WEBVTT mathematics/algebra-1/smith
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Welcome to www.educator.com.
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In this lesson we are to take a look at basic types of numbers.
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We will see that there are many different ways that we can take numbers and start to classify them.
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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.
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You will also see how you can represent these numbers on a number line.
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Be handy for say comparing numbers and figure out what it means to take the absolute value of a number.
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We will also see some symbols on how you can compare numbers meaning to our inequalities.
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When it comes to numbers you can really break them down into various different groups and classify them according to their properties.
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The most common types of groups that we can use to classify numbers are the natural numbers, whole numbers,
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integers, rational, irrational and imaginary numbers.
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We will go over each of these groups in more detail.
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In our first group will take a look at the natural numbers.
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These are the numbers that do not contain any fractions or any decimals.
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In fact they are sometimes called the counting numbers because they are some of the first number you learn when counting.
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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.
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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.
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Since we have all of the same numbers that we have before these natural numbers and we have that number 0,
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you could say that all natural numbers are a type of whole number.
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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.
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An important part of this was that they contain a natural numbers and 0 to be a whole number.
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Alright continuing around, we can also expand on those numbers by looking at the integers.
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The integer is not only includes say the natural numbers but the negatives of all of our natural numbers and 0.
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Again this makes all of our natural numbers and our whole numbers a type of integer.
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You will see from the list that we got some nice numbers on here like -3, -2, -1, 0.
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There is 0, 1, 2, 3, 4 all the way up on that side.
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The rational numbers are probably one of our most important groups.
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These include all numbers that can be written as a fraction.
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Now there is many different types of numbers that you can write as a fraction.
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In fact all the numbers that we just covered previously can easily be turned into a fraction by putting them over 1.
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A harder one says you determine whether you can write them as a fraction or not, or the one's that involve decimals.
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Here is how you can tell if they are rational or not.
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If that decimal terminates that means that stops, then you know you can write it as a fraction therefore it is rational.
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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.
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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.
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The first one I'm looking at here is 3/17, we know how this one is already a fraction.
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It is a pretty obvious choice that you can write as a fraction, it is rational.
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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.
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Since we can write as a fraction we know it is a type of rational number.
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Some of the more difficult one, these are the ones that involve decimals.
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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.
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That is what I mean by repeated block of numbers.
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Since it has a repeated block, it can be written as a fraction.
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In fact, this one is written as 16/99, I know that it is a type of rational number.
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The next one, 0.245 and then it stops, because it stops this is a type of terminating decimal.
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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.
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It is tens, hundreds, thousands, written as a fraction.
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Look for these types of numbers when determining out your rational numbers.
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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.
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These types of numbers are irrational numbers.
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We saw many different types of numbers that could be written as a fraction.
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They seem like they are might not be a whole lot that you can not write as a fraction
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but it turns out there is many common numbers that simply cannot be written as a fraction.
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More of the common ones are roots that cannot be reduced to any further.
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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.
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There are also many famous constants which happen to be irrational numbers.
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They show up in many different areas.
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Looking at my examples below to see why they are irrational numbers.
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Here when looking at the square root of 57, I know that this does not reduce.
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That gives you decide to punch this one into the calculator.
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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.
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That is how I know that that one is irrational.
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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.
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It is a very curious number and the variable it is one of those famous constants.
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π is equal to 3.141592 and then it keeps going on and on forever.
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And it does not have a repeated block of numbers in it, I know that it is irrational.
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All the types of numbers we have cover those far are actually types of real numbers.
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There is another group that is completely distinct from those real numbers.
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Those are the imaginary numbers and you can usually recognize those ones because no contain an imaginary part with i in it.
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The reasons why these will be so important is some equations might only have imaginary numbers as solutions.
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We will learn more about these imaginary numbers in some future lessons.
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As I said before, they are completely separate from our real numbers.
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You would not have an imaginary number that also ends up on our list for real numbers, completely different things.
