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Lecture Comments (29)

1 answer

Last reply by: Professor Eric Smith
Fri Aug 26, 2016 7:01 PM

Post by Summer Breeze on June 22 at 09:43:37 PM

Hello Eric, you said in one of the slides that imaginary numbers are numbers with i and constant like 'pie', but in this exercise, you categorize 'pie' as a real number. Can you please tell me which numbers fall under immaginary and where constants like 'pie' and 'e' go? thanks!

1 answer

Last reply by: Professor Eric Smith
Fri Aug 26, 2016 7:02 PM

Post by Summer Breeze on June 21 at 08:39:11 PM

Hello Eric, Can I consider all numbers to be real numbers except for imiginary numbers or the square roo of a negative number?

1 answer

Last reply by: Professor Eric Smith
Fri Aug 26, 2016 7:03 PM

Post by Summer Breeze on June 21 at 08:30:50 PM

Hello Eric, at the highest level, we have real and imaginary numbers; in your diagram, you have irrational under Real umbrella. Should irrational numbers fall under the imaginary category?

1 answer

Last reply by: Professor Eric Smith
Fri Aug 26, 2016 7:04 PM

Post by Summer Breeze on June 21 at 08:21:06 PM

Hello Eric! Can you please share the easiest way to convert a repeated decimal number like 0.161616 into a fraction?

1 answer

Last reply by: Professor Eric Smith
Fri Aug 26, 2016 7:06 PM

Post by Summer Breeze on June 21 at 08:19:02 PM

Hello! In your explanation, you mentionned that rational numbers are those that can be writen as a fraction and are terminating numbers, but 3/17 when converted to fraction does not terminate, why is rational?

2 answers

Last reply by: francisco marrero
Mon Aug 24, 2015 6:24 PM

Post by francisco marrero on July 22, 2015

Do you recommend a book to practice more equations?

1 answer

Last reply by: Professor Eric Smith
Tue Dec 30, 2014 3:39 PM

Post by Brad Cure on December 28, 2014

I like your diagram of "Basic Number Types"  It provides a nice simple global view of numbers that is easy to understand.  

3 answers

Last reply by: Professor Eric Smith
Wed Mar 18, 2015 10:25 PM

Post by Mohammed Jaweed on December 25, 2014

How do you plot a fraction on a number line.

3 answers

Last reply by: Douglas Williams
Tue Jan 14, 2014 8:58 AM

Post by Douglas Williams on January 7, 2014

Dear Mr. Smith, I just joined educator.com and have been studying a lot, I took a break and I was looking at some watches, the first watch I saw was water resistant to 30 meters, so I wanted to convert that to feet. so I found out that 1 m = 3.280 ft. So I got to wondering, what is the .280 ft. in inches? I know that .280 = (280/1000) but who ever heard of a thousandths of an inch? So I like painted myself into a corner, and I have been trying to just logically think it through, I drew a picture of a two number lines, etc, I just can not yet figure out how to get a precise conversion. Logically I know that 30 meters * 3.280 ft = water depth in ft. but how do I convert the hundredths of an inch to something more like a US fraction? Google says the answer is 3 and 3/8 inches what is the correlation between (280/1000) and (27/8) Do I factor? I can not grasp the concept, totally lost ugh. -Doug

1 answer

Last reply by: Professor Eric Smith
Mon Dec 2, 2013 8:47 PM

Post by Juan Manuel Gallardo on November 29, 2013

this course can help me for college admission tests?

1 answer

Last reply by: Professor Eric Smith
Tue Sep 3, 2013 6:38 PM

Post by Jonathan Traynor on August 31, 2013

You are an outstanding teacher. Thanks for all your help!!!!!

1 answer

Last reply by: Professor Eric Smith
Mon Aug 19, 2013 1:44 PM

Post by Theresa Sharp on August 19, 2013

How do I write a decimal with a repeating block of numbers as a fraction? what is the formula for that?

