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### Properties of Real Numbers

- The commutative property of addition and multiplication says that the order of addition or multiplication does not matter.
- The associate property of addition and multiplication says that the grouping of addition or multiplication does not matter.
- The distributive property says that we can distribute multiplication over addition.
- When 0 is divided by a number the result is zero, however we may not divide by zero, it remains undefined.
- When you divide a number by 1 you get the same number back. A number divided by itself is one.
- When multiplying by zero, the result is zero. Multiplying by 1, the number remains unchanged.
- When adding zero to a number, the number remains unchanged.

### Properties of Real Numbers

- 8

− [22/60]

√5

π, [28/9] , 3.5 , √{10}

- 2

[5/15]

√{50}

- 67

− [1/8]

√{100}

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

### Properties of Real Numbers

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro
- Objectives
- The Properties of Real Numbers
- Commutative Property of Addition and Multiplication
- Associative Property of Addition and Multiplication
- Distributive Property of Multiplication Over Addition
- Division Property of Zero
- Division Property of One
- Multiplication Property of Zero
- Multiplication Property of One
- Addition Property of Zero
- Why Are These Properties Important?
- Example 1
- Example 2
- Example 3
- Example 4

- Intro 0:00
- Objectives 0:07
- The Properties of Real Numbers 0:23
- Commutative Property of Addition and Multiplication
- Associative Property of Addition and Multiplication
- Distributive Property of Multiplication Over Addition
- Division Property of Zero
- Division Property of One
- Multiplication Property of Zero
- Multiplication Property of One
- Addition Property of Zero
- Why Are These Properties Important?
- Example 1 9:16
- Example 2 13:04
- Example 3 14:30
- Example 4 16:57

### Algebra 1 Online Course

### Transcription: Properties of Real Numbers

*Welcome back to www.educator.com.*0000

*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

*Thanks for watching www.educator.com.*1130

1 answer

Last reply by: Professor Eric Smith

Wed Nov 12, 2014 11:19 AM

Post by levon guyumjan on November 11, 2014

I truly enjoy your explanations, you do have the ability to be a teacher thank you for your hard work.

1 answer

Last reply by: Professor Eric Smith

Wed Jul 30, 2014 2:41 PM

Post by Kitt Parker on July 18, 2014

[28/9] , Ï€, âˆš{10} , 3.5

I was about to ask how do we know the size of N, but I believe its Pi. On the practice question it looks like an N.

0 answers

Post by Kitt Parker on July 18, 2014

I might have missed it, but I believe this learning experience could be improved with printable work sheets of questions and also questions that refer back a number of sections. Id like to print out a sheet of questions, work them and then refer back to sections as necessary. Just an idea.