Basic Math & Pre-Algebra For Dummies, 2nd Edition (2014)
Part II. Getting a Handle on Whole Numbers
Chapter 5. A Question of Values: Evaluating Arithmetic Expressions
In This Chapter
Understanding the Three E's of math — equations, expressions, and evaluation
Using order of precedence to evaluate expressions containing the Big Four operations
Working with expressions that contain exponents
Evaluating expressions with parentheses
In this chapter, I introduce you to what I call the Three E's of math: equations, expressions, and evaluation. You'll likely find the Three E's of math familiar because, whether you realize it or not, you've been using them for a long time. Whenever you add up the cost of several items at the store, balance your checkbook, or figure out the area of your room, you're evaluating expressions and setting up equations. In this section, I shed light on this stuff and give you a new way to look at it.
You probably already know that an equation is a mathematical statement that has an equals sign (=) — for example, 1 + 1 = 2. An expression is a string of mathematical symbols that can be placed on one side of an equation — for example, 1 + 1. And evaluation is finding out the value of an expression as a number — for example, finding out that the expression 1 + 1 is equal to the number 2.
Throughout the rest of the chapter, I show you how to turn expressions into numbers using a set of rules called the order of operations (or order of precedence). These rules look complicated, but I break them down so you can see for yourself what to do next in any situation.
Seeking Equality for All: Equations
An equation is a mathematical statement that tells you that two things have the same value — in other words, it's a statement with an equals sign. The equation is one of the most important concepts in mathematics because it allows you to boil down a bunch of complicated information into a single number.
Mathematical equations come in a lot of varieties: arithmetic equations, algebraic equations, differential equations, partial differential equations, Diophantine equations, and many more. In this book, I look at only two types: arithmetic equations and algebraic equations.
In this chapter, I discuss only arithmetic equations, which are equations involving numbers, the Big Four operations, and the other basic operations I introduce in Chapter 4 (absolute values, exponents, and roots). In Part V, I introduce you to algebraic equations. Here are a few examples of simple arithmetic equations:
And here are a few examples of more-complicated arithmetic equations:
Three properties of equality
Three properties of equality are reflexivity, symmetry, and transitivity:
· Reflexivity says that everything is equal to itself. For example,
1 = 1 23 = 23 1,000,007 = 1,000,007
· Symmetry says that you can switch the order in which things are equal. For example,
· Transitivity says that if something is equal to two other things, then those two other things are equal to each other. For example,
Because equality has all three of these properties, mathematicians call equality an equivalence relation. The inequalities that I introduce in Chapter 4 (≠, >, <, and ≈) don't necessarily share all these properties.
Hey, it's just an expression
An expression is any string of mathematical symbols that can be placed on one side of an equation. Mathematical expressions, just like equations, come in a lot of varieties. In this chapter, I focus only on arithmetic expressions,which are expressions that contain numbers, the Big Four operations, and a few other basic operations (see Chapter 4). In Part V, I introduce you to algebraic expressions. Here are a few examples of simple expressions:
And here are a few examples of more-complicated expressions:
Evaluating the situation
At the root of the word evaluation is the word value. In other words, when you evaluate something, you find its value. Evaluating an expression is also referred to as simplifying, solving, or finding the value of an expression. The words may change, but the idea is the same — boiling down a string of numbers and math symbols to a single number.
When you evaluate an arithmetic expression, you simplify it to a single numerical value — in other words, you find the number that it's equal to. For example, evaluate the following arithmetic expression:
How? Simplify it to a single number:
· 35
Putting the Three E's together
I'm sure you're dying to know how the Three E's — equations, expressions, and evaluation — are all connected. Evaluation allows you to take an expressioncontaining more than one number and reduce it to a single number. Then you can make an equation, using an equals sign, to connect the expression and the number. For example, here's an expression containing four numbers:
· 1 + 2 + 3 + 4
When you evaluate it, you reduce it to a single number:
· 10
And now you can make an equation by connecting the expression and the number with an equals sign:
· 1 + 2 + 3 + 4 = 10
Introducing Order of Operations
When you were a kid, did you ever try putting on your shoes first and then your socks? If you did, you probably discovered this simple rule:
1. Put on socks.
2. Put on shoes.
Thus, you have an order of operations: The socks have to go on your feet before your shoes. So in the act of putting on your shoes and socks, your socks have precedence over your shoes. A simple rule to follow, right?