Related Articles:

Basic Types of Numbers

  • The basic types of numbers are
    • Natural numbers – Also called the counting numbers and contain the numbers 1, 2, 3, 4, 5, 6, …
    • Whole numbers – These include the natural numbers and the number zero. In other words the numbers 0, 1, 2, 3, 4, 5, 6, …
    • Integers – These contain the positive, and negative natural numbers as well as zero. In other words the numbers …-3, -2, -1, 0, 1, 2, 3, …
    • Rational – The rationals contain all real numbers that can be written as a fraction. This includes numbers with an infinite decimal that have a repeated block of numbers, and numbers with a decimal that stops.
    • Irrational – These contain all real numbers than cannot be written as a fraction. This includes numbers with an infinite decimal that do not have a repeated block of numbers.
    • Imaginary – These contain an imaginary part “i”
  • The natural, whole, rational, and irrational numbers are all types of real numbers.
  • A particular number may belong to more than one type. For example, the number 2 is a natural numbers, as well as an integer, a rational, and a real number.
  • You can plot numbers onto a number line. In a number line the smaller numbers are written on the left of the larger numbers.
  • The absolute value of a number is its distance from zero on a number line.

Basic Types of Numbers

To what sets of numbers does the following number belong?
- 8
Real, Integer, Rational
To what sets of numbers does the following number belong?
− [22/60]
Real, Rational
To what sets of numbers does the following number belong?
√5
Real, Irrational
Order the numbers from largest to smallest value:
π, [28/9] , 3.5 , √{10}
[28/9] , π, √{10} , 3.5
To what sets of numbers does the following number belong?
- 2
real, integer, rational
To what sets of numbers does the following number belong?
[5/15]
real, rational
To what sets of numbers does the following number belong?
√{50}
real, irrational
To what sets of numbers does the following number belong?
- 67
real, rational, integer
To what sets of numbers does the following number belong?
− [1/8]
real, rational
To what sets of numbers does the following number belong?
√{100}
real, irrational

*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

Basic Types of 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 0:00
  • Objectives 0:07
  • Basic Types of Numbers 0:36
    • Natural Numbers
    • Whole Numbers
    • Integers
    • Rational Numbers
    • Irrational Numbers
    • Imaginary Numbers
  • Basic Types of Numbers Cont. 8:09
    • The Big Picture
    • Real vs. Imaginary Numbers
    • Rational vs. Irrational Numbers
  • Basic Types of Numbers Cont. 10:55
    • Number Line
    • Absolute Value
    • Inequalities
  • Example 1 13:16
  • Example 2 17:30
  • Example 3 21:56
  • Example 4 24:27
  • Example 5 27:48

Transcription: Basic Types of Numbers

Welcome to www.educator.com.0000

In this lesson we are to take a look at basic types of numbers.0002

We will see that there are many different ways that we can take numbers and start to classify them.0008

I will go over all these types of numbers and more in detail as we see how a number gets into each of the groups.0012

You will also see how you can represent these numbers on a number line.0020

Be handy for say comparing numbers and figure out what it means to take the absolute value of a number.0024

We will also see some symbols on how you can compare numbers meaning to our inequalities.0030

When it comes to numbers you can really break them down into various different groups and classify them according to their properties.0039

The most common types of groups that we can use to classify numbers are the natural numbers, whole numbers, 0046

integers, rational, irrational and imaginary numbers.0052

We will go over each of these groups in more detail.0057

In our first group will take a look at the natural numbers.0064

These are the numbers that do not contain any fractions or any decimals.0068

In fact they are sometimes called the counting numbers because they are some of the first number you learn when counting.0073

They contain the numbers 1, 2, 3, 4, 5 and it does go up from there so you know how we do not have a fractions and decimals and no negative numbers here.0078

In the next group we would start expanding on a lot of that last list just little bit and we also include the number 0.0092

Since we have all of the same numbers that we have before these natural numbers and we have that number 0,0100

you could say that all natural numbers are a type of whole number.0107

Watching a step in a few different times as we go through these groups of numbers, some numbers end up in more than 1 group.0112

An important part of this was that they contain a natural numbers and 0 to be a whole number.0118

Alright continuing around, we can also expand on those numbers by looking at the integers.0126

The integer is not only includes say the natural numbers but the negatives of all of our natural numbers and 0.0133

Again this makes all of our natural numbers and our whole numbers a type of integer.0141

You will see from the list that we got some nice numbers on here like -3, -2, - 1.0148

There is 0,1, 2, 3, 4 all the way up on that side.0154

The rational numbers are probably one of our most important groups.0161

These include all numbers that can be written as a fraction.0166

Now there is many different types of numbers that you can write as a fraction.0171

