2.3 - Multiplication of Numbers

First a word on notation:
• a · b means multiply a times b. So does a b and so does a*b.
For example 3 · 2 = 6. The numbers 3 and 2 are called factors and the number 6 is called the product.

• − ( ) means to take the opposite or negative of whatever is in the brackets.
For example − (3) = −3. Literally the negative of the number 3 is the number −3.
Also − (−3) = 3. Literally the negative of the number −3 is the number 3.

Multiplication originated as a way to describe repeated addition. For example adding 6 to itself 4 times is the same thing as 6 multiplied by 4:
6 + 6 + 6 + 6 = 4 · 6
A more advanced way to interpret multiplication is as stretching. Thus the number 4 · 6 is 4 times as long as the number 6: Here are some important laws of multiplication.

 The commutative law of multiplication. This law states that for any numbers a and b, a · b = b · a In other words the result is the same no matter which number is considered to be multiplied onto which.

 The associative law of multiplication. If we need to multiply three or more numbers together then we do this by multiplying two at a time. The associative law says that the result is the same no matter which two we multiply first. It states that for any numbers a, b and c, (a · b) · c = a · (b · c)

Zero Times Anything is Zero

 No matter how complicated the factor a is, 0 · a = 0

The Multiplicative Identity

 The number 1 is called the multiplicative identity because multiplying any number a by 1 just gives back a : 1 · a = a

A Negative Number Times a Positive Number is Negative

Probably the hardest thing about multiplication is understanding how to deal with negative numbers. We mentioned that multiplication originated as a way to do repeated addition. Then, as the following picture shows, it must be true that 3 · (−2)  = (−2) + (−2) + (−2) = − 6 : Notice that, as the next picture shows, it is also true that (−3) · 2 = − 6 : Finally it is also true that − (3 · 2) = −6. Now put together all these ways of writing −6: If you compare these three forms you see that you can move the − sign in front of either factor or in front of the entire product. This is true in general:

 Moving the − sign around in a product. For any numbers a and b, a · (− b) = (− a) · b = − (a · b)

The Product of Two Negative Numbers is Positive

To derive this rule look at this example: The first step was to move the − sign from the second factor to the first (as explained in the previous section).

The second step was to use the fact that the opposite or negative of − 3 is 3.

The last step was to just multiply 3 times 2. We conclude that the product of two negative numbers is a positive number.

A Product with Several Negative Factors

We can generalize the above results:

 The sign of a product having several negative factors. If a product has several negative factors then count how many negative factors there are. If there are an odd number then the product is negative. If there are an even number then the product is positive.

Here are some examples: Definition: Even and odd numbers. The even numbers are the integers …, −6, −4, −2, 0, 2, 4, 6, … and the odd numbers are all the other integers, namely …, −5, −3, −1, 1, 3, 5, …. The even numbers are evenly divisible by 2 and the odd numbers are not.

The many roles of the − sign

Have you ever wondered why some calculators have more than one − key? The reason is that the − sign plays several roles in algebra, as this table shows. Each expression evaluates to − 5.
 The expression What the − sign means in this expression − 5 This is just a number to the left of 0 on the number line. 0 + (− 5) This is an addition. We are adding −5 onto zero. 0 − 5 This is a subtraction. We are subtracting 5 from zero. −1 · 5 This is a multiplication. We are multiplying 5 by −1. − ( 5 ) This means “the opposite of 5”.
In many algebra problems it is important for you to be able to switch the − sign from one role to another.

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