Algebra 1 Properties of Real Numbers

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In this lesson we will take a look at the properties of real numbers.0002

Some of these properties involve the commutative, associative, distributive, identity, inverse and 0 properties.0008

It seems like that is quite a bit to keep track of but once we get into the nuts and bolts of it, you will see it is not so bad.0015

There are several properties that can help you put different numbers together.0028

Some of the very first properties are the commutative property and the associative property.0034

We will first deal with that commutative property first.0041

What the commutative property says is that the order of addition simply does not matter.0044

It also says that the order of multiplication does not matter.0050

It seems like you should, but a quick example shows that actually it does not make much of a difference.0054

For example, if I have 2 × 3 × 4, I'm just multiplying all those numbers together.0060

I actually do this from left to right, this would be 6 × 4, I will get 24.0070

Since the order does not matter, I can feel free to scramble up that order a little bit just like that.0079

3 × 4 × 2 and again do these from left to right, 3 × 4=12 × 2.0087

12 × 2 you will see that the answer is still the same.0095

When you have addition or multiplication, the order you have a little bit of freedom with, you can scramble them up just a little bit.0102

With the associative property, we are also changing something and we are saying that the grouping does not matter.0110

If you are dealing with the associative property of addition then changing the grouping of addition does not matter.0117

The associative property for multiplication then the grouping of multiplication does not matter.0124

Let me take a quick look at a nice quick example to see what I mean by the grouping.0129

Let us look at (2 + 3) + 4, here I'm adding a bunch of different numbers.0135

Up with parentheses to show that I'm grouping the first two together.0143

According to our order of operations, we would have 2 and 3 together first.0147

This would give us a result of 5 + 4, giving us a final result of 9.0153

If I take those same numbers and the same exact order and this time I decide to group together the 3 and the 4 together,0160

you will see that you will get exactly the same answer.0170

We will do what is inside parentheses first, 3 + 4 is 7 and then we will go ahead and we will put that 2 with the 7.0173

I'm sure enough you will notice that our answers are exactly the same.0184

The associative property says that the grouping for addition and multiplication does not matter.0188

The distributive property is a neat one, it deals with multiplication and addition.0203

It says that we can distribute our multiplication over addition.0207

We looked at several different examples of this but one of the classic things is something like this.0212

I have 2 × (3 + 4), according to the distributive property I will take this multiplication with the 2 and distribute it over addition.0221

I actually multiplied it by the 3 and then multiply it by 4.0233

This will be 2 × 3, this will be 2 × 4 and I can take care of each of those individually.0239

Get a 6 + 8, combine those together and get a 14.0248

Just to highlight that this property is valid, I will also do the same problem using my order of operations to take care of the part inside parentheses first.0256

3 + 4 is 7 and 2 × 7 is 14, I'm sure enough just like we should we get the same answer.0268

This is a unique one involves multiplication and addition.0281

Some other properties that you want to keep in mind have to deal with 0 and 1.0288

With the division of property of 0, we say that 0 divided by any number we get the results is 0.0295

If we are trying to divide by 0, that is something that we can not do.0305

For that one we say that it is undefined.0311

Be very careful for this one, many people will try and divide by 0 later on, simply can not do it.0315

When it comes to 1, 1 is a little bit easier to deal with.0323

When you divide a number by 1, the number remains unchanged, nothing happens.0326

Think of 5 ÷ 1, you will still get 5, or 7 ÷ 1 =7.0332

A number divided by itself is 1, that will be like 5 ÷ 5, number divided by itself results to 1.0338

All of these properties seem like they might be arbitrary, but all of them play a very important role.0348

On to the multiplication property of 0 and let us see what we can do with that.0360

With the multiplication property of 0, when you multiply it by 0 the result is simply 0.0366

Think of stuff like 3 × 0 result 0.0372

Multiplication property of 1 says that when you multiply a number by 1, the number remains unchanged.0377

3 × 1 now 3 stays exactly the same, it is still 3.0383

With the addition property of 0, when you add 0 to a number, the number remains unchanged, 3 + 0 is still 3.0389

These last few properties that deal with 0 and 1, they can be a little confusing to keep straight.0400

But again the most important is you can not divide it by 0, watch out for that one.0407

You may be thinking that all of these properties and say okay what is the big deal?0416

Why do I necessarily need to know these?0419

Will they become extremely important when you are manipulating large expressions, especially when you get into a lot of the solving stuff later on.0421

What they do is they allow us to take an expression change in many different ways without changing what it is.0429

For example, if I want to look at an example of the students work,0436

what allows them to do many of these steps is some of those of properties that we covered earlier.0441

In fact, let us take a close look at this and see what properties we will use.0446

In the very first part of this I see (7 + 2) × (x + 9) and in the next part I know that the 2 is moved inside parentheses and now I have 18.0451

What happened there is they took the 2 and they distributed it over the x and the 9.0462

What allows a person to do that part of this work is our distributive property.0469

In the next little bit of work, I see that they have actually switch the order of the 2x and the 18.0478

They have moved around where things are.0483

Are they allowed to do this? The answer is yes.0487

We are only dealing with addition right there so they can change the order just fine.0489

This is our commutative property, let us see what is going on in the next step.0496

In the next step, everything is in the same exact order but now the parentheses have moved into a different spot.0504

Now they are actually around the 7 and 8.0510

Since only addition is being shown here, then I know that this is a valid step, this is our associative property.0514

It looks like they did some simplifying to combine the 7 and 18 together and they switched the location of the 25 and 2x.0527

