What Is Algebra?

Algebra is a branch of mathematics that uses letters and other symbols to represent numbers.

Dictionary Definition

The Oxford English Dictionary defines algebra as "the part of mathematics in which letters and other general symbols are used to represent numbers and quantities in formulae and equations."
Here is an example of an algebraic equation. In this example, the letter x represents a number. Our job is find the value of x.

algebra 2

Equations ("Tetris" Game)

Here is an interactive game to help you learn about equations.

Tip: If the game is playing slowly, you can use the controls below to remove the background text and image to speed up the game.

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Where Does the Word "Algebra" Come From?

Algebra comes from the Arabic word al-jebr, which means "reunion of broken parts".

Albert Einstein and Algebra

Albert_Einstein The famous scientist Albert Einstein learned algebra from a young age. His Uncle Jakob gave him books on the subject and called algebra "a merry science". He compared algebra to hunting a little animal. You didn't know the name of the animal, so you called it "x". When you finally caught the animal you gave it the correct name.

The Curriculum

The lessons are grouped into mini-curriculum to help you organise your learning. A brief description is given for each mini-curriculum. Click the MORE button to learn more.

Algebra Definitions

algebra definitions

There are many different words used in algebra: such as equation, variable, coefficient, term. Understanding algebra is easier when you understand what each of these words mean. In this mini-curriculum, you will learn all the definitions.

Equation
An equation has an equals sign (=) and tells us two values are equal. Here is an example of an equation:
algebra equation example

Constant
A constant have a fixed value. In the equation below, the 2, 1, and 9 are constants.
algebra constants example

Variable
A variable is a symbol that stands for a number. Its value is not fixed - it can take any value. In the equation below, the x is a variable.
algebra variables example

Coefficient
A coefficient is a number that is placed in front of a letter. The coefficient is multiplying the variable. In the equation below, the 2 is a coefficient.
algebra coefficients example

Exponent
An exponent tells you how many times a number (or symbol) is multiplied by itself. It is written to the right and above the number. In the equation below, the 2 is an exponent.
algebra exponents example

Operator
An operator defines an operation, such as +, , ×, and ÷. In the equation below, the + is an operator.
algebra operators example

Term
A term is a collection of one or more numbers, letters and brackets written next to each other (they are multiplied together). Terms are separated by + or operators. In the equation below, the 2x2, 1 and 9 are terms.
algebra terms example

Expression
An expression is one term or several terms that are added (+) or subtracted (−) together. In the equation below, the 2x + 1 and 9 are expressions. They are separated by the = sign.
algebra expressions example

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Arithmetic with Letters

algebra arithmetic letters

Algebra uses letters and other symbols to represent numbers. Just like numbers, the letters and symbols can be added, subtracted, multiplied and divided. In this mini-curriculum, you will learn arithmetic with letters.

Adding Letters
Adding letters works when the letter is added to numbers, other letters and the same letter.
algebra add letters

Subtracting Letters
Subtracting letters works when the letter is subtracted by numbers, other letters and the same letter.
algebra subtract letters

Multiplying Letters
Multiplying letters works when the letter is multiplied by numbers, other letters and the same letter.
algebra multiply letters

Dividing Letters
Dividing letters works when the letter is divided by numbers, other letters and the same letter.
algebra divide letters

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how to add letters
how to subtract letters
how to multiply letters
how to divide letters

Arithmetic with Terms

algebra arithmetic terms

Terms are a collection of one or more numbers, letters and brackets written next to each other. Terms can be added, subtracted, multiplied and divided. In this mini-curriculum, you will learn arithmetic with terms.

Adding Terms
Terms can be added.
algebra add terms

Subtracting Terms
Terms can be subtracted.
algebra subtract terms

Multiplying Terms
Terms can be multiplied. The coefficients are multiplied, and each letter must appear in the result as many times as they appear in the multiplication.
algebra multiply terms

Dividing Terms
Terms can be divided.
algebra divide terms

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how to add terms
how to subtract terms
how to multiply terms
how to divide terms

Laws of Exponents in Algebra

algebra powers

An exponent tells you how many times a number, letter or term is multiplied by itself. In this mini-curriculum, you will learn how to use the law of exponents in algebra.

Laws of Exponents
The laws of exponents in algebra let us understand how a letter can be multiplied by itself.
algebra exponent notation

Exponent of −1
A letter with an exponent of −1 is equal to 1 divided by the letter.
algebra exponent_-1

Multiplying Powers
When the same letters with different exponents are multiplied together, the exponents can be added.
algebra multiply exponent

Dividing Powers
When the same letters with different exponents are divided, the exponents can be subtracted.
algebra divide exponent

Powers of Powers
A power of a power is where a letter with an exponent is itself raised to an exponent. When a letter with an exponent is raised to another exponent, the exponents can be multiplied.
algebra raise exponent

Powers of Products
A power of a product is where a term containing letters multiplied together is raised to an exponent. Each letter in the term is raised to the exponent.
algebra product exponent

Negative Exponent
A letter with a negative exponent is equal to 1 divided by the letter with the exponent made positive.
algebra negative exponent

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the laws of exponents
an exponent of −1
how to multiply powers
how to divide powers
how to find a power of a power
how to find a power of a product
how to find a negative exponent

Like Terms

algebra like terms

Like terms are terms which have the same combination of letters, each with the same exponent. The only difference allowed between terms that are like terms are the numbers and signs in front of them. In this mini-curriculum, you will learn how to find like terms and how to collect like terms to simplify expressions.

