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.