In fact all the numbers that we just covered previously can easily be turned into a fraction by putting them over 1.0176

A harder one says you determine whether you can write them as a fraction or not, or the one's that involve decimals.0183

Here is how you can tell if they are rational or not.0189

If that decimal terminates that means that stops, then you know you can write it as a fraction therefore it is rational.0193

If your decimal goes on and on forever and has a repeated block of numbers, then you may also write those as fractions, they are rational.0200

To help you figure out some of these, let us look at a few examples and see why they are all types of rational numbers.0209

The first one I'm looking at here is 3/17, we know how this one is already a fraction.0217

It is a pretty obvious choice that you can write as a fraction, it is rational.0224

This number 4 could have been one of our numbers on our natural number list and it is also one that we can write as a fraction fairly quickly by simply putting it over 1.0231

Since we can write as a fraction we know it is a type of rational number.0246

Some of the more difficult one, these are the ones that involve decimals.0251

In .161616 repeating of this one goes on and on forever and ever but it has about 16 to just keep repeating over and over again.0255

That is what I mean by repeated block of numbers.0266

Since it has a repeated block, it can be written as a fraction.0269

In fact, this one is written as 16/99, I know that it is a type of rational number.0272

The next one, 0.245 and then it stops, because it stops this is a type of terminating decimal.0279

It can be written as a fraction as well that we can count up the number of places in it and just put it over that number.0289

It is tens, hundreds, thousands, written as a fraction.0296

Look for these types of numbers when determining out your rational numbers. 0300

If we know what numbers can be written as a fraction, then we must also talk about the numbers that cannot be written as a fraction.0309

These types of numbers are irrational numbers. 0315

We saw many different types of numbers that could be written as a fraction.0319

They seem like they are might not be a whole lot that you can not write as a fraction0322

but it turns out there is many common numbers that simply cannot be written as a fraction.0327

More of the common ones are roots that cannot be reduced to any further.0331

If you have a decimal that goes on forever and does not have a repeated block of numbers in it, then that is a type of irrational number.0336

There are also many famous constants which happen to be irrational numbers.0345

They show up in many different areas.0349

Looking at my examples below to see why they are irrational numbers.0351

Here when looking at the square root of 57, I know that this does not reduce.0357

That gives you decide to punch this one into the calculator.0365

You will see that is has a decimal that just keep going on and on forever, if it does not have a repeated block of number.0368

That is how I know that that one is irrational.0374

That one is a little bit more clear to see because I can actually look at its decimal and see it has no repeated blocks and yet it goes on and on forever, it is irrational.0378

It is a very curious number and the variable it is one of those famous constants.0388

Pi is equal to 3.141592 and then it keeps going on and on forever.0393

And it does not have a repeated block of numbers in it, I know that it is irrational.0401

All the types of numbers we have cover those far are actually types of real numbers.0409

There is another group that is completely distinct from those real numbers.0414

Those are the imaginary numbers and you can usually recognize those ones because no contain an imaginary part with i in it.0418

The reasons why these will be so important is some equations might only have imaginary numbers as solutions.0426

We will learn more about these imaginary numbers in some future lessons.0433

As I said before, they are completely separate from our real numbers.0439

You would not have an imaginary number that also ends up on our list for real numbers, completely different things.0443

Here are some examples of some imaginary numbers.0449

I'm looking at 2i, I see that it has the (i) right next to it.0452

Definitely imaginary, 1/2 + 5/7i, I can see that (i) is in there.0456

This is one of our complex numbers but you know it is an imaginary number for sure.0464

At the end here I have the square root of -1, I do not see any (i) in there and why it could be an imaginary number.0470

We will learn that imaginary numbers come from taking the square root of negative quantities.0478

In fact, the square root of -1 is equal to (i), it is actually is an imaginary number. 0484

To understand why some numbers get to be on multiple groups, you have to take a step back and look at the big picture for this classification.0491

I'm trying out a nice diagram so you can see what numbers end up in which groups.0500

The most important distinction that you could make between numbers is probably whether they are real or imaginary.0506

Since those groups are completely separate.0512

Those in the real category we can go further and start breaking that down into many other different types of numbers.0515

Again, we do that according to the properties.0521

The most important distinction we make is whether we can write it as a fraction, we call these rational.0524