In this section, I outline a similar set of rules for evaluating expressions, called the order of operations (sometimes called order of precedence). Don't let the long name throw you. Order of operations is just a set of rules to make sure you get your socks and shoes on in the right order, mathematically speaking, so you always get the right answer.
Note: Through most of this book, I introduce overarching themes at the beginning of each section and then explain them later in the chapter instead of building them and finally revealing the result. But order of operations is a bit too confusing to present that way. Instead, I start with a list of four rules and go into more detail about them later in the chapter. Don't let the complexity of these rules scare you off before you work through them!
Evaluate arithmetic expressions from left to right according to the following order of operations:
1. Parentheses
2. Exponents
3. Multiplication and division
4. Addition and subtraction
Don't worry about memorizing this list right now. I break it to you slowly in the remaining sections of this chapter, starting from the bottom and working toward the top, as follows:
· In “Applying order of operations to Big Four expressions,” I show Steps 3 and 4 — how to evaluate expressions with any combination of addition, subtraction, multiplication, and division.
· In “Using order of operations in expressions with exponents,” I show you how Step 2 fits in — how to evaluate expressions with Big Four operations plus exponents, square roots, and absolute value.
· In “Understanding order of operations in expressions with parentheses,” I show you how Step 1 fits in — how to evaluate all the expressions I explain plus expressions with parentheses.
Applying order of operations to Big Four expressions
As I explain earlier in this chapter, evaluating an expression is just simplifying it to a single number. Now I get you started on the basics of evaluating expressions that contain any combination of the Big Four operations — adding, subtracting, multiplying, and dividing. (For more on the Big Four, see Chapter 3.) Generally speaking, the Big Four expressions come in the three types in Table 5-1.
Table 5-1 The Three Types of Big Four Expressions
Expression |
Example |
Rule |
Contains only addition and subtraction |
12 + 7 – 6 – 3 + 8 |
Evaluate left to right. |
Contains only multiplication and division |
18 ÷ 3 × 7 ÷ 14 |
Evaluate left to right. |
Mixed-operator expression: contains a combination of addition/subtraction and multiplication/division |
9 + 6 ÷ 3 |
1. Evaluate multiplication and division left to right.2. Evaluate addition and subtraction left to right. |
In this section, I show you how to identify and evaluate all three types of expressions.
Expressions with only addition and subtraction
Some expressions contain only addition and subtraction. When this is the case, the rule for evaluating the expression is simple.
When an expression contains only addition and subtraction, evaluate it step by step from left to right. For example, suppose you want to evaluate this expression:
· 17 − 5 + 3 − 8
Because the only operations are addition and subtraction, you can evaluate from left to right, starting with 17 − 5:
· = 12 + 3 − 8
As you can see, the number 12 replaces 17 − 5. Now the expression has three numbers instead of four. Next, evaluate 12 + 3:
· = 15 − 8
This step breaks down the expression to two numbers, which you can evaluate easily:
· = 7
So 17 − 5 + 3 − 8 = 7.
Expressions with only multiplication and division
Some expressions contain only multiplication and division. When this is the case, the rule for evaluating the expression is pretty straightforward.
When an expression contains only multiplication and division, evaluate it step by step from left to right. Suppose you want to evaluate this expression:
Again, the expression contains only multiplication and division, so you can move from left to right, starting with :
Notice that the expression shrinks one number at a time until all that's left is 2. So .
Here's another quick example:
Even though this expression has some negative numbers, the only operations it contains are multiplication and division. So you can evaluate it in two steps from left to right (remembering the rules for multiplying and dividing with negative numbers that I show you in Chapter 4):
Thus, .
Mixed-operator expressions
Often an expression contains
· At least one addition or subtraction operator
· At least one multiplication or division operator
I call these mixed-operator expressions. To evaluate them, you need some stronger medicine.
