Use the Language of Algebra
Read this text. Pay special attention to the Order of Operations portion that summarizes the rules for performing multi-step calculations.
Many students use a silly phrase to remember the order of operations: Please Excuse My Dear Aunt Sally, where the first letter of each word corresponds to a mathematical operation: parentheses, exponents, multiplication and division, addition and subtraction.
Do Examples 2.8 – 2.12, and check your answers. If you need more practice, you can try Try It 2.15 – 2.24.
Use the Language of Algebra
Use Variables and Algebraic Symbols
Greg and Alex have the same birthday, but they were born in different years. This year Greg is
In the language of algebra, we say that Greg's age and Alex's age are variable and the three is a constant. The ages change, or vary, so age is a variable. The years between them always stays the same, so the age difference is the constant.
In algebra, letters of the alphabet are used to represent variables. Suppose we call Greg's age . Then we could use to represent Alex's age. See Table 2.1.
|Greg's age||Alex's age|
Variables and Constants
||Notation||Say:||The result is…|
||plus||the sum ofand|
||minus||the difference ofand|
||, , ,||times||The product ofand|
||divided by||The quotient ofand|
In algebra, the cross symbol, , is not used to show multiplication because that symbol may cause confusion. Does mean (three times ) or (three times )? To make it clear, use • or parentheses for multiplication.
We perform these operations on two numbers. When translating from symbolic form to words, or from words to symbolic form, pay attention to the words of or and to help you find the numbers.
- The sum of and means add plus , which we write as .
- The difference of and means subtract minus , which we write as .
- The product of and means multiply times , which we can write as .
- The quotient of and means divide by , which we can write as .
The symbol = is called the equal sign.
is to the right of on the number line
When we write an inequality symbol with a line under it, such as , it means or . We read this is less than or equal to . Also, if we put a slash through an equal sign, , it means not equal.
We summarize the symbols of equality and inequality in Table 2.2.
|is equal to|
|is not equal to|
|is less than|
|is greater than|
|is less than or equal to|
|is greater than or equal to|
Symbols < and >
larger side > smaller side
The smaller side of the symbol faces the smaller number and the larger faces the larger number.
Grouping symbols in algebra are much like the commas, colons, and other punctuation marks in written language. They indicate which expressions are to be kept together and separate from other expressions. Table 2.3 lists three of the most commonly used grouping symbols in algebra.
Common Grouping Symbols
Identify Expressions and Equations
In algebra, we have expressions and equations. An expression is like a phrase. Here are some examples of expressions and how they relate to word phrases:
|plus||the sum of three and five|
|minus one||the difference ofand one|
|the product of six and seven|
|divided by||the quotient ofand|
Notice that the phrases do not form a complete sentence because the phrase does not have a verb. An equation is two expressions linked with an equal sign. When you read the words the symbols represent in an equation, you have a complete sentence in English. The equal sign gives the verb. Here are some examples of equations:
|The sum of three and five is equal to eight.|
|minus one equals fourteen.|
|The product of six and seven is equal to forty-two.|
|is equal to fifty-three.|
|plus nine is equal to two minus three.|
Expressions and Equations
Simplify Expressions with Exponents
For powers of and , we have special names.
is read as " squared"
is read as " cubed"
|Exponential Notation||In Words|
|7 to the second power, or 7 squared|
|5 to the third power, or 5 cubed|
|9 to the fourth power|
|12 to the fifth power|
Simplify Expressions Using the Order of Operations
|Some students say it simplifies to 49.||Some students say it simplifies to 25.|
|Sincegives .||Sinceis .|
|Andis .||Andmakes .|
Order of Operations
1. Parentheses and other Grouping Symbols
- Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.
- Simplify all expressions with exponents.
3. Multiplication and Division
- Perform all multiplication and division in order from left to right. These operations have equal priority.
4. Addition and Subtraction
- Perform all addition and subtraction in order from left to right. These operations have equal priority.
Students often ask, "How will I remember the order?" Here is a way to help you remember:
Take the first letter of each key word and substitute the silly phrase.
Please Excuse My Dear Aunt Sally.
|Order of Operations|
|My Dear||Multiplication and Division|
|Aunt Sally||Addition and Subtraction|
It's good that "My Dear" goes together, as this reminds us that multiplication and division have equal priority. We do not always do multiplication before division or always do division before multiplication. We do them in order from left to right.
Similarly, "Aunt Sally" goes together and so reminds us that addition and subtraction also have equal priority and we do them in order from left to right.
Source: Rice University, https://openstax.org/books/prealgebra/pages/2-1-use-the-language-of-algebra
This work is licensed under a Creative Commons Attribution 4.0 License.