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# Terminology Used in Algebra

by Ron Kurtus (revised 18 January 2022)

You should know and understand the * terminology used in algebra*, which are variable, constant, operation, expression and equation.

A variable is also called an unknown and can be represented by letters from the alphabet. Operations in algebra are the same as in arithmetic: addition, subtraction, multiplication and division. An expression is a group of numbers and variables, along with operations. An equation is the equality of two expressions.

Questions you may have include:

- What are variables and constants?
- What are algebraic operations?
- What are expressions and equations?

This lesson will answer those questions.

## Variables and constants

Algebra started off as the addition and subtraction of similar objects. For example, you could add three apples plus two apples to equal five apples. If you substituted the letter "**a**" for apples, you would have **3 a + 2 a = 5 a** in place of **3** **apples** plus **2** **apples** equals **5** **apples**.

Thus, a letter such as "**a**" could be used for any object. In fact, it could even represent another number or even group of numbers and objects.

### Variables

The basis of algebra is that the various mathematical operations can be applied for no matter what you have for "**a**", "**b**", or "**x**".

These letters are called **variables**, because they can vary and be almost anything. They are also often called **unknowns**.

### Constants

In some situations a quantity may be known, but it is convenient to designate it as a letter. In such a case, the letter represents a **constant**.

In Einstein's famous equation **E = mc ^{2}**, the letter

**c**represents the speed of light and is thus a constant. It is easier to use

**c**than to write out the speed of light as 186,000 miles per second (or 300,000 km per second).

In some cases, a constant can be an unknown, but it does not necessarily vary.

### Common convention

Although any alphabetical letter can be a variable, a common convention used is to designate letters toward the end of the alphabet to be variables and letters to the front as constants.

Thus, variables are usually represented by the letters **x**, **y** or **z** and constants are **a**, **b** or **c**.

In Einstein's equation, **c** is a constant, but **m** is a variable because it represents the unknown **m**ass of an object (thus the reason to use **m** instead of some other letter).

## Operations

Mathematical operations are addition, subtraction, multiplication and division. Their common symbols are:

- Addition
**+** - Subtraction
**−** - Multiplication
**×** - Division
**÷**

### Addition

The following are designated by **x + y**:

Add

xandyFind the sum of

xandyIncrease

xbyy

### Subtraction

The following are all designated by **x − y**:

Subtract

xfromyFind the difference between

xandy

xis decreased byy

xtake awayy

xminusy

xlessy

The following are designated by **y − x**:

xless thany

xfromy

### Multiplication and division have some variations.

### Multiplication sign

Since the letter "**x**" is often use as a variable or unknown in algebra, this can cause confusion with the multiplication sign **×**. Thus, an asterisk (*****) or a dot (**·**) is often used to indicate multiplication instead of **×**. Instead of **x × y**, multiplication is designated as **x*y** or **x·y**.

But another problem then pops up. If you use the dot when dealing with numbers, it may be confused with a decimal point. For example, writing **2** times **5 ** as **2·5** might be confused with **2½** or **2.5**. With numbers, it is better to use *****, as in **2*5**, or even the **×** sign.

### Using no multiplication sign

Mathematicians decided to completely drop the multiplication sign altogether, when multiplying variables or constantsr. Instead of writing **x** times **y** times **z** as **x*y*z**, it is usually written as **xyz** in algebra.

#### Numbers first

When including numbers in the multiplication, the number is written first and no multiplication sign is used. Thus, **x** times **3** is written as **3x**, and **x** times **2** times **a** is written as **2ax**.

#### Multiplying numbers together

But if you are multiplying two or more numbers together, you **must include** a multiplication sign. **2** times **5** times **3** is **not** written as **253** but as **2*5*3**.

If there is a letter involved, try to make it as clear as possible. **2*5x** or **2*5*x** are both acceptable.

### Division sign

The slash (**/**) is also used to denote division. Thus, **a ÷ b** and **a/b** both mean "**a** divided by **b**."

** a ****
—
b ** also can denote the division of

**a**by

**b**or a fraction. Since it is difficult to write on a web page, it is seldom used in websites.

### Other algebraic operations

Other algebraic operations, such as square root and exponent will be explained in other lessons.

## Expressions and equations

Expressions lead into equations.

### Expressions

An expression is any group or collection of algebraic numbers and variables, including mathematical operations such as addition, division, etc. Examples of expressions include:

3 + 5

2a + 3x − 6/7

5abc

Some expressions may be long and complex, even including parentheses:

3x + (2z − y)/x + 125y − (x + y)/(z +2)

### Terms

An expression consists of one or more terms that are separated by an addition or subtraction operation. The expression **2a + 3x − 6/7** consists of the terms (separated by commas):

2a, 3x, − 6/7

### Equations

An equation consists of expressions separated by an equal sign. The assumption is that the expressions on the left side of the equal sign are equal to those on the right side.

3 + 5 = 8

5x − 3y = 4z

27 + x = 17/3x

Since some of these equations contain unknowns or variables, they require a solution to verify the equation is valid.

3x − 7 = 2is valid with whenx = 3.

Some equations are equalities when the values are known or the solution is trivial. **3 + 5 = 8** and **x = 3** are considered equalities.

## Summary

Major terms used in algebra include: variables, constants, operations, expressions and equations. A variable can be represented by letters from the alphabet.

While the addition and subtraction operations are the same in algebra as in arithmetic, there are some different designations of multiplication and division used in web pages.

An expression is a group of numbers and variables, along with operations. An equation is the equality of two expressions.

Understand what the words mean

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## Terminology Used in Algebra