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Here are some examples of some imaginary numbers.
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I'm looking at 2i, I see that it has the (i) right next to it.
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Definitely imaginary, 1/2 + 5/7i, I can see that (i) is in there.
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This is one of our complex numbers but you know it is an imaginary number for sure.
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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.
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We will learn that imaginary numbers come from taking the square root of negative quantities.
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In fact, the square root of -1 is equal to (i), it is actually is an imaginary number.
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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.
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I'm trying out a nice diagram so you can see what numbers end up in which groups.
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The most important distinction that you could make between numbers is probably whether they are real or imaginary.
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Since those groups are completely separate.
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Those in the real category we can go further and start breaking that down into many other different types of numbers.
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Again, we do that according to the properties.
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The most important distinction we make is whether we can write it as a fraction, we call these rational.
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Or whether we can not write those as fractions, we call these irrationals.
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That is how I'm connecting things with arrows here.
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I'm doing that to show how these categories break down.
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Rational numbers are types of real numbers and irrational numbers are types of real numbers.
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Continue on with those numbers that can be written as fractions, those are the rational.
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We move on to integers.
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You will notice at that stage at we can drop with all our fractions, we do not have decimals anymore.
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Now we have numbers like -2 , -1, 0, 1 and we go on from there.
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As we continue classifying them, we get to our whole numbers.
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In these ones now, we do not have any more negatives.
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We start at 0, we have 1, 2, 3 and we go up from there.
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On to our primal simplest list, those are the natural numbers.
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They start at 1,2, 3 and they go up from there.
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Remember, these ones are known as our counting numbers.
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One way that you can use this diagram to help you classify numbers
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is to know that if a number ends up in one of these categories it is also in all of the categories above it.
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We can see this happen for some of our numbers.
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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.
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In addition, is a type of integer and I can take 2 and write it as a fraction.
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It is a type of rational number which is of course a type of real number.
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2 gets to be in all of those categories above it.
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I will also take one that is not in quite as many groups.
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For example let us just take the square root of 3, it is an irrational number.
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But it is also in a category above it, it is a square root of 3 and it is a type of real number.
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Okay, not bad.
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Now we know a little bit more about the different types of numbers.
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We will show you how you can visualize a great way to compare them using what is known as a number line.
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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.
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We put the numbers that are smaller on the left and the larger numbers on the right.
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In this way I can make good comparisons between numbers.
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You can see that 0 is on the left side of 3, we could say that 0 is less than 3.
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It is handy to be able to visualize numbers in this way when looking at their absolute value.
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The absolute value of the number is its distance from 0 on a number line.
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It is a quick example may be looking at the absolute value of 2.
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Since 2 is exactly 2 away on a number line, I know that the absolute value of 2 is 2.
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We will start another one, how about the absolute value of -3.
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That one I can see is exactly 3 away on a number line, its absolute value is a +3.
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We might develop some shortcuts and say wait a minute, the absolute value just takes the number and always makes a (+).
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That is okay, that is the way it should work that is because our distances are always (+).
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As long as we can go ahead and compare the numbers, we might as well pick up some new notation for doing this.
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You can compare numbers using inequalities and use the following symbols.
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You can use greater than, less then, greater than or equal to and less than or equal to.
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The way these symbols work, is you want put the smaller number with the smaller end of the inequality sign.
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And the larger end of the inequality sign with the larger number.
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It could say something like -3 < 5, that would be a good comparison between the two.
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We have seen a lot about classifying numbers and comparing them.
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Let us go ahead and practice these ideas by classifying the following numbers.
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Let us say from the list that all of the groups that the following numbers belong to.
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Let me start with 2/3, first I think is 2/3 a real number or an imaginary number.
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I do not see any (i) on it so I will call this a real number.
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Now, I need to decide can I write it as a fraction or not.
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This one is already a fraction I know that I can write as a fraction for sure.
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I will call this a rational number.
