RWM101: Foundations of Real World Math

Use Variables and Algebraic Symbols

Greg and Alex have the same birthday, but they were born in different years. This year Greg is $20$ years old and Alex is $23$, so Alex is $3$ years older than Greg. When Greg was $12$, Alex was $15$. When Greg is $35$, Alex will be $38$. No matter what Greg's age is, Alex's age will always be $3$ years more, right?

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 $3$ 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 $g$. Then we could use $g+3$ to represent Alex's age. See Table 2.1.

Variables and Constants

• The sum of $5$ and $3$ means add $5$ plus $3$, which we write as $5+3$.
• The difference of $9$ and $2$ means subtract $9$ minus $2$, which we write as $9−2$.
• The product of $4$ and $8$ means multiply $4$ times $8$, which we can write as $4 ⋅ 8$.
• The quotient of $20$ and $5$ means divide $20$ by $5$, which we can write as $20÷5$.

Equality Symbol

Inequality

Algebraic Notation	Say
$a = b$	$a$ is equal to $b$
$a ≠ b$	$a$ is not equal to $b$
$a < b$	$a$ is less than $b$
$a > b$	$a$ is greater than $b$
$a ≤ b$	$a$ is less than or equal to $b$
$a ≥ b$	$a$ is greater than or equal to $b$

Table 2.2

Symbols < and >

The symbols < and > each have a smaller side and a larger side.

smaller side < larger side
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

parentheses	( )
brackets	[ ]
braces	{ }

Table 2.3

Here are some examples of expressions that include grouping symbols. We will simplify expressions like these later in this section.

$8 (14−8)$

$21−3 [2+4 (9−8) ]$

$24÷{13−2[1(6−5)+4]}$

Identify Expressions and Equations

What is the difference in English between a phrase and a sentence? A phrase expresses a single thought that is incomplete by itself, but a sentence makes a complete statement. "Running very fast" is a phrase, but "The football player was running very fast" is a sentence. A sentence has a subject and a verb.

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:

Expression	Words	Phrase
$3+5$	$3$ plus $5$	the sum of three and five
$n−1$	$n$ minus one	the difference of $n$ and one
$6 ⋅ 7$	$6$ times $7$	the product of six and seven
$\frac{x}{y}$	$x$ divided by $y$	the quotient of $x$ and $y$

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:

Equation	Sentence
$3+5=8$	The sum of three and five is equal to eight.
$n−1=14$	$n$ minus one equals fourteen.
$6⋅7=42$	The product of six and seven is equal to forty-two.
$x=53$	$x$ is equal to fifty-three.
$y+9=2y−3$	$y$ plus nine is equal to two $y$ minus three.

Expressions and Equations

An expression is a number, a variable, or a combination of numbers and variables and operation symbols.

An equation is made up of two expressions connected by an equal sign.

Simplify Expressions with Exponents

To simplify a numerical expression means to do all the math possible. For example, to simplify $4⋅2+1$ we'd first multiply $4⋅2$ to get $8$ and then add the $1$ to get $9$. A good habit to develop is to work down the page, writing each step of the process below the previous step. The example just described would look like this:

$4⋅2+1$
$8+1$
$9$

Suppose we have the expression $2⋅2⋅2⋅2⋅2⋅2⋅2⋅2⋅2$. We could write this more compactly using exponential notation. Exponential notation is used in algebra to represent a quantity multiplied by itself several times. We write $2⋅2⋅2$ as $2^3$ and $2⋅2⋅2⋅2⋅2⋅2⋅2⋅2⋅2$ as $2^9$. In expressions such as $2^3$, the $2$ is called the base and the 3 is called the exponent. The exponent tells us how many factors of the base we have to multiply.

means multiply three factors of 2

We say $2^3$ is in exponential notation and $2⋅2⋅2$ is in expanded notation.

Exponential Notation

For any expression $a^n$, $a$ is a factor multiplied by itself $n$ times if $n$ is a positive integer.

$a^n$ means multiply $n$ factors of $a$

The expression $a^n$ is read $a$ to the $n^{th}$ power.

For powers of $n=2$ and $n=3$, we have special names.

$a^2$ is read as "$a$ squared"

$a^3$ is read as "$a$ cubed"

Table 2.4 lists some examples of expressions written in exponential notation.

Exponential Notation	In Words
$7^2$	7 to the second power, or 7 squared
$5^3$	5 to the third power, or 5 cubed
$9^4$	9 to the fourth power
$12^5$	12 to the fifth power

Table 2.4

To simplify an exponential expression without using a calculator, we write it in expanded form and then multiply the factors.

Simplify Expressions Using the Order of Operations

We've introduced most of the symbols and notation used in algebra, but now we need to clarify the order of operations. Otherwise, expressions may have different meanings, and they may result in different values.

For example, consider the expression:

$4+3⋅7$

Some students say it simplifies to 49.		Some students say it simplifies to 25.
	$4+3⋅7$		$4+3⋅7$
Since $4+3$ gives $7$.	$7⋅7$	Since $3⋅7$ is $21$.	$4+21$
And $7⋅7$ is $49$.	$49$	And $21+4$ makes $25$.	$25$

Imagine the confusion that could result if every problem had several different correct answers. The same expression should give the same result. So mathematicians established some guidelines called the order of operations, which outlines the order in which parts of an expression must be simplified.

Order of Operations

When simplifying mathematical expressions perform the operations in the following order:

1. Parentheses and other Grouping Symbols

• Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.
  

2. Exponents

• 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
Please	Parentheses
Excuse	Exponents
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.

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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.