Or whether we can not write those as fractions, we call these irrationals.0529

That is how I'm connecting things with arrows here.0535

I'm doing that to show how these categories break down.0539

Rational numbers are types of real numbers and irrational numbers are types of real numbers.0543

Continue on with those numbers that can be written as fractions, those are the rational.0550

We move on to integers.0556

You will notice at that stage at we can drop with all our fractions, we do not have decimals anymore.0559

Now we have numbers like -2 , -1, 0, 1 and we go on from there.0563

As we continue classifying them, we get to our whole numbers.0571

In these ones now, we do not have any more negatives.0576

We start at 0, we have 1, 2, 3 and we go up from there.0579

On to our primal simplest list, those are the natural numbers.0585

They start at 1,2, 3 and they go up from there.0590

Remember, these ones are known as our counting numbers.0594

One way that you can use this diagram to help you classify numbers0599

is to know that if a number ends up in one of these categories it is also in all of the categories above it.0602

We can see this happen for some of our numbers.0609

Let us take the number 2, I see that it is definitely on my natural number lists, but it is also a type of whole number.0612

In addition, is a type of integer and I can take 2 and write it as a fraction.0621

It is a type of rational number which is of course a type of real number.0627

2 gets to be in all of those categories above it.0633

I will also take one that is not in quite as many groups.0638

For example let us just take the square root of 3, it is an irrational number.0640

But it is also in a category above it, it is a square root of 3 and it is a type of real number.0646

Okay, not bad.0653

Now we know a little bit more about the different types of numbers.0657

We will show you how you can visualize a great way to compare them using what is known as a number line.0660

On a number line, we draw out a straight line and mark out some key values such as like -3, -2, all the way up from there.0668

We put the numbers that are smaller on the left and the larger numbers on the right.0679

In this way I can make good comparisons between numbers.0688

You can see that 0 is on the left side of 3, we could say that 0 is less than 3.0692

It is handy to be able to visualize numbers in this way when looking at their absolute value.0701

The absolute value of the number is its distance from 0 on a number line.0707

It is a quick example may be looking at the absolute value of 2.0713

Since 2 is exactly 2 away on a number line, I know that the absolute value of 2 is 2.0718

We will start another one, how about the absolute value of -3.0727

That one I can see is exactly 3 away on a number line, its absolute value is a +3.0733

We might develop some shortcuts and say wait a minute, the absolute value just takes the number and always makes a (+).0741

That is okay, that is the way it should work that is because our distances are always (+).0746

As long as we can go ahead and compare the numbers, we might as well pick up some new notation for doing this.0753

You can compare numbers using inequalities and use the following symbols.0758

You can use greater than, less then, greater than or equal to and less than or equal to.0763

The way these symbols work, is you want put the smaller number with the smaller end of the inequality sign.0770

And the larger end of the inequality sign with the larger number.0780

It could say something like -3 < 5, that would be a good comparison between the two.0787

We have seen a lot about classifying numbers and comparing them.0798

Let us go ahead and practice these ideas by classifying the following numbers.0801

Let us say from the list that all of the groups that the following numbers belong to.0806

Let me start with 2/3, first I think is 2/3 a real number or an imaginary number.0810

I do not see any (i) on it so I will call this a real number.0817

Now, I need to decide can I write it as a fraction or not.0824

This one is already a fraction I know that I can write as a fraction for sure.0829

I will call this a rational number.0833

Moving on from there, in my integers those containing numbers like -3, -2, -1, 0 end up from there.0837

That is how the integers, we do not have fractions, we do not have decimals.0846

This one does not get to be in the inter group or anything below that for that matter.0849

I could say 2/3 is a real number and I could say that 2/3 is a rational number.0854

Let us try another one of these, 2.666 repeating.0861

I do not see an imaginary part so I will say that this is definitely a real number.0865

We can not write it as a fraction, why do you see it has a repeated block of numbers that goes on and on forever, it is a type of rational number.0871

What else can I say? Is it an integer? No, it has the decimal part on it, it is not an integer.0883

I will leave that one as it is, moving on, the square root of 3.0892

This is a type of real number, it does not have any imaginary part on.0898

Can we write this one as a fraction or not? This one I can not.0903

In fact, when you look at the decimal, it goes on and on forever and it does not have that repeated block of numbers, irrational.0907

