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Terminology in Algebra

by Ron Kurtus (10 January 2008)

Major terms used in algebra that you should know and understand include: variable, operations, expression and equation. A variable is also called an unknown and can be represented by letters from the alphabet. Operations in are the same as in arithmetic: addition, subtraction, multiplication and division. There are some different designation of those operations in algebra. An expression is a group of numbers and variables, along with operations. An equation is the equality of two expressions.

Questions you may have are:

• What is a variable?
• What are algebraic operations?
• What is an expression and equation?

This lesson will answer those questions. There is a mini-quiz near the end of the lesson.

Variable

Algebra started off as the addition and subtraction of 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.

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.

Operations

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

• Addition +
• Subtraction −
• Multiplication ×
• Division ÷

Multiplication sign

Since the letter "x" is often use as a variable or unknown in algebra, an asterisk (*) or a dot (·) is often substituted for the multiplication sign ×. Thus, instead of a × b, you could write a*b or a·b.

The problem with using the dot when dealing with numbers is that 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.

No multiplication sign

In algebra, when multiplying variable together, they have completely dropped the multiplication sign altogether. Instead of writing a times b times c as a*b*c, it is usually written as abc in algebra.

When multiplying a variable and a number, the number is written first and no multiplication sign is used. Thus, b times 3 is written as 3b, and x times 2 times z is written as 2xz.

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. a÷b and a/b are both 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.

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)

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

3x − 7 = 3 is valid with when x = 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: variable, operations, expression and equation. A variable can be represented by letters from the alphabet. Operations in are the same as in arithmetic. There are some different designation of those operations in algebra. An expression is a group of numbers and variables, along with operations. An equation is the equality of two expressions.

Answers to Readers' Questions

Understand what the words mean

Resources

The following resources provide information on this subject:

Websites

Algebra Resources

Books

Top-rated Algebra books

Miscellaneous

Mini-quiz to check your understanding

1. What could x represent?

2. Why is xyz used in algebra to denote multiplication instead of x*y*z?

3. Is 3x + 7y = 0 an expression?

If you got all three correct, you are on your way to becoming a Champion in Algebra. If you had problems, you had better look over the material again.

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