# Diophantus: “Father of Algebra” Influenced Rebirth of Number Theory

By Jen Breitegan

Diophantus was a Hellenistic Greek mathematician who lived in Alexandria, Egypt from ca. 201-285.

Diophantus is often referred to as the “father of algebra.” He is considered most famous for his series of books entitled Arithmetica, where he was the first mathematician to present algebra in a form we would recognize today.

Diophantus’s use of symbols for variables, positive and negative numbers, and fractions was ahead of its time. Many of the concepts he presented in Arithmetica are still used in modern mathematics. His ideas blazed a trail for future mathematicians’ big discoveries, even centuries later.

Read on to learn more about Diophantus’s ability to express and solve complex math problems—and why his concepts are still used in modern mathematics!

## Diophantus (ca. 201-285), Hellenistic Greek Mathematician

Diophantus lived in Alexandria, Egypt, but was considered a Greek mathematician. Why?

Alexander the Great conquered much of the Mediterranean region, particularly the area around Alexandria, in the 4th century BC. As a result of these conquests, Greek culture, ideas and language spread to people living far outside Greece and remained influential for centuries.

Little is known about the life of Diophantus. It is believed that he was married at age 33, died at age 84, and had a son who died a few years before him. Interestingly, this information was gleaned from an epitaph written as a word puzzle in the late fifth century. The 1941 translation by Ivor Thomas reads:

This tomb holds Diophantus. Ah, what a marvel.

And the tomb tells scientifically the measure of his life.

God vouchsafed that he should be a boy for the sixth part of his life;

When a twelfth was added, his cheeks acquired a beard;

He kindled for him the light of marriage after a seventh,

And in the fifth year after his marriage He granted him a son.

Alas! late-begotten and miserable child, when he had reached the measure of half his father’s life, the chill grave took him.

After consoling his grief by this science of numbers for four years, he reached the end of his life.

The puzzle, when converted into an algebraic expression, appears to reveal Diophantus’s age at certain points of his life including marriage, the birth of his son, his son’s death and his own death.

x=x6 + x12+x7+5+x2+4

The solution to the problem is x = 84. It can then be deduced that Diophantus:

• Was a boy for 14 years
• Grew a beard when he was 21
• Got married at 33
• Was 38 when his son was born
• Was 80 when his son died
• Died at age 84

While the epitaph may or may not be true, it is certainly intriguing.

What we do know for sure about Diophantus is the legacy of mathematical writings he left behind.

## The Significance of Diophantus’s Arithmetica Throughout History

Cover of Diophantus’s Arithmetica, via Wikimedia

Diophantus dedicated Arithmetica to St. Dionysius, the bishop of Alexandria. Diophantus wrote that he wanted to help Dionysius solve mathematical problems with the concepts he presented.

Arithmetica is considered the first set of writings to discuss algebra in a way we recognize today. It addresses expressing and solving equations to find one or more unknowns. Diophantus used symbols for the unknown (like “x”), something that had not been seen before in mathematical writings.

In fact, Diophantus was so ahead of his fellow mathematicians that this type of algebraic symbolism would not be seen again until the 15th century.

He introduced the idea that math problems like x + y = 7 could have many solutions (for instance: x = 2, y = 5 and x = 1, y = 6).

Diophantus also introduced the world to positive and negative numbers and wrote about raising numbers to a higher power.

Over a thousand years after Arithmetica was written, its concepts influenced notable mathematicians like Pierre de Fermat, a French scholar and judge from the 17th century.

Fermat is considered the founder of modern number theory. He was largely inspired by Diophantus’s work and wrote Fermat’s Last Theorem in the margin of a 1621 French translation of Arithmetica

xn + yn = zn (where n is greater than or equal to 3) has no integer solutions

Fermat noted he didn’t have enough space in the margin of his book to write the proof. (Centuries later, British mathematician Andrew Wiles published a proof of Fermat’s theorem—in 1995.)

In the 1700s, renowned Swiss mathematician Leonhard Euler found enjoyment from tackling Arithmetica’s more challenging problems.

Euler wrote in 1761 that Diophantus’s third-century problem-solving methods were still commonly used in the 18th century.

Another famous 18th-century mathematician, Joseph Lagrange, proved a postulation in Arithmetica that every number can be written as the sum of four squares. No one else had been able to produce proof before that time.

This was just one more example of the many ways Diophantus was ahead of his time, demonstrating why he’s still known as the “father of algebra.”

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