In this video we're going to talk about different sets of numbers and each of these sets has a particular name vocabulary word that you're going to want to memorize so you might want to get a piece of paper and pencil to write these words down if you're like me it helps to write them down as far as remembering them the first set of numbers we're going to talk about is called the natural numbers or also they're called the counting numbers because they're the first numbers that you use when you learn how to count they start with the number one that's the smallest natural number and they go one two three four five six seven like counting numbers now notice at the end here we've got a dot dot dot what that means is that this set goes on forever so if we kept counting eventually we'd get to 1 million so therefore 1 million is a natural number 1 million in 1 is a natural number and so on and so on and so on the next set of numbers is called the whole numbers now the whole numbers are pretty much exactly like the natural numbers with one exception and that is the number 0 the whole number start at the number 0 and then they just contain all the natural numbers from there so the whole numbers and the natural numbers are pretty much exactly the same except for the number 0 the next set of numbers is called the integers notice what the integer is what we've added are the negative numbers we've got all the whole natural numbers that we had before 1 2 3 4 5 etc we've also got zero in there like the whole numbers but now we've added negative 1 negative 2 negative 3 negative 4 and we have dot dot dot because it continues that way forever as well negative 5 negative 6 negative 7 so negative 843 is an integer now positive 843 is also an integer positive 843 is a whole number and positive 843 is a natural number so it's possible for one number to belong to many different sets and we're going to see how all those all these sets of numbers relate to each other in a little bit one place that you would see integers is on a number line many of you have seen a number line like this before where their numbers are marked 1 2 3 4 5 to positive infinity is eight on its side means infinity and then to the left we have the negative numbers negative 1 negative 2 negative 3 negative 4 negative 5 forever like that the integers do not include any of the numbers between 1 & 2 or between 3 & 4 or between 2 & 3 you'll notice that so far we've got positive numbers we've got negative numbers but we don't have fractions and we don't have decimals yet and those are coming up before we get to those let's look at some places where we would definitely need negative numbers in our lives those temperature is one place here unfortunately I'm not a big fan of negative temperatures but we need negative numbers to express negative temperatures I found this little example from an article the Falcons longest run against the Steelers was just seven yards and Turner got just 19 carries four of his carries went for negative yardage and he has not received more than 20 carries since week five in 2009 what does negative yardage mean in football well those of you that are familiar with football know what the line of scrimmage is that's where the ball is placed when it's hiked if the guy gets tackled behind the line of scrimmage it's considered negative yardage or a loss of yards and in totaling his total yardage for the day they would count that as a negative number if he got tackled behind the line of scrimmage this is an inappropriate use of negative numbers I think somebody I found this picture is kind of funny I think somebody accidentally punched in negative one is a baseball game here negative one balls in five strikes I don't think so some places you wouldn't use negative numbers if we're talking balls and strikes we would want hole numbers for that don't need the integers to talk about balls and strikes earlier I mentioned that we didn't have fractions and decimals so here comes the rational numbers this gives us our fractions and our decimals we can't list out all the rational numbers like we did with the other sets of numbers so we have to describe them and they're described as a rational number is any number that can be written as a fraction of two integers so you've got to remember what your integers are remember the integers was negative four well it was dot dot negative 4 negative 3 negative 2 negative 1 0 1 2 3 4 5 .

Those were our integers I've thrown in some other integers here that are not on the list negative 6 negative 10 negative 240 when I say not on the list I mean that they're sort of in the dot-dot-dot part I didn't write them out that far but they certainly are integers so all these numbers here are integers to make a rational number we're going to make a fraction of these two integers so we're going to take one of these integers and put it in the numerator let's pick two then we're going to take another one of these integers and we're going to put it in the denominator say 3 by doing that we've created a new number that number here is 2/3 so 2/3 is a rational number this is how we're going to create all of our fractions by taking two integers and making a fraction out of them one on the top one on the bottom now if you remember 2/3 as a fraction can also be expressed as a decimal the decimal for 2/3 is 0.6 repeating so 0.6 repeating is a rational number because it can be expressed as a fraction it doesn't have to be written as a fraction to be rational it just has to be equal to some fraction so you've got to remember that your decimals many of your decimals can be written as fractions almost all of them as a matter of fact if that decimal repeats like this point 6 repeating point 6 that bar right there means that that this bar right there means that 6 goes on forever so this is the same as zero points I can't write very well with this 6 6 6 6 that's terrible dot dot forever you get the idea let's try another one let's make another rational number I'm going to pick a negative number for the numerator and 5 for the denominator negative 1 / 5 that's the same as negative 1/5 and the decimal for that is point 2 so here's another example of a rational number with these two examples we see the two different kind of decimals we can have to make a rational number we can either have a decimal that stops like point two and it's done or a decimal that repeats point six six six six if the decimal stops or repeats then you're going to be able to write it as a fraction of two integers and therefore it would be rational let's look at another example what if the top number is bigger than the bottom number 40 over 3 well we know that is an improper fraction improper fractions can be written as mixed numbers so those mixed numbers are also rational we could also write the decimal form of that mixed number which is 13.

