Lesson 1: How to Define, Classify, and Use Numbers

Numbers and Their Properties

How to Define, Classify, and Use Numbers

Number Taxonomy

• Numbers - Self-explanatory

• Real Numbers - Any numbers which repeat with a pattern. Specifically, they can be written as quotients of integers or are decimals which repeat or terminate. (ex. $-\frac{1}{3}$ ,  $\frac{3}{4}$ )

• Integers - Whole numbers and their opposites. (ex. -3, 12, 91)

• Whole Numbers - Positive numbers without fractions. Includes zero. (ex. 0, 8, 2040)

• Natural (Counting) Numbers - Positive numbers without fractions. Does not include zero. (ex. 1, 9, 25)

Multiple Select

Which of the following categories describes "-2"?

• Real Numbers

• Rational Numbers

• Integers

• Whole Numbers

• Counting Numbers

Multiple Select

Which of the following categories describes " $\frac{3}{4}$ "?

• Real Numbers

• Rational Numbers

• Integers

• Whole Numbers

• Counting Numbers

Real Numbers Can Be Graphed

Graph the following on a number line:

 $\sqrt{3}$  
 $-\frac{5}{4}$  

The Commutative Property

To commute means "to move." So, in the commutative property, we move terms around without changing the meaning of the statement. In order to utilize this property, we have to think of the terms we are moving as addition or multiplication, but not both at the same time.

NOTE: Any negative signs MUST move with the term to which they belong.

Associative Property

The associative property shows us that we can move grouping symbols whenever we consider terms as being added or multiplied, but not both at the same time. If all the terms are added or multiplied, then the grouping symbols can be moved or even removed.

NOTE: "grouping symbols" means "parentheses and anything used in place of parentheses such as brackets."

The Identity Property

The Identity Properties represent expressions in which the number given keeps its identity.

For all addition, if a number is added to zero, then the result is the number that you added to zero

(ex. 2 + 0 = 2)

For all multiplication, if a number is multiplied by 1, then the result is the number you multiplied by 1

(ex. 9 x 1 = 9)

Inverse Property

The Inverse Property of Addition states that any number added to the opposite* of that number equals zero.

*(In math, "opposite" means that same number with the opposite sign. So, 7 and -7 are "opposites")

Inverse Property

The Inverse Property of Multiplication states that any number multiplied by its reciprocal* equals 1.

*(In math, a "reciprocal" is the fraction form of the number, but flipped so that the numerator becomes the denominator and the denominator becomes the numerator. For instance, 3/4 is the reciprocal of 4/3).

The Distributive Property

For the distributive property, you can multiply the number outside a grouping symbol to each term within the grouping symbol without changing the value or meaning of the expression.

For instance, a(b + c) = ab + ac

Try plugging in any numbers for a, b, and c and simplify it using the distributive property and then using the Order of Operations and see if you achieve a different result.

Multiple Choice

 $9+7=7+9$  demonstrates which of the following properties of addition?

• Associative

• Inverse

• Commutative

• Identity

Multiple Choice

 $22\cdot1=22$  demonstrates which of the following properties of multiplication?

• Associative

• Inverse

• Commutative

• Identity

To Approach Algebra Correctly, We Need to Reconsider Our Definitions of...

Subtraction - It is "adding the opposite" rather than an operation in and of itself.
(ex. " $2-8$ " should be thought of as " $2+\left(-8\right)$ ")

Division - It is "multiplying by the reciprocal" rather than its own operation.
(ex. " $28\div2$ " should be thought of as " $28\cdot\frac{1}{2}$ ")

We can use these definitions and properties to prove various mathematical statements

Prove  $a+\left(2-a\right)=2$  

•  $a+\left(2-a\right)=a+\left[2+\left(-a\right)\right]$        Definition of Subtraction

•  $a+\left[2+\left(-a\right)\right]=a+\left[\left(-a\right)+2\right]$  Commutative Property of Addition

•  $a+\left[\left(-a\right)+2\right]=\left[a+\left(-a\right)\right]+2$  Associative Property of Addition

•  $\left[a+\left(-a\right)\right]+2=0+2$                   Inverse property of Addition

•  $0+2=2$                                           Identity Property of Addition

Unit Analysis

In real life situations, you can use unit analysis to check if you selected the correct operations for conversions

Consider the Following Examples:

• You work 4 hours and earn $36. What is your earning rate?

• Answer: $9 per hour

• You travel 2.5 miles at 50 miles per hour. How far did you go?

• Answer: 125 miles

• You drive 45 miles per hour. What is your speed in feet per second?

• Answer: 66 feet per second