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If you have ever written "let x be the number" and turned a word problem into an equation, you are using a habit that took centuries to form. One of the first people to write algebra problems in a compact, half-symbolic style was Diophantus of Alexandria. This post explains what we know about him, works three of his problems in modern notation, and shows the mistakes students make when they try them.
Who was Diophantus?
Almost nothing is known about his life. The MacTutor biography from the University of St Andrews explains how dates are pinned down. He quotes Hypsicles, so he wrote after about 150 BC. Theon of Alexandria, Hypatia's father, quotes him, so he wrote before about 350 AD. A later argument put him around 250 AD, but historians dispute it. He probably worked in Alexandria.
The best-known "fact" about his life comes from a puzzle in the Greek Anthology, compiled around 500 AD, and MacTutor warns it may be invented. It says his boyhood lasted a sixth of his life, he married after a further seventh, his beard grew after a further twelfth, his son was born five years later, the son lived half as long as his father, and Diophantus died four years after his son.
What is in the Arithmetica?
MacTutor describes the Arithmetica as a collection of 130 problems with numerical solutions to determinate and indeterminate equations. Of 13 original books, six survive in Greek. In 1968 an Arabic manuscript in Meshed, Iran, was found that claims to be a translation of Books IV to VII; scholars still debate whether it is Diophantus or a commentator.
Two features make the book unusual. First, Diophantus accepted only positive rational answers. He called the equation 4 = 4x + 20 "absurd", because x would be −4. Second, he used an abbreviation for the unknown and its powers, and even for "equals". MacTutor quotes the historian Kurt Vogel calling this a fundamental step from verbal algebra towards symbolic algebra. He had only one symbol for an unknown, though, so problems with two unknowns had to be rewritten in terms of one before he started.
Three problems, worked in full
1. The epitaph puzzle. Let his age be x years. Then
x/6 + x/12 + x/7 + 5 + x/2 + 4 = x
Multiply through by 84, the lowest common multiple of 6, 12, 7 and 2: 14x + 7x + 12x + 420 + 42x + 336 = 84x. So 75x + 756 = 84x, which gives 9x = 756 and x = 84. He married at 14 + 12 = 26, and his son lived 42 years.
2. Two numbers with sum 10 and product 9. This is MacTutor's example of his style. Rather than use two unknowns, write the numbers as 5 + x and 5 − x, so their sum is already 10. Then (5 + x)(5 − x) = 25 − x² = 9, so x² = 16 and x = 4. The numbers are 9 and 1. Choosing the right single unknown did all the work.
3. Divide 16 into two squares (Book II, problem 8). In Thomas Heath's 1910 translation, Diophantus lets the first square be x², so the second is 16 − x². He then says the second square should have a side of the form 2x − 4. So
(2x − 4)² = 16 − x² 4x² − 16x + 16 = 16 − x² 5x² = 16x, so x = 16/5.
The squares are (16/5)² = 256/25 and 16 − 256/25 = 144/25 = (12/5)². Check: 256/25 + 144/25 = 400/25 = 16.
Which mistakes should you avoid on Diophantine problems?
Dropping x = 0. In problem 3, 5x² = 16x also has x = 0. Diophantus ignored it because it gives no useful split; you should state why you reject it rather than lose it silently.
Expecting whole numbers. His answers are fractions like 16/5. Do not abandon a correct method because the answer is not an integer.
Using two unknowns too early. Writing 5 + x and 5 − x is often faster than solving a pair of equations.
Why does Fermat appear in this story?
Claude Gaspard Bachet published a Greek and Latin edition of the Arithmetica in 1621. Pierre de Fermat read it and wrote notes in the margins. Heath records that next to problem 8 of Book II, Fermat claimed that no cube can be split into two cubes, nor any higher power into two like powers, adding, in Heath's translation, "I have discovered a truly marvellous proof of this, which however the margin is not large enough to contain." Heath notes that Fermat's son published these notes with a reprint of Bachet's edition in 1670. The claim was finally proved in the 1990s, a story told in our post on Andrew Wiles and Fermat's Last Theorem.
The Arithmetica also shaped the Renaissance. Read how Rafael Bombelli borrowed many of its problems for his own algebra book.
What can a student take from Diophantus?
He shows that algebra is mostly about choosing the unknown well. Before you expand anything, ask which substitution will make the equation collapse. When you are stuck on that choice, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning rather than printing an answer.
Try one now: using Diophantus's method with a side of the form 3x − 5, split 25 into two rational squares and check your answer adds back to 25.