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Moving on from there, in my integers those containing numbers like -3, -2, -1, 0 end up from there.
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That is how the integers, we do not have fractions, we do not have decimals.
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This one does not get to be in the inter group or anything below that for that matter.
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I could say 2/3 is a real number and I could say that 2/3 is a rational number.
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Let us try another one of these, 2.666 repeating.
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I do not see an imaginary part so I will say that this is definitely a real number.
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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.
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What else can I say? Is it an integer? No, it has the decimal part on it, it is not an integer.
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I will leave that one as it is, moving on, the square root of 3.
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This is a type of real number, it does not have any imaginary part on.
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Can we write this one as a fraction or not? This one I can not.
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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.
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Since we do not have any more distinct groups of below irrational, we will go ahead and stop classifying that one.
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Onto some other numbers, -5 that is a type of real number.
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It looks good, can we write as a fraction?
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You bet we will simply put it over 1, it is rational.
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Is it an integer? it does not have any fractions, it does not have any decimals, I will say that it is an integer.
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Is it a type of whole number? that is where I need to stop.
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Whole numbers do not contain negative numbers.
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-5 is real, it is rational and it is an integer.
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On to the number 0, this one used to be in a lot of different groups.
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0 is a type of real number.
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You can write it as a fraction, we will say that it is rational.
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It is a type of integer, since it is in between our negative numbers and our positive numbers.
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It is definitely a whole number.
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That is where this one stops getting classified because the natural numbers start at 1 and then go up from there.
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One more, let us classify 9, this one is a type of real number.
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We can write it as a fraction, I know that it is rational, it is definitely on our list of integers.
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It is also a type of whole number and we can go just a little bit further with this one.
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This is a type of natural number.
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9 used to be in a lot of different groups.
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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.
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Let us try this in a slightly different way.
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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.
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That way we can think of visualizing, classifying them in just a slightly different way.
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Let us start out with the first one.
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I want to figure out all the groups that -7 belongs to.
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I know that it is a type of real number, let us go ahead and put it into that group.
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Can we write this as a fraction or not, yes I can write it as a fraction.
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Let us put it in our rational category.
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Is it a type of an integer? Yes it is on my integer lists.
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Is it a type of whole number? No, because our whole numbers do not contain negative.
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We will stop classifying that number.
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Let us try another one, negative the square root of 3, that is another type of real number.
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However, that one I can not write as a fraction.
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I better put it in the irrational category and then that one stop.
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Moving on, -0.7 it is a type of real number.
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This one can be written as a fraction, it is -7/10.
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Let us go ahead and put it in our rational category.
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Can we go any further from there?
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Unfortunately not, because it contains those decimals and integers some contain decimals.
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We can stop classifying that one.
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Moving on to 0, 0 is a type of real number.
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It is a type of rational, it is a type of integer and it is a type of whole number.
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It gets to be in a lot of different groups.
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Remember, it is not a natural number since that starts at 1 and goes up.
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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.
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It is not an integer since it is a fraction, 2/3.
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The square root of 11, it is a real number, it does not contain an imaginary parts.
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This one cannot be written as a fraction and I will put it in the irrational category and then stop classifying that one.
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On to our famous number here, π.
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π is a type of real number, even though it is a little unusual, it does go on and on forever.
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It is a type of real number and it is irrational since I cannot write it as a fraction.
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We will stop classifying them since there is no two groups below irrational.
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On to the number 8, this one is going to be in a lot of different groups.
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It is a type of real number, I can write as a fraction by putting it over 1.
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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.
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On to 15/2, I will say that that is a type of real number.
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I can write it as a fraction, let us put it in our rational category.
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Unfortunately it is not an integer so I will not put it in that one.
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Then number12, 12 is a type of real number.
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We can write it as a fraction by putting it over 1, let us put in rational.
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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.
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All we have from there I know that it is a natural number.
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Just one more to do, the number 3i.
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I have to throw an imaginary number on my list so it will immediately drop that into the imaginary bin.
