Terminology Used in Algebra
by Ron Kurtus (updated 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.
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.
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 = mc2, 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.
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 mass of an object (thus the reason to use m instead of some other letter).
Mathematical operations are addition, subtraction, multiplication and division. Their common symbols are:
- Addition +
- Subtraction −
- Multiplication ×
- Division ÷
The following are designated by x + y:
Add x and y
Find the sum of x and y
Increase x by y
The following are all designated by x − y:
Subtract x from y
Find the difference between x and y
x is decreased by y
x take away y
x minus y
x less y
The following are designated by y − x:
x less than y
x from y
Multiplication and division have some variations.
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.
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.
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.
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
Some expressions may be long and complex, even including parentheses:
3x + (2z − y)/x + 125y − (x + y)/(z +2)
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
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 = 2 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.
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
Resources and references
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Terminology Used in Algebra