It looks like in that one they have used the commutative property again.0536

The big thing to take away from this is that we will be manipulating expressions quite a bit.0541

These properties that work in the background that allows to do many of these manipulations.0546

Switching the order of things or switching our grouping as we go along.0551

It is time to get into some examples and see what we can do about identifying many of these great properties.0557

Here I have lots of different examples and we just simply want to identify the property that is being used.0564

The first one I have 6 + - 4 = -4 + 6, carefully look at that and see what is changed.0570

One thing that picked up in my mind is I'm looking at the order of things and the order has been switched.0579

I will call this my commutative property.0586

Since this is dealing with addition, I could take this a little bit further and say this is the commutative property of addition.0598

The next one let us see what we got, 7 × 11 × 18 = (11 × 8) × 7, it is tricky here.0617

Let us see what is changed, the grouping is actually exactly the same.0626

The 11 and 8 are being grouped together and over here the 11 and the 8 are still being grouped together.0632

It has nothing to do with grouping.0637

This is another one where the order has changed.0639

Since the order has changed, it is still our commutative property.0642

This one has nothing to do with addition.0651

It is actually multiplication so I will say commutative property of multiplication.0652

Nice, I like it, next problem (11 × 7) + 4 = (11 × 7) + (11 × 4) more interesting.0663

This one has multiplication and addition, in fact it looks like they took the 11 and multiplied it by the 7 and by the 4.0672

We can recognize that as our distributive property.0681

That is the unique one, it has both multiplication and addition.0694

Continuing on, pi × √2 =√2 × pi, someone has been messing around with the order of things, another one of my commutative properties.0699

Since I have multiplication here, commutative property of multiplication.0720

One more for this example 6 + 2 + 7 = 6 + 2 + 7.0728

We noticed with that one all the numbers are in exactly the same order.0736

The order is untouched but what is different here is that the parentheses are showing up in an entirely different spot.0741

They are changing the grouping, this is our associative property.0749

I have addition, associative property of addition, not bad.0765

If you change the order of the numbers present, then you are dealing with your commutative property.0773

If you change the grouping, you have your associative property.0780

With these ones we have some sort of property present and we want to fill in the blank so that the property ring is true.0786

The first one it looks like they are trying to display the commutative property.0796

I have 3 + 4 + 5= 3 + 5+ _, the commutative property changes the order of things.0799

It looks like we are changing the order of the 4 and the 5 here.0808

I see I got the 5, we will put the 4 in there and that would be a good example of the commutative property.0812

I technically, the commutative property of addition.0818

The next one is the associative property that one changes grouping so the order should stay exactly the same.0821

Let us write the same exact numbers in the same exact order,4, 5, and 3, it looks good.0829

Now I can see that since the parentheses are in a different spot, the grouping has changed.0837

Alright distributive property, that one involves multiplication over division.0844

I mean multiplication over addition.0850

I can see that the two has been taken to the 7 and the 3, I'm just missing the 7.0856

We have a good example of all 3 properties.0866

Example 3 deals with our division by 0, remember that we can not divide 0.0873

It is important to recognize what numbers can we sometimes not use, those are known as restricted values.0878

With these ones we are going to try and come up with a number that our letter x in this first one can not be.0886

Let us say we do not want the bottom to be 0, if x was 4 then what we will have on the bottom is 4 - 4.0896

That would definitely give us a 0.0911

In terms of our restricted value, we would say that x can not equal 4, since 4 is restricted.0915

It can not be 4 when we get that 0, let us try the next one.0926

Looking at the bottom of that fraction, I have 5 + a, what would (a) have to be to give me a 0 on the bottom?0931

I'm adding to 5 and thinking about 0 but it is possible if we start thinking negative numbers.0942

In fact what happens if a =-5, based on what you have seen there on the bottom would be 5 - 5 and that would give you 0.0948

My restricted value, we would say that (a) can not be -5, I want that 0 on the bottom.0959

Let us try this one, it looks a little bit more difficult.0968

What can (y) not being what is restricted?0973

I will be subtracting 1 from it but I'm also multiplying by 4.0980

The one thing that is going to do it, I have to borrow my fractions but 1/4 is actually my restricted value.0988

To see this, I'm thinking of taking 1/4 and multiply it by 4, that gives you 1.0995

When we subtract 1 you get 0, you know that (y) can not equal a 1/4.0999

Keep in mind that you can not divide by 0 and sometimes there are simply values that are restricted, you can not use them.1009

One last example here, we will look at the distributive property and how we can use it to simplify some expressions.1020

You may have see me use these lines on top of the number here, actually show the multiplication of what I need to do.1028

Definitely a good practice you should adopt.1035

In the first one, I looked at 2 being multiplied by (a) and 2 being multiplied by (k).1039

This will give me (2 × a) + (2 × k), I will leave that one as it is.1046

Moving on to the next one, here is -5 × (4 - 2x), this one used multiplied by both parts inside those parentheses.1056

-5 × 4 =-20, we will do that first, -5 × -2x.1069

Negative x negative is a positive, we will say10x.1077

One more, you may be looking at that one the same way then why are we not using the distributive property.1085

All I see is a negative sign but imagine that as -1 and you will see that the distributive property will come in handy just fine.1090

We will take -1 and distribute it among both parts on the insides of those parentheses.1098

-1 × -2 = 2 and -1 × 10q= -10q.1104

In all of these examples, we want to take the multiplication and distribute it out over addition.1117

It even works for some of these subtraction problems here because subtraction is just like adding a negative number.1122

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