3 bar all these different ways to write 40 over 3 are all rational numbers because they are equivalent to 40 over 3 which is a fraction of two integers let's look at another example negative 10 over 4 this fraction can be reduced to negative 5 over 2 so therefore it is a rational number negative 2 and 1/2 is the mixed number for that improper fraction and as a decimal negative 2 point 5 so all these four different ways are just ways to express the same amount which is an amount that can be written as a fraction of two integers negative 10 over 4 or negative 5 over 2 therefore all these are rational numbers let's do one last example 826 over 1 what's 826 divided by 1 hopefully you said 826 now if you remember 826 was a natural number and also a whole number and also an integer so all those natural numbers whole numbers and integers you could write them as that number over 1 and therefore they are all rational and we're going to see how they all relate to each other coming up here's some examples of where you might see rational numbers in your real life this is the stock market even though this is expressed as a decimal all these are expressed as decimals point 30 point 9 6 that's the same as 30 and 96 one hundredths so therefore it can be written as a fraction and therefore it is a rational number you've got positive rational numbers and negative rational numbers this day the Nasdaq is down that's why I usually see it in red rational numbers another place other places you see rational numbers tape measures have the all these fractions those are rational numbers fractions we've got measuring liquids weight measuring weight things are not always exactly a pound you can have fractions or decimals in there we need those rational numbers and of course money 25 cents is 0.

25 that's a rational number okay we're going to digress here for a minute and talk about square roots in order to introduce the idea of numbers that are not rational if you haven't seen a square root before let's go ahead and look at this you've got square root of 25 that means what time's itself is 25 what time's itself is 25 5 times itself is 25 so the square root of 25 is 5 the square root of 9 is 3 because 3 times 3 is 9 the square root of 49 is 7 because 7 times 7 is 49 what's the square root of 11 what times itself would be 11 3 times 3 is 9 4 times 4 is 16 it's got to be somewhere between 3 & 4 it's going to turn out that it is an irrational number here's the square root of 11 it's going to turn out to be three point three one six six two four on and on and on forever is not a nice number the decimal does not repeat and it does not end that those are called irrational numbers if you look up irrational number it's going to say that it's number that's not rational well that seems obvious right let's think about that a little bit further remember rational numbers had decimals that end it or repeat it so an irrational number is going to be a decimal that never ends and never repeats these are all examples common examples any square root that's not nice all these guys right here any square roots that are not nice are going to be irrational numbers you'd think you wouldn't see irrational numbers out there in nature but it turns out if you have a square that's one foot on each side this diagonal length across here is the square root of two feet even a simple shape like a square has got an irrational number for the length of its diagonal another place where you see irrational number is PI I know you guys have heard of Pi before pi is circumference divided by diameter so if you have the diameter the distance across a circle how many times will that diameter go around your circle it'll go around once twice three times and a little bit more as a matter of fact it'll go around 3.

1415926 times irrational number goes on forever never repeats and never stops so here's how all these things relate to each other we've got our natural numbers our whole numbers are integers rationals and our air rationals all these numbers together make up what's called the real numbers notice how I've got these piled on top of each other in what's called the hierarchy every natural number is a whole number that's why it's underneath the whole numbers and every whole number is in the integers and every integer is a rational number so each number belongs to every set above it another way to look at it would be in this sort of a circle diagram all the natural numbers live in here and they live inside the whole number so every natural numbers a whole number and every whole number is an integer and every integer is rational and every rational number is real and then over here we've got this other set of real numbers which is your irrational numbers so they all relate to each other let's look at a couple different examples of problems that you might see to which set or sets does the given number belong so let's say we have negative 6 and here's the question is it a natural whole integer rational irrational or real number which one of those does it belong well let's find negative 6 is it natural that's for sure it's not that it's not a whole number because there's no negatives there it is definitely an integer because it's going to be in this set now since it's an integer it's also rational and it's also real because it belongs to every set above it let's look at another example three and five-eighths what do you think where would that go well there's no fractions in the natural numbers there's no fractions in the whole numbers there's no fractions in the integers here we go rational numbers there it is it is a rational number because it can be written as 27 over 8 which is the fraction of two integers and since it's rational it's also real any of these numbers you work with they're going to be real there's a whole nother set of numbers that are not real but we're not going to talk about that you might run into that later on in your math career but don't worry about it now a couple more examples square root of 7 you remember square roots where do they go that's right they're irrational they have decimals that never repeat and never end irrational numbers are also real they belong to the real numbers so square root of seven would be irrational and real six point eight seven five well 0.

875 is the same as 875 over a thousand notice this decimal terminates it finishes it stops so therefore it is a rational number and since it's rational it's also real last one eighty seven well eighty seven would definitely be in this list right here of the natural numbers if we started counting four five six seven eight nine that we'd eventually get to eighty-seven now since its natural it's going to belong to everything above it it's going to be a whole number an integer a rational number and a real number so it's possible for one never to belong to multiple sets of numbers well I hope this gives you a good start on different sets of numbers and be sure to memorize those definitions.