachet became well known in 1621. Pierre de Fermat studied it and wrote notes in the margins. He included his famous "Last Theorem". This theorem was not solved for many years until Andrew Wiles proved it in 1994. Diophantus was one of the first to use positive rational numbers as numbers, allowing fractions in his answers.