2.6: Multiplication of Rational Numbers
Suppose you wrote a computer program that multiplies
Guidance
When you began learning how to multiply whole numbers, you replaced repeated addition with the multiplication sign
Multiplying rational numbers is performed the same way. We will start with the Multiplication Property of –1.
The Multiplication Property of –1: For any real number
This can be summarized by saying, "A number times a negative is the opposite of the number."
Example A
Evaluate
Solution:
Using the Multiplication Property of
This property can also be used when the values are negative, as shown in Example B.
Example B
Evaluate
Solution:
Using the Multiplication Property of
A basic algebraic property is the Multiplicative Identity. Similar to the Additive Identity, this property states that any value multiplied by 1 will result in the original value.
The Multiplicative Identity Property: For any real number
A third property of multiplication is the Multiplication Property of Zero. This property states that any value multiplied by zero will result in zero.
The Zero Property of Multiplication: For any real number
Multiplication of fractions can also be shown visually, as you can see in the example below.
Example C
Find
Solution:
By placing one model (divided in thirds horizontally) on top of the other (divided in fifths vertically), you divide one whole rectangle into smaller parts.
The product of the two fractions is the
Video Review
Guided Practice
Simplify
Solution: By drawing visual representations, you can see that
Practice
Multiply the following rational numbers.

12⋅34 
−7.85⋅−2.3 
25⋅59 
13⋅27⋅25 
4.5⋅−3 
12⋅23⋅34⋅45 
512×910 
275⋅0 
23×14 
−11.1(4.1)
Multiply the following by negative one.
 79.5

π 
(x+1) 
x  25
 –105

x2 
(3+x) 
(3−x)
Quick Quiz
 Order from least to greatest:
(56, 2326, 3132, 314) .  Simplify
59×274.  Simplify
−5+11−9−37 .  Add
215 and78.
Multiplication Property of –1
For any real number .multiplicative identity property
The product of any number and one is the number itself.Zero Property of Multiplication
For any real number .Associative Property
The associative property states that you can change the groupings of numbers being added or multiplied without changing the sum. For example: (2+3) + 4 = 2 + (3+4), and (2 X 3) X 4 = 2 X (3 X 4).Commutative Property
The commutative property states that the order in which two numbers are added or multiplied does not affect the sum or product. For example .distributive property
The distributive property states that the product of an expression and a sum is equal to the sum of the products of the expression and each term in the sum. For example, .Integer
The integers consist of all natural numbers, their opposites, and zero. Integers are numbers in the list ..., 3, 2, 1, 0, 1, 2, 3...Mixed Number
A mixed number is a number made up of a whole number and a fraction, such as .Image Attributions
Concept Nodes:
Multiplication Property of –1
For any real number .multiplicative identity property
The product of any number and one is the number itself.Zero Property of Multiplication
For any real number .Associative Property
The associative property states that you can change the groupings of numbers being added or multiplied without changing the sum. For example: (2+3) + 4 = 2 + (3+4), and (2 X 3) X 4 = 2 X (3 X 4).Commutative Property
The commutative property states that the order in which two numbers are added or multiplied does not affect the sum or product. For example .distributive property
The distributive property states that the product of an expression and a sum is equal to the sum of the products of the expression and each term in the sum. For example, .Integer
The integers consist of all natural numbers, their opposites, and zero. Integers are numbers in the list ..., 3, 2, 1, 0, 1, 2, 3...Mixed Number
A mixed number is a number made up of a whole number and a fraction, such as .