Multiplication of Real Numbers

1. Some Definitions

When multiplying two numbers such as

2 x 3  =  6

The numbers on the left (the 2 and the 3) are called factors and the result (the 6) is called the product.

2. Properties of Multiplication

1. The Zero Property

If you receive one million dollars exactly zero times, how much money do you get?  Clearly the answer is zero.  In general any number times zero is zero.

 The Zero Property of Multiplication  Any number times zero is zero.

Examples

3 x 0  =  0    55 x 0  =  0    0 x 4  =  0    0 x 0  =  0

1. The Identity Element for Multiplication

Just as it is easy to multiply a number by zero, multiplying by one is also very simple.  If you buy twelve eggs exactly one time, then you have bought a total of twelve eggs.  In general, any number times one is the original number.

 The Identity Property of Multiplication  Any number times one is one.

Examples

3 x 1  =  3    55 x 1  =  55    1 x 4  =  4    1 x 1  =  1    0 x 1  =  0

2. The Commutative Property of Multiplication

For multiplication, it does not matter what order the factors appear.  Two six packs of beer has the same number of bottles as six two packs of beer.

 The Commutative Property of Multiplication  Numbers multiplied in different orders give identical products

Examples

2 x 6  =  6 x 2    43 x 12  =  12 x 43    1 x 7  =  7 x 1    0 x 5  =  5 x 0

3. The Associative Property of Multiplication

When we multiply three numbers, we can multiply the product of the first two by the third or we can multiply the first by the product of the last two.  The result will always be the same.

Suppose you have three books, each having two chapters such that every chapter has ten pages.  There are two ways of determining the total number of pages in the books.

Method 1:  Since there are three books with two chapters each, there are 3 x 2  =  6 chapters.  Each chapter has ten pages so there are 6 x 10  =  60 total pages.

Method 2:  Since each book has two chapters each with ten pages, each book has  2 x 10  =  20 pages.  There are three books each with 20 pages, so there are 3 x 20  =  60 total pages.

 The Associative Property of Multiplication  When three numbers are multiplied, the multiplication can be grouped in any way.

Examples

3 x (2 x 10)  =  (3 x 2) x 10  =  60         (12 x 1) x 3  =  12 x (1 x 3)  =  36

4. The Distributive Property

There is a property that relates to when multiplication and addition are combined.  If two people have ten fingers and four limbs each, then we can determine the total number of fingers and limbs by either of the following two methods:

Method 1:  Each person has ten fingers and four limbs, hence

10 + 4  =  14

body parts.  Since there are two people, the total number of body parts is

2 x 14  =  28

Method 2:  The are two people with ten fingers each, hence there are

2 x 10  =  20

total fingers.  Similarly, there are two people with four limbs each, hence there are

2 x 4  =  8

total limbs.  Adding the body parts gives

20 + 8  =  28

total body parts.

 The Distributive Property  Multiplication can be distributed over addition without changing the result.

Examples

2 x (10 + 4)  =  2 x 10 + 2 x 4  =  28     3 x (5 + 6)  =   3 x 5 + 3 x 6  =  33

1. Practice with Multiplication

Example

145
x    37
1015
(245 x 7  =  1015)
4350
(245 x 3  =  435)  Remember to place an extra 0 in the right digit
5385

Exercises

1. 423 x 27  =          (Hold mouse over the yellow rectangles to check your answer)

2. 314 x 103  =

2. Applications

1. You own a bicycle rental shop near the lake.  Your bikes rent for \$29 per rental and you had 327 customers in the first week of July.  What was your total revenue for bicycle rentals during the first week of July?

Solution

Since

Revenue  =  price x  number of customers

We multiply

327
x    29
2943
(327 x 9  =  2943)
6540
(327 x 2  =  6540)  Remember to place an extra 0 in the right digit
9483

Your total revenue during the first week of July was \$9,483.

2. Your computer screen measures 13 inches by 9 inches.  What is the total area of your computer screen?

Solution

Since

Area  =  Width x Height

We multiply

13
x     9
117

The area of the computer screen is 117 square inches.

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