# Algebra Intro 7: Properties of Multiplication

### Properties Of Multiplication

Do the patterns that applied to addition also apply to multiplication… do the following all produce the same result?

$3 \cdot 5 \cdot 7 \\*3 \cdot (5 \cdot 7) \\*(3 \cdot 5) \cdot 7$

After carefully following the order of operations, we see that they all result in a value of 105. Therefore, it is reasonable to conclude that multiplication is associative, similar to addition.

Do the following all produce the same result?

$3 \cdot 5 \cdot 7 \\*3 \cdot 7 \cdot 5 \\*7 \cdot 3 \cdot 5 \\*7 \cdot 5 \cdot 3 \\*5 \cdot 7 \cdot 3 \\* 5 \cdot 3 \cdot 7$

once again, after dutifully evaluating all of these from left to right, we see that they all result in a value of 105 once again. Therefore, it is also reasonable to conclude that multiplication is commutative, similar to addition.

It is interesting to note that while multiplication of numbers (scalar quantities) is always commutative, the real-world situation being modeled by those numbers is not always commutative. Three rows of two chairs facing forward looks very different than two rows of three chairs facing forward.  Both situations do indeed involve the same number of chairs (the commutative property of multiplication holds). Yet, even though their quantity is the same, they are not identical layouts… multiplication describes only one aspect of this situation: the quantity of chairs involved.

### Distributive Property

Since multiplication affects “everything in the bag” (see the previous posting), the expression

$3(x+6)$

means that everything in the parentheses, the “x” as well as the “6”,  must be tripled. So

$3(x+6)=3 \cdot x+3 \cdot 6$

This is more formally known as the distributive property of multiplication over addition. The “3” must be “distributed to” or “multiplied by” every term in the factor that follows (every term in the parentheses is considered one factor in the product). Similarly

$3(x-6)=3 \cdot x-3 \cdot 6$

illustrates the distributive property of multiplication over subtraction. Once the “x” has been tripled, we need to take away three sixes to get the final result to be exactly triple the original quantity in parentheses.

This leads to another question that often arises… if

$3(x+6)=3 \cdot x+3 \cdot 6$

then should

$3 \cdot (5 \cdot 7)=^?(3 \cdot 5) \cdot (3 \cdot 7)$

The answer is NO! The distributive property of multiplication applies over addition and/or subtraction, but not multiplication.

Why is this the case?  If you follow the order of operations rules, the result of product on the left must be 105. But the computation on the right would produce a result of 315… three times too big, because three was used as a factor too many times (twice).

Another way to think the above through is: once the three has been multiplied by the five, that product is now three times greater than the five was, and will be used to multiply the seven. So the final result will be three times greater than the original five times seven would have been, as intended. If both the five and the seven were tripled, the final result would be nine times greater instead of only three times greater. By applying the three to one of the factors in the product, it will end up being applied to all of them by the time the calculation is finished.

### Multiplying By A Negative

How should we deal with negative numbers in a product? Let’s start by considering

$(3)(-1)=(-1)(3)$

The commutative property of multiplication tells us that the two expressions above must be equal.

Using the “scaling” model of multiplication, the expression on the left tells us  to stretch (-1) until it is three times as far from zero (staying on the same side of zero) as it was, which makes the result -3.

The expression on the right tells us to scale a positive three by a factor of negative one. If the result must be a -3, then the original factor of positive three does not change its magnitude, but must be rotated about zero on the number lines from the positive side of zero to the negative side.  This also illustrates that multiplying a number by negative one changes it sign:

$(-1)(3)=-3=-(3)$

So, multiplying a positive number (in this case, three) by a negative number causes it to rotate 180 degrees to the other side of zero on the number line, then be scaled by the appropriate ratio (in this case, 1).

A number of students find the notation used on the far right above confusing… a negative sign in front of parentheses just seems a bit out of place, particularly when there is another number to the left of the negative sign.  If you are bothered by this, try replacing the negative sign with a negative one in parentheses as shown on the left side above.  This is often useful in situations such as this, where the negative one in parentheses seems to do a better job of reminding people to distribute the negative to all terms inside the parentheses:

$3-(2-1)=3+(-1)(2-1)=3-2+1$

Returning to the rotation idea above,

$(3)(-2)=(-2)(3)$

tells us to either scale a (-2) by a factor of three, for a result of -6; or scale a (3) by a factor of negative two… which tells us to rotate the three by 180 degrees to the other side of zero, making it a negative three, then stretch it by a factor of 2 for a final result of -6.

So far so good. Now what happens when both factors are negative?

$(-2)(-3)$

To scale (-3) by a factor of (-2), first rotate it by 180 degrees to the other side of zero to obtain (+3), then scale it by a factor of two for a final result of +6.  This is often summarized as “multiplying two negatives makes a positive”, which is similar to the double negative grammar rule in English:

$-(-3)=(-1)(-3)=3$

### The Sign Of A Product

The double negative rule can provide a very convenient shortcut for determining the sign of a product with many factors: pairs of negatives in a product will cancel one another out.  An even number of negatives will result in a positive product; an odd number of negatives will result in a negative product:

$(-2)(-3)(-4)(-5)(6)(-7)=(-720)$

will be negative because there are an odd number of negative factors.

$(-2)(-3)(-4)(-5)(-6)(-7)=+720$

will be positive because there are an even number of negative factors.

### Mental Math Tricks

When multiplying more than two factors in your head, it is often easiest to break the task into two parts:

1. What is the sign of the product (are there an even or odd number of negative quantities in the product)?
2. What is the magnitude of the product (what is the product if you ignore any negative signs)?

By breaking the task into two simpler tasks, and writing the results of each down as soon as you have figured them out, you reduce the chances of your making mistakes:

$(-3)(-2)(4)(-5)$

1. Three of the factors above are negative.  Three is an odd number, therefore the final result will be negative.
2. Using the commutative property of multiplication, I will choose to multiply the 2 by the 5 first, for a partial product of 10. The remaining partial product, of 3 and 4, is 12.  10 times 12 is 120.

So, the answer to the above is negative one hundred and twenty.  No calculator necessary.