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
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.