Like Terms
Like terms have the same letters with the same exponents. x, 2x and −x are like terms, they all contain x; only the number or sign in front of them is different. a2b, 5a2b and 12a2b are like terms, they all contain a2b; only the number or sign in front of them is different.
algebra like terms example

Identifying Like Terms
Identify like terms in an expression by finding terms with the same letters and exponents.
algebra collect like terms 1 example

Adding Like Terms
Like terms can be added. Add the coefficients together.
algebra add like terms

Subtracting Like Terms
Like terms can be subtracted. Subtract the coefficients from each other.
algebra subtract like terms

Collecting Like Terms
Collect like terms by adding and subtracting the like terms you have identified. This simplifies the expression.
algebra collect like terms 2 example

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like terms
identifying like terms
adding like terms
subtracting like terms
collecting like terms

Expanding Brackets

algebra expand brackets

Expanding brackets means multiplying terms inside of a pair of brackets in order to remove the brackets from an expression. In this mini-curriculum, you will learn how to expand brackets.

Expanding Brackets
Expanding a single pair of brackets means multiplying the term outside of the brackets with each term inside the brackets. This removes the brackets.
algebra expand single brackets

Expanding Brackets with the Grid Method
Brackets can be expanded using the grid method. This is a useful method if you prefer using pictures to understand a method.
algebra expand single brackets grid method

Expanding Double Brackets
Expanding double brackets means multiplying the term in two pairs of brackets that are written next to each other. This removes the brackets.
algebra expand double brackets

The FOIL Method
FOIL stands for Firsts, Outsides, Insides, Lasts. It helps you remember an order to multiply terms when expanding two pairs of brackets.
algebra FOIL method

Multiplying Expressions
Multiplying expressions means writing each expression in a pair of brackets and expanding them.
algebra multiply expressions

Expanding Double Brackets with the Grid Method
Double brackets can be expanded using the grid method.
algebra expand double brackets grid method

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expanding brackets
expanding single brackets using the grid method
expanding double brackets
the FOIL method
multiplying expressions in algebra
expanding double brackets using the grid method

Solving Equations with Brackets

algebra about solve equations with brackets

Some equations in algebra contain brackets. In this mini-curriculum, you will learn about how to solve equations with brackets.

Solve Equations with Brackets
To solve equations with brackets, expand the brackets and rearrange the equation to find what x equals.
algebra solve equations with brackets

Solve Equations with Two Sets of Brackets
To solve equations with two sets of brackets, expand both brackets and rearrange the equation to find what x equals.
algebra solve equations with two sets of brackets

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how to solve an equation with brackets
how to solve an equation with two sets of brackets

Factoring

algebra factoring

Factoring is the opposite of expanding brackets. It is a way of simplifying an expression. In this mini-curriculum, you will learn about factors in algebra, the greatest common factor and how to factor expressions.

Factors
A factor is one of the numbers, letters and brackets (or a product of them) that are multiplied together to make a term.
algebra factors

Finding Factors
You can find factors of terms by looking at the numbers, letters and brackets (with any exponents) that appear in the term. The factors of 2x are 1, 2, x and 2x.
algebra find factors

Greatest Common Factor
The greatest common factor is the largest factor that is common to two or more terms. The greatest common factor of 2xy and 2xz is 2x.
algebra greatest common factor

Finding the Greatest Common Factor
Find the greatest common factor by looking at the numbers, letters and brackets that appear in two or more terms.
algebra find greatest common factor

Factoring Expressions
Factoring simplifies an expression by writing it as a product of factors. It lets you write an expression where terms are added together to one where they are multiplied together to make a single term.
algebra factor expression

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factors in algebra
how to find factors in algebra
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how to find the greatest common factor in algebra
factoring an expression

Algebraic Fractions

algebra algebraic fractions

Algebraic fractions are fractions which contain letters and other symbols. In this mini-curriculum, you will learn about algebraic fractions, including arithmetic and simplifying.

Algebraic Fractions
An algebraic fraction is a fraction which contains letters and other symbols.
algebra algebraic fraction

Adding Algebraic Fractions
Algebraic fractions can be added.
algebra add algebraic fractions

Subtracting Algebraic Fractions
Algebraic fractions can be subtracted.
algebra subtract algebraic fractions

Multiplying Algebraic Fractions
Algebraic fractions can be multiplied. algebra multiply algebraic fractions

Dividing Algebraic Fractions
Algebraic fractions can be divided.
algebra divide algebraic fractions

Simplifying Algebraic Fractions
Algebraic fractions can be simplified by cancelling out terms that appear on the top and bottom of the algebraic fraction.
algebra simplify algebraic fractions

Powers of an Algebraic Fraction
An algebraic fraction can be raised to an exponent. The top and bottom of the fraction are both raised to the same exponent.
algebra power of algebraic fraction

Reciprocal of an Algebraic Fraction
The reciprocal of an algebraic fraction is found by turning the algebraic fraction upside down.
algebra reciprocal of algebraic fraction

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algebraic fractions
adding algebraic fractions
subtracting algebraic fractions
multiplying algebraic fractions
dividing algebraic fractions
simplifying algebraic fractions
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how to find the reciprocal of an algebraic fraction