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Authors: Mark Zegarelli
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of Values: Evaluating Arithmetic Expressions
In This Chapter
Understanding the Three E's of math â 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 (=) â for example, 1 + 1 = 2. An
expression
is a string of mathematical symbols that can be placed on one side of an equation â for example, 1 + 1. And
evaluation
is finding out the
value
of an expression as a number â 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 â in other words, it's a statement with an equals sign. Theequation 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 = 1â23 = 23â1,000,007 = 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 (â , >, <, and â) 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
In This Chapter
Understanding the Three E's of math â 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 (=) â for example, 1 + 1 = 2. An
expression
is a string of mathematical symbols that can be placed on one side of an equation â for example, 1 + 1. And
evaluation
is finding out the
value
of an expression as a number â 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 â in other words, it's a statement with an equals sign. Theequation 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 = 1â23 = 23â1,000,007 = 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 (â , >, <, and â) 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
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