History of algebra

Algebra is a part of mathematics where we solve problems using symbols. It is like arithmetic, but instead of using numbers, we use letters and other symbols.

Historically, people used algebra mostly to solve equations. This means they wanted to find answers to math problems written in a special way.

The fundamental theorem of algebra is an important idea. It says that every equation with powers of a number has at least one solution. Learning about this idea needs knowledge of real numbers, which is a bit more than just algebra.

This article looks at how algebra started and grew into its own special area of math. It shows how our ways of solving equations changed over time and how these changes created the algebra we use today.

Etymology

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

In this book, al-jabr meant moving numbers from one side of an equation to the other. Another term, muqabalah, meant simplifying the equation by taking the same numbers away from both sides. These ideas are still important in algebra today. Later, the word appeared in stories like Don Quixote, where it described someone who fixes broken bones.

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 unknown numbers. He made a rule called the "bloom of Thymaridas" to help solve certain groups of equations. This rule shows how to find one unknown number when you know the totals of pairs of numbers.

Euclid of Alexandria

Euclid was a famous Greek math expert who lived in Alexandria, Egypt. He is called the "father of geometry" because of his book Elements. This book has been very important for teaching math. In Elements, Euclid used shapes to show many basic math ideas, like adding and multiplying numbers, and he solved many problems using shapes instead of letters.

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 found by Menaechmus around 380–320 BC. By studying these curves, math experts could solve harder problems, like some that are like 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 a long time ago who lived around 250 AD. He wrote a book called Arithmetica. This book had thirteen parts, but only six of them still exist today. This book is important because it shows early uses of algebra to solve math problems.

Diophantus was the first to use symbols for unknown numbers and special short ways to show powers and operations. This is called "syncopated algebra." It was different from algebra today because it did not have symbols for addition or multiplication. Instead, he used lines and letters to show numbers and their relationships. Even though his methods were different, Diophantus's work helped shape algebra as we know it 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 wrote about rules for adding 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 figured out ways to solve 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 tricky equation called Pell's equation. In his books, Lilavati and Vija-Ganita, he used special symbols to stand for 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. Still, his methods were important steps in developing algebra.

Abu Kamil and al-Karaji

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 important mathematicians in Alexandria during Late Antiquity. When the Western Roman Empire ended, work on math slowed down in Europe. Many scholars went east to places like Persia, where they kept studying.

Later, in the Late Middle Ages, Fibonacci shared new math ideas from the Islamic world with Europe in his book Liber Abaci. In 1545, Gerolamo Cardano wrote Ars Magna, a book that helped solve many algebra problems, including cubic and quartic equations. During this time, Europe started making its own discoveries in math.

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 wanted to solve equations that were very hard, but they could not find a way for all of them. Then, in the late 1700s, Carl Friedrich Gauss proved a big idea called the fundamental theorem of algebra. This idea showed that equations always have answers.

By the early 1800s, Paolo Ruffini and Niels Henrik Abel showed that some very hard equations cannot be solved in a simple way. Around this time, Évariste Galois created Galois theory. This helped people understand solutions better and started the study of group theory.

In the mid-1800s, algebra started looking at more general ideas instead of just solving equations. This led to new types of algebra like Boolean algebra, vector algebra, and matrix algebra. Later, mathematicians like Alfred North Whitehead and Garrett Birkhoff made algebra connect to many other parts 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." They note that algebra was also used by merchants and surveyors before their time.