Algebra Definition

This branch of mathematics puts real-life variables into equations

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Algebra is a branch of mathematics that substitutes letters for numbers. Algebra is about finding the unknown or putting real-life variables into equations and then solving them. Algebra can include real and complex numbers, matrices, and vectors. An algebraic equation represents a scale where what is done on one side of the scale is also done to the other and numbers act as constants.

The important branch of mathematics dates back centuries, to the Middle East.


Algebra was invented by Abu Ja'far Muhammad ibn Musa al-Khwarizmi, a mathematician, astronomer, and geographer, who was born about 780 in Baghdad. Al-Khwarizmi's treatise on algebra, al-Kitab al-mukhtasar fi hisab al-jabr waʾl-muqabala (“The Compendious Book on Calculation by Completion and Balancing”), which was published about 830, included elements of Greek, Hebrew, and Hindu works that were derived from Babylonian mathematics more than 2000 years earlier.

The term al-jabr in the title led to the word "algebra" when the work was translated into Latin several centuries later. Although it sets forth the basic rules of algebra, the treatise had a practical objective: to teach, as al-Khwarizmi put it:

"...what is easiest and most useful in arithmetic, such as men constantly require in cases of inheritance, legacies, partition, lawsuits, and trade, and in all their dealings with one another, or where the measuring of lands, the digging of canals, geometrical computations, and other objects of various sorts and kinds are concerned."

The work included examples as well as algebraic rules to help the reader with practical applications.

Uses of Algebra

Algebra is widely used in many fields including medicine and accounting, but it can also be useful for everyday problem-solving. Along with developing critical thinking—such as logic, patterns, and deductive and inductive reasoning—understanding the core concepts of algebra can help people better handle complex problems involving numbers.

This can help them in the workplace where real-life scenarios of unknown variables related to expenses and profits require employees to use algebraic equations to determine the missing factors. For example, suppose an employee needed to determine how many boxes of detergent he started the day with if he sold 37 but still had 13 remaining. The algebraic equation for this problem would be:

  • x – 37 = 13

where the number of boxes of detergent he started with is represented by x, the unknown he is trying to solve. Algebra seeks to find the unknown and to find it here, the employee would manipulate the scale of the equation to isolate x on one side by adding 37 to both sides:

  • x – 37 + 37 = 13 + 37
  • x = 50

So, the employee started the day with 50 boxes of detergent if he had 13 remaining after selling 37 of them.

Types of Algebra

There are numerous branches of algebra, but these are generally considered the most important:

Elementary: a branch of algebra that deals with the general properties of numbers and the relations between them

Abstract: deals with abstract algebraic structures rather than the usual number systems 

Linear: focuses on linear equations such as linear functions and their representations through matrices and vector spaces

Boolean: used to analyze and simplify digital (logic) circuits, says Tutorials Point. It uses only binary numbers, such as 0 and 1.

Commutative: studies commutative rings—rings in which multiplication operations are commutative.

Computer: studies and develops algorithms and software for manipulating mathematical expressions and objects

Homological: used to prove nonconstructive existence theorems in algebra, says the text, "An Introduction to Homological Algebra"

Universal: studies common properties of all algebraic structures, including groups, rings, fields, and lattices, notes Wolfram Mathworld

Relational: a procedural query language, which takes a relation as input and generates a relation as output, says Geeks for Geeks

Algebraic number theory: a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations

Algebraic geometry: studies zeros of multivariate polynomials, algebraic expressions that include real numbers and variables

Algebraic combinatorics: studies finite or discrete structures, such as networks, polyhedra, codes, or algorithms, notes Duke University's Department of Mathematics.