Since we do not have any more distinct groups of below irrational, we will go ahead and stop classifying that one.0918

Onto some other numbers, -5 that is a type of real number.0924

It looks good, can we write as a fraction?0931

You bet we will simply put it over 1, it is rational.0934

Is it an integer? it does not have any fractions, it does not have any decimals, I will say that it is an integer.0941

Is it a type of whole number? that is where I need to stop.0952

Whole numbers do not contain negative numbers.0956

-5 is real, it is rational and it is an integer.0959

On to the number 0, this one used to be in a lot of different groups.0965

0 is a type of real number.0970

You can write it as a fraction, we will say that it is rational.0975

It is a type of integer, since it is in between our negative numbers and our positive numbers.0983

It is definitely a whole number.0993

That is where this one stops getting classified because the natural numbers start at 1 and then go up from there.1002

One more, let us classify 9, this one is a type of real number.1009

We can write it as a fraction, I know that it is rational, it is definitely on our list of integers.1016

It is also a type of whole number and we can go just a little bit further with this one.1028

This is a type of natural number.1035

9 used to be in a lot of different groups.1039

It is a type of real number, a rational number, it is an integer, it is a whole number and is a type of natural number.1041

Let us try this in a slightly different way.1050

Here I have a giant group of numbers, we want to list out whether the numbers in some of our various different groups like imaginary, real, or irrational.1052

That way we can think of visualizing, classifying them in just a slightly different way.1060

Let us start out with the first one.1067

I want to figure out all the groups that -7 belongs to.1068

I know that it is a type of real number, let us go ahead and put it into that group.1072

Can we write this as a fraction or not, yes I can write it as a fraction.1078

Let us put it in our rational category.1081

Is it a type of an integer? Yes it is on my integer lists.1086

Is it a type of whole number? No, because our whole numbers do not contain negative.1091

We will stop classifying that number.1096

Let us try another one, negative the square root of 3, that is another type of real number.1098

However, that one I can not write as a fraction.1107

I better put it in the irrational category and then that one stop.1111

Moving on, -0.7 it is a type of real number.1117

This one can be written as a fraction, it is -7/10.1125

Let us go ahead and put it in our rational category.1129

Can we go any further from there?1134

Unfortunately not, because it contains those decimals and integers some contain decimals.1136

We can stop classifying that one.1142

Moving on to 0, 0 is a type of real number.1146

It is a type of rational, it is a type of integer and it is a type of whole number.1151

It gets to be in a lot of different groups.1160

Remember, it is not a natural number since that starts at 1 and goes up.1162

On the 2/3, that one is definitely a real number and since it is already a fraction, I know it is a rational number.1167

It is not an integer since it is a fraction, 2/3.1178

The square root of 11, it is a real number, it does not contain an imaginary parts.1187

This one cannot be written as a fraction and I will put it in the irrational category and then stop classifying that one.1194

On to our famous number here, pi.1202

Pi is a type of real number, even though it is a little unusual, it does go on and on forever.1207

It is a type of real number and it is irrational since I cannot write it as a fraction.1212

We will stop classifying them since there is no two groups below irrational.1220

On to the number 8, this one is going to be in a lot of different groups.1226

It is a type of real number, I can write as a fraction by putting it over 1.1231

It is on our integer lists, it is on our whole number list and it is a type of natural number, a lot of different things.1237

On to 15/2, I will say that that is a type of real number.1247

I can write it as a fraction, let us put it in our rational category.1254

Unfortunately it is not an integer so I will not put it in that one.1259

Then number12, 12 is a type of real number.1266

We can write it as a fraction by putting it over 1, let us put in rational.1271

It is a type of integer, it is a type of whole number and since the natural number starts at 1 and then goes 2, 3, 4.1276

All we have from there I know that it is a natural number.1285

Just one more to do, the number 3i.1291

I have to throw an imaginary number on my list so it will immediately drop that into the imaginary bin.1295

And that is all the more classifying we will do with that one.1301

Since again imaginary numbers and real numbers are completely distinct from one another.1305

What you will know is that most of these categories are all types of real numbers.1310

We have classified numbers, what gets better about comparing them on a number line or just being to plot them out.1318

The way we plot out a number on a number line is we find it.1325

Say using one of our markers below and put a big (dot) to where it is.1330

If I want to graph something out like 3 on a number line, I will find 3 and I will place a big old dot right at 3.1333

