Hello. I’m Professor Von Schmohawk

and welcome to Why U. In our last lecture we saw that addition and

multiplication are both commutative operations. The order of numbers which are added or multiplied

can be rearranged without affecting the result. As we saw, addition also has

an “associative” property. According to the

Associative Property of Addition three or more numbers which are added

can be grouped in any way without affecting the result. Does this also apply to multiplication? Let’s start with our stack of 24 boxes and group them in different ways before multiplying. For instance if we group the two and the three we get four groups of six which is still equal to 24. Or we can group the four and the two to get three groups of eight which is still 24. Either way we group the numbers

we still get the same result. If we use the letters A, B, and C

instead of numbers then we can write this property in a

more general way. So the associative property applies to both

addition and multiplication. But what if we have a group of numbers which

are added and multiplied? Does the associative property still apply? Let’s take an example where we group

two plus three times four in two different ways. When we group numbers in parentheses the operation inside the parentheses

is performed first. So when two plus three is written in parentheses

we do the addition first and then we multiply the result times four. In the second case, the parentheses are around

the three times four so the multiplication is performed first and then two is added to the result. The result in the first case is 20

while the result in the second case is 14. So the order of addition and multiplication

does matter. The associative property does not apply to

combinations of addition and multiplication. So how could this expression be written without

parentheses without changing the result? We can see from the diagram that both the

two and the three must be multiplied by four. So if we remove the parentheses we must multiply

each number in the parentheses by four. In other words, the multiplier must be “distributed”

to each number in the parentheses. This is called “The Distributive Property

of Multiplication over Addition” or for short,

The Distributive Property. Once again if we use the letters A, B, and C

instead of numbers we can write this property in a

more general way. The distributive property says that

when a sum of numbers in parentheses are multiplied by a number

outside of the parentheses if we remove the parentheses, then each number

inside must be multiplied individually by the number outside. We can also use the distributive property

in reverse. If we have two numbers, A and B which are both multiplied by C

and then added then we can group these two products

in parentheses and move the common multiplier C

outside of the parentheses. Of course, because of the commutative property the multiplier could be placed either before

or after the parentheses. The distributive property can be applied to

any quantity of numbers which all have a common multiplier. For example, let’s say that four numbers

which we will call A, B, C, and D are all multiplied by five, and then added. Then we can group these numbers as a sum

in the parentheses and multiply the parentheses by

the common multiplier five. As you can see, the commutative, associative

and distributive properties are powerful tools which we will add to our

tool chest of mathematical tricks. With them, we can manipulate groups of numbers

and mathematical operations and change their form into equivalent forms

which may be simpler or more useful.