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.