Once I applied it out, I can do some good comparisons.1342

We can see that since 3 is to the left of 4, that 3 is less than 4.1346

Since 3 is on the right side of -1, 9, 0, 3 is greater than -1.1355

Let us spot out a few more, -2 on our number line.1361

We would find -2 and go ahead and put up the big old dot there.1366

When it gets in to fractions and decimals it does get a little bit more difficult but you can still put these on the number line as well.1371

This one is 5/3 and I do not see any 5/3 in my markers here on the bottom.1378

What I can do is I can break down each little section into thirds and mark out the 5th one.1384

1/3 and more thirds and more thirds.1390

We are looking for 5/3, 1,2, 3, 4, 5, we put that big dot right here.1395

Now we can better compare where 5/3 is into other numbers.1403

5/3 < 2 but it is greater than 1.1407

Alright, -3.75 that would be the same as -3 and 75/100.1414

That can also be written as -3 and 3/4.1425

That tells me I need to break down my number line into quarters.1430

1/4, 1/4, 1/4 and 1/4.1436

I'm looking to mark out 3 whole sections and then 3 quarters.1446

And we are going the negative directions 3,1, 2, 3 and we will put up the old dot there.1451

We can see that -3.75 > -4 and it is also less than a -3.1458

Let us use our number lines so that we can actually line up various different numbers and see which ones are smaller than the other ones.1470

Be just a rough sketch of the number lines, I'm not going to be too accurate with my thick marks.1481

But I just used it so I know how they compare to one another.1487

Let us go ahead and start with our first number here and put -7 on a number line.1493

Since it is a negative number, I'm going to aim for somewhere on the left side here -7.1498

-3 is a little bit more than that, I will put it on the positive side over here.1508

Let me put a spot there for 3.1515

-0.7, that is not very big and is definitely larger than -7 and less than 3.1519

Let us go ahead and put it right here - 0.7.1527

0 is a good number and put it greater than -0.73.1539

2/3 is larger than 0, I will put on the right side.1550

Alright on to something little bit trickier, the square root of 15.1561

I know that that is less than 4, since the square root of that 16th is something a bit larger will be on the right side.1566

It is greater than 3, since the square root of 9 would be 3.1573

I'm going to put this one larger than 3, square root of 15.1577

It is a good one, definitely larger than square root of fifteenths.1587

-7/2, that one is about -7 1/2, I mean -3 1/2.1597

Let us put that one down here -7 1/2, -5 and one more number pi.1606

3.1415 a little bit larger than 3, put it a little bit larger than 3.1625

Now that we have used our number line, it gets some comparisons among all these.1636

We will simply list them from smallest all the way up to largest.1641

-7, -5, -7/2, -0.7, 0, 2/3, 3, pi, square root of 15 and 8, not bad.1646

For this last example, we will go ahead and use our inequality symbols like less than or greater than to go ahead and compare these 2 numbers.1670

If you want you can use a number line to plot them out before using these symbols.1676

Let us try the first one, comparing 6 and 2.1684

When I plot these out, 2 is on the left side of 6.1690

I know that 2<6, it is my smaller number, I will drop my inequality symbols so that I show that 2 is less than 6.1695

I can also say that 6>2.1705

Let us try another one, -7 and 5.1711

It is tempting to say that -7 is bigger but our negatives are on the left side and our positives are on the right side.1722

You can see that -7 is less than 5.1730

Let us write that out, -7<5.1735

-5 and -3, -5 is further down the -3, I know that it will be less than -3.1745

One more 2.3 and 5.7, 2.3, 5.7 will be much larger.1765

I know the 5.7 > 2.3 or in the order that they are in 2.3 <5.7.1780

These symbols are handy and in showing the comparison especially where they are on a number line.1790

One thing I did not use here is the or equals to symbol.1796

I could have put that in for all of these spots, 6 is greater than or equal to 2.1802

Or I could have said -7 is less than or equal to 5.1807

That is because it also takes into the possibility that the numbers could have been equal.1812

The reason why I did these is I can see that all of the numbers are not equal.1819

And it is a little bit more flexible when using this other one.1826

Watch for the or equal to symbol to show up when we are doing a lot of our inequalities, these ones are good.1831

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