History of algebra

Algebra is a branch of mathematics that deals with solving equations and manipulating symbols, much like arithmetic but with non-numerical objects. Historically, algebra was mainly focused on the theory of equations until the 19th century. This means early mathematicians were primarily concerned with finding solutions to mathematical problems expressed through equations.

The fundamental theorem of algebra, for instance, is a key part of the theory of equations. It states that every polynomial equation has at least one solution, though proving this requires concepts from real numbers that are not strictly algebraic.

This article explores the development of algebra from its beginnings to when it became its own distinct area of mathematics. It traces how our understanding of solving equations evolved over time, shaping the algebraic methods we use today.

Etymology

The word "algebra" comes from an Arabic word, al-jabr, which was used in a famous book written around the year 830 by a Persian mathematician named Al-Khwārizmī. His book, titled The Compendious Book on Calculation by Completion and Balancing, explained ways to solve certain math problems called linear and quadratic equations.

In this book, al-jabr referred to moving numbers from one side of an equation to the other, like putting missing pieces back into place. Another term, muqabalah, meant simplifying the equation by removing the same numbers from both sides. Even today, these ideas are important in algebra. Later, the word appeared in literature, such as in Don Quixote, where it described someone who fixes broken bones, meaning to restore or complete something.

Stages of algebra

Babylon

Ancient Egypt

A portion of the Rhind Papyrus

Greek mathematics

Bloom of Thymaridas

Iamblichus wrote about Thymaridas, who lived around 400–350 BC. Thymaridas worked with equations that had many unknowns. He created a rule called the "bloom of Thymaridas," which helps solve certain sets of equations. This rule shows how to find a particular unknown number when you know the sums of pairs of numbers.

Euclid of Alexandria

Euclid was a famous Greek mathematician who lived in Alexandria, Egypt. He is known as the "father of geometry" because of his book Elements, which has been very important in teaching math. In Elements, Euclid used geometry to show many basic math rules, like how to add and multiply numbers, and he solved many equations using shapes instead of symbols.

Conic sections

A conic section is a curve that comes from cutting a cone with a plane. There are three main types: ellipses (which include circles), parabolas, and hyperbolas. These curves were discovered by Menaechmus around 380–320 BC. By studying these curves, mathematicians could solve more complex equations, like some that are similar to equations with cubed numbers.

China

Diophantus

See also: Diophantine equation and Arithmetica

Cover of the 1621 edition of Diophantus' Arithmetica, translated into Latin by Claude Gaspard Bachet de Méziriac

Diophantus was a mathematician from the Hellenistic period who lived around 250 AD. He wrote a book called Arithmetica, which had thirteen parts, but only the first six have survived today. This book is important because it shows early uses of algebra to solve arithmetic problems. Even though Diophantus did not invent algebra, he used special methods to solve problems that helped shape the future of the subject.

Diophantus was the first to use symbols for unknown numbers and abbreviations for powers and operations. This is called "syncopated algebra." It was different from the algebra we use today because it did not have special symbols for operations like addition or multiplication. Instead, he used lines and letters to represent numbers and their relationships. For example, a problem we might write today as "x³ − 2x² + 10x − 1 = 5" would look very different in his notation. Despite these differences, Diophantus's work was an important step toward the algebraic methods we use now.

Symbol	What it represents
α ¯ {\displaystyle {\overline {\alpha }}}	1
β ¯ {\displaystyle {\overline {\beta }}}	2
ε ¯ {\displaystyle {\overline {\varepsilon }}}	5
ι ¯ {\displaystyle {\overline {\iota }}}	10
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 (i.e. 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

( a 2 + b 2 ) ( c 2 + d 2 ) {\displaystyle (a^{2}+b^{2})(c^{2}+d^{2})}	= ( a c + d b ) 2 + ( b c − a d ) 2 {\displaystyle =(ac+db)^{2}+(bc-ad)^{2}}
	= ( a d + b c ) 2 + ( a c − b d ) 2 {\displaystyle =(ad+bc)^{2}+(ac-bd)^{2}}

India

Aryabhata

Aryabhata (476–550) was a famous Indian mathematician. In his book Aryabhatiya, he described special rules for adding up squares and cubes of numbers.

