Hello. I'm Professor Von Schmohawk and welcome to Why U.
In the first lecture, we explored the origins of the first number systems.
We also saw how the people on my primitive island of Cocoloco
first learned about the decimal number system.
Once the Cocoloconians discovered decimal numbers
we could do much more than count coconuts.
We could do arithmetic calculations with coconuts!
The first arithmetic operations we invented were addition and subtraction
which came in very handy when dealing with coconuts.
For instance, if you have three coconuts
and then your neighbor gives you five more
you will have eight coconuts.
Interestingly, if you start out with five coconuts
and then your neighbor gives you three more
you will also have eight coconuts.
For some reason, five plus three gives you the same answer as three plus five.
Eventually we figured out
that it doesn't matter in which order you acquire your coconuts.
You still end up with the same number of coconuts.
Since the two numbers on either side of the addition symbol
can switch positions without changing the answer
we say that they “commute” since commute means to travel back and forth.
Mathematical operations in which the numbers operated on
can be switched without affecting the result are said to be “commutative”.
This illustrates what the Cocoloconians call
The Commutative Property of Addition of Coconuts.
Apparently, this property applies to adding anything
so we will just call it The Commutative Property of Addition.
So addition is a commutative operation.
Instead of talking about specific numbers like three and five
if we call these two quantities A and B
then we can write this property in a more general way.
Although addition is a commutative operation, subtraction is not.
For example, four minus three is one.
However, if you switch the order of the three and the four, the result will not be the same.
Three minus four is negative one.
As we saw in the previous lecture
we can write four minus three as an addition of positive four plus negative three.
Now, since addition is commutative
we can switch the four and the negative three without changing the result.
This is a trick which can come in handy in algebra problems.
Addition is a “binary operation”.
Binary operations are mathematical calculations which involve two numbers.
These numbers are called “operands” and in the case of addition
these operands are added together to produce a result called the “sum”.
In addition operations, the operands are sometimes referred to as the “addends”.
Even though addition is defined as a binary operation
you may often see additions involving more than two operands.
This is possible because pairs of operands can be added one at a time
with each sum replacing the pair.
In this way, an unlimited number of operands can be added sequentially.
The commutative property can be applied to addition operations
involving more than two numbers.
By switching adjacent pairs of numbers, operands can be reordered in any way we please.
For instance, in this addition involving four operands
the two at the end could be moved up to the front.
Or the five could be moved to the back.
In addition to the commutative property
here is another interesting property of addition that we discovered.
Let's say that you have five coconuts
and your neighbor on the left has three.
Both of you get together and pool your coconuts into one group of eight.
Then your neighbor on the right gives you four more.
Now you will have your group of eight plus four more, for a total of twelve coconuts.
On the other hand, let's say you started out
by pooling your five coconuts with your neighbor on the right who had four coconuts.
So you start out with a group of nine coconuts.
Then your neighbor on the left gives you his three
so you end up with three plus your group of nine
or a total of twelve coconuts.
You still end up with the same number of coconuts.
In these two scenarios the coconuts were grouped in different ways before they were added.
However, we ended up with the same number.
This illustrates what is called The Associative Property of Addition
because it doesn't matter in which way the coconuts are grouped
or associated with each other before they are added.
In the end, they all add up to the same number.
If we call these quantities A, B, and C
then we can write this property in a more general way.
The commutative property of addition involves moving around the numbers to be added
whereas the associative property of addition involves grouping them differently.
Using the associative and commutative properties, we can rearrange groups of numbers.
Let's see what this looks like on a number line.
As an example, we will take an addition problem involving positive and negative numbers.
Let's start at the origin and add positive two
plus positive three
plus negative six
plus negative two
plus positive four.
This all totals up to one.
However, because of the commutative property
we are free to rearrange this sequence of numbers in any order we like.
For instance, we could add all the negative numbers first
and we will still get the same result.
We could also use the associative property to group some of the numbers to be added.
For instance, the negative two and the positive two could be grouped.
Since this group adds up to zero, we could replace it with a zero
or eliminate it altogether.
Having a familiarity with the properties of addition
allows us to start building a tool chest of mathematical tricks
which we can use later to simplify complicated problems.
In the next few lectures, we will explore the properties of more arithmetic operations
such as multiplication and division.