Evaluate mixed-operator expressions as follows:
1. Evaluate the multiplication and division from left to right.
2. Evaluate the addition and subtraction from left to right.
For example, suppose you want to evaluate the following expression:
As you can see, this expression contains addition, multiplication, and division, so it's a mixed-operator expression. To evaluate it, start by underlining the multiplication and division in the expression:
Now evaluate what you’ve underlined from left to right:
At this point, you're left with an expression that contains only addition, so you can evaluate it from left to right:
Thus, .
Using order of operations in expressions with exponents
Here's what you need to know to evaluate expressions that have exponents (see Chapter 4 for info on exponents).
Evaluate exponents from left to right before you begin evaluating Big Four operations (adding, subtracting, multiplying, and dividing).
The trick here is to turn the expression into a Big Four expression and then use what I show you earlier in “Applying order of operations to Big Four expressions.” For example, suppose you want to evaluate the following:
First, evaluate the exponent:
At this point, the expression contains only addition and subtraction, so you can evaluate it from left to right in two steps:
So .
Understanding order of precedence in expressions with parentheses
In math, parentheses — ( ) — are often used to group together parts of an expression. When it comes to evaluating expressions, here's what you need to know about parentheses.
To evaluate expressions that contain parentheses,
1. Evaluate the contents of parentheses from the inside out.
2. Evaluate the rest of the expression.
Big Four expressions with parentheses
Similarly, suppose you want to evaluate . This expression contains two sets of parentheses, so evaluate these from left to right. Notice that the first set of parentheses contains a mixed-operator expression, so evaluate this in two steps, starting with the division:
Now evaluate the contents of the second set of parentheses:
Now you have a mixed-operator expression, so evaluate the multiplication () first:
· = 4 + −15
Finally, evaluate the addition:
· = −11
So .
Expressions with exponents and parentheses
As another example, try this out:
Start by working with only what's inside the parentheses. The first part to evaluate there is the exponent, :
Continue working inside the parentheses by evaluating the division :
Now you can get rid of the parentheses altogether:
At this point, what's left is an expression with an exponent. This expression takes three steps, starting with the exponent:
So .
Expressions with parentheses raised to an exponent
Sometimes the entire contents of a set of parentheses are raised to an exponent. In this case, evaluate the contents of the parentheses before evaluating the exponent, as usual. Here's an example:
· (7 − 5)^{3}
First, evaluate 7 − 5:
· = 2^{3}
With the parentheses removed, you're ready to evaluate the exponent:
· = 8
Once in a rare while, the exponent itself contains parentheses. As always, evaluate what's in the parentheses first. For example,
This time, the smaller expression inside the parentheses is a mixed-operator expression. I’ve underlined the part that you need to evaluate first:
Now you can finish off what's inside the parentheses:
· = 21^{1}
At this point, all that's left is a very simple exponent:
· = 21
So .
Note: Technically, you don't need to put parentheses around the exponent. If you see an expression in the exponent, treat it as though it has parentheses around it. In other words, means the same as .
Expressions with nested parentheses
Occasionally, an expression has nested parentheses, or one or more sets of parentheses inside another set. Here I give you the rule for handling nested parentheses.
When evaluating an expression with nested parentheses, evaluate what's inside the innermost set of parentheses first and work your way toward the outermost parentheses.
For example, suppose you want to evaluate the following expression:
· 2 + (9 − (7 − 3))
I underlined the contents of the innermost set of parentheses, so evaluate these contents first:
· = 2 + (9 − 4)
Next, evaluate what's inside the remaining set of parentheses:
· = 2 + 5
Now you can finish things off easily:
· = 7
So 2 + (9 − (7 − 3)) = 7.
As a final example, here's an expression that requires everything from this chapter:
This expression is about as complicated as you're ever likely to see in pre-algebra: one set of parentheses containing another set, which contains a third set. To start you off, I underlined what's deep inside this third set of parentheses. This is where you begin evaluating:
What's left is one set of parentheses inside another set. Again, work from the inside out. The smaller expression here is , so evaluate the exponent first, then the multiplication, and finally the subtraction:
Only one more set of parentheses to go:
· = 4 + 56
At this point, finishing up is easy:
· = 60
Therefore, .
As I say earlier in this section, this problem is about as hard as they come at this stage of math. Copy it down and try solving it step by step with the book closed.