properties.

Basically these things are common sense, and you probably know them already.

Probably the only hard part is remembering the names for them.

So let's start with the commutative property.

The commutative property says that if you have 2 numbers...

let's say 5 and 10...

you can add them in two different ways. You can either say '5 plus 10'

or you can say

'10 plus 5'

Kinda makes sense.

The same thing will work for variables.

So if you have x plus y

that would be the same as y plus x.

So why is it called commutative?

Well, when two things commute, when people commute, like if they commute to work

they move, they change places,

and what we're doing is we're taking these numbers, the 5 and the 10,

for instance, or the x and y

and we're changing their places.

So this is the commutative property of addition

because we're dealing with addition.

We've also got a commutative property of multiplication

and all that says

is if we have two numbers, let's use 5 and 10 again

we can say 5 times 10

or we can say

10 times 5

and and we'll get the same results either way.

And if we have variables

we can say x times y

or we can say

one times x

y times x.

All that's happening is the numbers or the variables are moving,

they're changing places and so they're communicating,

and this becomes the commutative property.

Okay so we have the commutative property of addition

and the commutative property of multiplication.

Let's go on to the associative property.

So

Let's say we have three numbers, let's say we have

2

and 3 and 4, and we want to add them.

Well we could either add

the 2 and the 3 together first

and then the 4,

or we could take the same three numbers,

2, 3 and 4,

we can add the 2

to the sum of the 3 and the 4.

and we're going to get the same result either way.

When things associate, when you have an association of people, you have groups of people,

so this is the associative property of addition.

Once again it will also work for variables.

So I could have

x plus y

in parentheses

plus z

and that would be the same as

x

plus

and then my parentheses

y plus z.

You realize, it's pretty obvious these things are equal.

Okay, and that's the associative property of addition,

associative because

these things are forming associations,

they're forming little groups.

We also have an associative property of multiplication,

and all that says is that if I have

2, 3 and 4 and I want to multiply them,

I could multiply 2 times 3 first

and then multiply that result by 4

or 4 could take 2, 3 and 4

and

multiply the 2

times the product of the 3 and 4.

Once again these things will be exactly the same,

and once again we can do the same thing with variables.

So I can have x times y

times z,

and I can multiply the x and y first

and then multiply the product times z,

or I could have x

and y and z,

and multiply the x

by the product

of y and z.

So this is the associative property of multiplication. Once again i think this

is common sense and you probably knew it already.

So the last property

is called the distributive property of multiplication over addition, which is a

great name. Here's all it means...

let's say I've got

2

and I want to multiply that by

3 plus 4.

Well, what the distributive property tells me is that I can distribute this

multiplication, the 2 times something,

to whatever is in the parentheses.

So I'm going to distribute the '2 times' to the 3

and distribute it to the 4.

2 times 3,

let's just write that as '2 times 3'

I'll take my plus sign and then 2 times 4.

Carrying out this multiplication

I'm going to get 2 times 3 is 6,

plus 2 times fo4r is 8

and that's going to be equal to 14.

The other way I could have done this, the way you might have been thinking, is I could take

2 times

3 plus 4, add the 3 and the 4 together,

in other words, turn this into

2 times ... 3 plus for is 7 ...

and 2 times 7 is 14. Either way I get the same answer.

So the distributive property of multiplication over addition

says that if I'm multiplying

a number or variable

times

the sum

of numbers or variables...

that's what's in this parentheses here...

what I can do is multiply that number times each of the parts of that sum

separately, and I can have more than two parts here,

so in other words I could have something like

3 times

let's say 4

uh... let's put a minus sign in...

minus

let's use a variable, 2x

plus

5y

and to distribute this multiplication what I'm gonna do is take the 3 times

the 4,

and that's going to get me... 3 times 4 is 12,

I take the 3 times the negative 2x,

so I have negative 6x,

3 times the 5y

so that will give me a positive

3 times 5 is 15.

Okay, and the result I get from distributing this 3

over this

4 minus 2x plus 5y

is going to be this 12 minus 6x plus

15y.

And that's it for that for those properties.

Take care, I'll see you next time.