1.5+Multiplication

Section 1.5 - Multiplication
Multiplication is repeated addition, but faster. If I had a picnic tablecloth that had five rows with seven watermelons in each row, I could add them up, or multiply.



How do you calculate 5 X 7? You don't - you memorize it! If you don't know your times tables, memorize them now!


 * Product - The answer to a multiplication problem. **


 * Factor - One of the two numbers being multiplied together to get a product. **



There are several different ways to represent multiplication: x - Not used so much in algebra because of the confusion with x as a variable. It is, however, used in Scientific Notation. - Parentheses are used to separate numbers or terms, but also for multiplication. 3(4) and (3)(4) both mean to multiply 3 and 4. No sign - If there is no sign between two terms, it means multiply. This is most common with variables (unknowns written as letters usually). 3z means 3 times z. 4ab means 4 times a times b.
 * - The dot is an older style, but still used often.
 * - Asterisk is used to represent the dot for a lot of devices.

Here's the Khan Academy video introducing basic multiplication. Sal also covers the Identity Property and multiplication involving zero.

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 * Commutative Property of Multiplication**

If you are multiplying two or more numbers, you can do them in any order. 6 x 3 = 18 and so does 3 x 6.

6 x 3 = 3 x 6

AB = BA

Although this seems fairly obvious, you will use this rule in a few important ways later in the course, particularly with area problems and scientific notation.


 * Associative Property of Multiplication**

As long as all of the numbers are being multiplied, and you can do that in any order, you could choose to multiply a couple here and a couple there, then multiply those products. Remember, parentheses mean "do this first."

( 6 x 4 ) x 2 = 6 ( 4 x 2 ) or A(BC) = (AB)C


 * Identity Property of Multiplication**

Anything times 1 is itself. 6 x 1 = 6. 45xyz times 1 = 45xyz. This is a very fundamental rule, but amazingly important for pre-algebra. Multiplying something by 1 does not change its value -- but can change the way it looks! We will really use this in the chapters on fractions and unit analysis.


 * Multiples of a Number**

Multiples of a number are that number multiplied by each whole number, starting with 1. So, the first few multiples of 7 are: 7, 14, 21, 28, 35. ..




 * Multiplying by Powers of Ten**

Because of the base ten system, it is very easy to multiply a number by 10 - just add a zero. So, 7 x 10 = 70 and 45 x 10 = 450. The same thing happens with powers of ten, such as 100 or 1000 - just add the number of zeros to the end. 14 x 1,000 = 14,000


 * Multiplying with Multiple Digits**

When multiplying larger numbers, they are generally written on top of each other, with the right-most digits lining up. If you are multiplying whole numbers or integers, the place values will line up, but even if you are multiplying decimals, line up the right-most digits. Following the Commutative Property of Multiplication, it doesn't matter if the larger number is on top or bottom, but generally it is on top.




 * The Distributive Property**

The Distributive Property of Multiplication over Addition is a very important concept in algebra for simplifying math expressions. Although it is primarily used with variables (like x and y), it is presented here with numbers to show how the property works.



You are "distributing" the 3 by multiplying it by each of the terms in parentheses, which are being added (or subtracted) together.

You will learn, or have learned, that you are supposed to add the numbers in parentheses before multiplying it by three. And you would, but this property is designed for times when you can't add the terms together first, such as when variables are involved. You will usually see it as: 4 ( x + 2) = 4(x) = 4(2)


 * Application Problems with Multiplication**

With multiplication, you are able to calculate the area of squares and rectangles and other shapes (Section 9.2).

**Where to from here?**
1.1 Place Value 1.2 Rounding and Estimating Numbers 1.3 Addition 1.4 Subtraction 1.5 Multiplication 1.6 Division 1.7 Exponents 1.8 Order of Operations 1.9 Prime Numbers
 * Chapter One Review**
 * Chapter One Homework**