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And that is all the more classifying we will do with that one.
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Since again imaginary numbers and real numbers are completely distinct from one another.
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What you will know is that most of these categories are all types of real numbers.
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We have classified numbers, what gets better about comparing them on a number line or just being to plot them out.
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The way we plot out a number on a number line is we find it.
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Say using one of our markers below and put a big (dot) to where it is.
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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.
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Once I applied it out, I can do some good comparisons.
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We can see that since 3 is to the left of 4, that 3 is less than 4.
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Since 3 is on the right side of -1, 9, 0, 3 is greater than -1.
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Let us spot out a few more, -2 on our number line.
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We would find -2 and go ahead and put up the big old dot there.
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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.
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This one is 5/3 and I do not see any 5/3 in my markers here on the bottom.
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What I can do is I can break down each little section into thirds and mark out the 5th one.
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1/3 and more thirds and more thirds.
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We are looking for 5/3, 1,2, 3, 4, 5, we put that big dot right here.
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Now we can better compare where 5/3 is into other numbers.
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5/3 < 2 but it is greater than 1.
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Alright, -3.75 that would be the same as -3 and 75/100.
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That can also be written as -3 and 3/4.
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That tells me I need to break down my number line into quarters.
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1/4, 1/4, 1/4 and 1/4.
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I'm looking to mark out 3 whole sections and then 3 quarters.
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And we are going the negative directions 3,1, 2, 3 and we will put up the old dot there.
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We can see that -3.75 > -4 and it is also less than a -3.
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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.
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Be just a rough sketch of the number lines, I'm not going to be too accurate with my thick marks.
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But I just used it so I know how they compare to one another.
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Let us go ahead and start with our first number here and put -7 on a number line.
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Since it is a negative number, I'm going to aim for somewhere on the left side here -7.
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-3 is a little bit more than that, I will put it on the positive side over here.
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Let me put a spot there for 3.
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-0.7, that is not very big and is definitely larger than -7 and less than 3.
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Let us go ahead and put it right here - 0.7.
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0 is a good number and put it greater than -0.73.
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2/3 is larger than 0, I will put on the right side.
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Alright on to something little bit trickier, the square root of 15.
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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.
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It is greater than 3, since the square root of 9 would be 3.
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I'm going to put this one larger than 3, square root of 15.
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It is a good one, definitely larger than square root of fifteenths.
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-7/2, that one is about -7 1/2, I mean -3 1/2.
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Let us put that one down here -7 1/2, -5 and one more number π.
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3.1415 a little bit larger than 3, put it a little bit larger than 3.
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Now that we have used our number line, it gets some comparisons among all these.
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We will simply list them from smallest all the way up to largest.
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-7, -5, -7/2, -0.7, 0, 2/3, 3, π, square root of 15 and 8, not bad.
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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.
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If you want you can use a number line to plot them out before using these symbols.
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Let us try the first one, comparing 6 and 2.
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When I plot these out, 2 is on the left side of 6.
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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.
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I can also say that 6 > 2.
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Let us try another one, -7 and 5.
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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.
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You can see that -7 is less than 5.
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Let us write that out, -7 < 5.
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-5 and -3, -5 is further down the -3, I know that it will be less than -3.
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One more 2.3 and 5.7, 2.3, 5.7 will be much larger.
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I know the 5.7 > 2.3 or in the order that they are in 2.3 < 5.7.
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These symbols are handy and in showing the comparison especially where they are on a number line.
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One thing I did not use here is the or equals to symbol.
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I could have put that in for all of these spots, 6 is greater than or equal to 2.
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Or I could have said -7 is less than or equal to 5.
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That is because it also takes into the possibility that the numbers could have been equal.
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The reason why I did these is I can see that all of the numbers are not equal.
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And it is a little bit more flexible when using this other one.
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Watch for the or equal to symbol to show up when we are doing a lot of our inequalities, these ones are good.
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