Brahma Sphuta Siddhanta

Brahmagupta (fl. 628) wrote a book called Brahma Sphuta Siddhanta. In it, he solved quadratic equations and found ways to solve certain types of puzzles with whole numbers. He was the first to find all whole-number answers to a special kind of equation.

Bhāskara II

Bhāskara II (1114 – c. 1185) was a leading mathematician in the 12th century. He solved a challenging type of equation known as Pell's equation. In his books, Lilavati and Vija-Ganita, he used creative symbols to represent unknown numbers in his math problems.

Islamic world

Al-jabr wa'l muqabalah

Pages from a 14th-century Arabic copy of the book, showing geometric solutions to two quadratic equations

The Persian mathematician Al-Khwarizmi is often called the father of algebra. He worked at the "House of Wisdom" in Baghdad. His famous book, Al-jabr wa'l muqabalah, described ways to solve equations with squares and numbers. He introduced ideas like moving terms from one side of an equation to the other, which we now call "balancing." This work gave algebra its name.

Al-Khwarizmi showed how to solve different types of equations, like ones where squares equal roots or numbers. He used pictures to explain his ideas but didn't use letters for numbers like we do today. Instead, he used words. Still, his methods were important steps in developing algebra.

Abu Kamil and al-Karaji

Omar Khayyám

Later mathematicians like Abū Kāmil Shujā ibn Aslam began using square roots and other tricky numbers in their equations. Al-Karaji was another key figure who started using arithmetic operations instead of just geometry in algebra. He looked at equations with powers and helped lay the groundwork for modern algebra.

Omar Khayyám, Sharaf al-Dīn al-Tusi, and al-Kashi

Omar Khayyám wrote about more complex equations, including ones with cubes. He used shapes to solve these, building on earlier work. Sharaf al-Dīn al-Tūsī also worked on cubic equations and developed ways to find solutions. In the 15th century, Jamshīd al-Kāshī worked on new methods for solving equations, including early ideas that would later become part of calculus.

Europe and the Mediterranean region

Theon of Alexandria and his daughter Hypatia were among the last mathematicians in Alexandria during Late Antiquity. When the Western Roman Empire ended, mathematical work slowed down in Europe. Many scholars moved east to places like Persia, where they continued their studies.

Later, in the Late Middle Ages, Fibonacci brought new mathematical ideas from the Islamic world to Europe in his book Liber Abaci. In 1545, Gerolamo Cardano wrote Ars Magna, a book that explained many algebra problems, including solving cubic and quartic equations. During this time, Europe began to grow in its own mathematical discoveries.

Symbolic algebra

Modern notation for arithmetic operations was introduced between the end of the 15th century and the beginning of the 16th century by Johannes Widmann and Michael Stifel. At the end of the 16th century, François Viète introduced symbols, now called variables, for representing unknown numbers. This created a new way of doing algebra by using symbols to stand for numbers.

Another important development was finding general solutions for certain types of equations, especially cubic and quartic equations, in the mid-16th century. The idea of a determinant was introduced by the Japanese mathematician Kowa Seki in the 17th century and later by Gottfried Leibniz, to help solve systems of equations using matrices. Gabriel Cramer also worked on matrices and determinants in the 18th century.

Modern

In the 1700s, mathematicians tried to solve equations with degrees higher than five but couldn't find a general way to do it. Then, in the late 1700s, Carl Friedrich Gauss proved an important idea called the fundamental theorem of algebra, showing that equations of any degree always have solutions.

By the early 1800s, Paolo Ruffini and Niels Henrik Abel showed that there is no general solution for equations of the fifth degree or higher. Around the same time, Évariste Galois created Galois theory, which helped understand these solutions better and started the study of group theory.

In the mid-1800s, algebra began studying more general structures instead of just equations. This led to the creation of new types of algebra like Boolean algebra, vector algebra, and matrix algebra. Later, mathematicians like Alfred North Whitehead and Garrett Birkhoff developed even broader ideas, helping algebra connect to many other areas of math.

Father of algebra

The title of "the father of algebra" is often given to the Persian mathematician Al-Khwarizmi. Some historians also suggest the Hellenistic mathematician Diophantus might deserve this title. Supporters of Al-Khwarizmi say he was the first to explain how to solve certain equations and taught algebra in a simple, clear way. He also introduced important ideas like moving terms from one side of an equation to the other. Others believe that neither should be called the sole "father of algebra," noting that algebra was also used by merchants and surveyors before their time.