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.