okay so in this video we’re going to talk about common sets of numbers so this is pretty important concept in math so these numbers there’s a lot of different types of numbers in math that show up at various places various different courses things like that so it’s important to be able to know the different types of numbers and be able to identify which ones which things like that so we’re gonna talk about them here talk about some related notation so for common sets of numbers the first type of number we’ll talk about is a natural number so the natural numbers here okay so these are important of so that they have their own special little notation here and that notation is this a sort of beefy looking n if you look into n right here so it’s just a letter in with like this beefy legs I guess I don’t like that so n for natural numbers and what are they two noted to be or they’re denoted like this but what are they equal to what are they so the natural numbers are one two three and then dot dot dot okay so again the dot dot dot means continue the pattern continue what pattern the pattern we established here with these first three numbers so one two three then if we keep coming where’d me at four or five six seven eight nine 10 11 12 13 14 and so on and so forth so the natural numbers basically all the counting numbers so they’re also sometimes called the counting numbers because you know these are the numbers we use to count so here’s a one marker two markers and so on and so forth so we use these numbers to count so they’re also called the counting numbers but more formally and more often in mathematical literature like the textbooks and things like that they’re called natural numbers so next the whole numbers okay so the whole numbers aren’t really important enough to have their own notation but for the purposes of this video we’ll just call them capital W and what are the whole numbers defined to be well pretty much all the natural numbers but now with a zero so zero and then all the natural numbers 1 2 dot dot okay so what so what’s the dot the thought mean it means continue the pattern continue what pattern the pattern I established here 0 1 2 3 4 5 6 7 so on and so forth okay so notice this is actually just all the natural numbers we actually already said it and all the natural numbers Union with 0 so be very careful when you talk about it like this you have to have these curly braces around the zero okay because you can only take the union of sets so even though it’s the natural numbers plus one more number stuck in there it’s actually wrong to say this because you can’t Union with just a number you can union with the set whose only element is a single number yeah you can do that yeah but be careful just when you take a union how to take a union of sets okay so this sets and this set Union together you get the whole numbers and I do want to point out that some people do count zero as a natural number but it’s it’s sort of uncommon to do that it depends on who you talk to you really I don’t know if anybody really does that nowadays anymore and it’s also I’ve seen it more often in computer science than I have in math but you know for for most mathematical purposes it’s better just to take the natural numbers to be 1 2 3 Delta thought and without 0 so if you tossing 0 then you’ll have all the whole numbers okay but again the whole numbers aren’t really important enough to have their own universally accepted notation none that I’ve ever seen any way or heard of but it’s more confident just talk about natural numbers and if you need 0 as well then we explicitly just throw in 0 so we say the natural numbers with 0 but anyway you know it all depends on who you talk to you what what you’re reading things like that so that’s natural numbers whole numbers so what do we notice here remember we talked about subsets so we while subsets number a is a subset of B if everything in a is also in B okay so do we notice a special relationship here yeah everything in the natural numbers is also in the whole numbers okay so what we can say so far is n is a subset of W and as a subset of W okay so we’ll come back to this also so what’s next so natural numbers whole numbers now let’s talk about the integers so this is a very important set of numbers here so the integer is very important set and it’s denoted with the capital B Fizi

like this so Z short for integer actually z comes from the german word azulon which means number so that’s that means number in German if I remember correctly so that’s where this notation comes from here okay so what are all the integers well there’s actually a couple different ways of describing this set so we could say we could stay like this dot dots negative 2 negative 1 0 1 2 comma dot dots okay so we’ve never seen anything like this before right with the dots in the beginning and at the end so what does it mean to have does in the beginning well let’s forget about that let’s just look here negative 2 negative 1 0 1 2 dot dot dot means continue the pattern continue what pattern the pattern I established here negative 2 negative 1 0 1 2 looks like I’m just increasing by 1 each time and that’s actually what I am doing so the next number will be 3 4 5 6 7 8 9 and so on and so forth what if I go back the other way to 1 0 negative 1 negative 2 negative 3 negative 4 negative 5 negative 6 negative 7 and so on and so forth okay so the integers basically it’s all of the natural numbers with 0 and all of the negative numbers okay so or we could say all of the whole numbers with all the negative natural numbers as well and now if you don’t like this double ellipses here you could say like this also so that’ll be 0 1 negative 1 2 negative 2 3 negative 3 dot dot dot candy-cane what does that mean it means continue the pattern continue what pattern the pattern I’ve established here 0 1 negative 1 no clear pattern emerges yet but let’s keep going to negative 2 3 negative 3 uh-huh 1 negative 1 2 negative 2 3 negative 3 I’ll bet next is 4 negative 4 5 negative 5 6 negative 6 and so on and so forth okay so that’s another way of describing the set of the integers here okay so these are all types these are all integers here and again you know we’re just listing all of the same elements just in a different order okay so if you don’t like using two of ez Lipsey’s two of the dots you could list it like or you could save describe the set of integers like this and you’ll only have to use one of them okay so that’s another option there anyway what do we notice here how does how do the whole numbers relate to the integers well we see here 0 1 2 0 1 2 knots at a plus other stuff so notice W is actually a subset of Z right because everything in W is also in Z okay everything in here 0 1 2 3 4 5 6 . to thought is also in here 0 1 2 3 4 5 6 right so W is actually a subset of Z another way of saying that is every whole number is an integer and so W is a subset of Z and you can just W it’s not really universally accepted notation here or there’s no universal standard as far as I know as far as I’ve ever seen or heard so we will make this one beefy just for the purposes of our video we’ll see use the W but N and see those are universal notations here for natural numbers and integers okay so notice every natural number is a whole number that’s what this says every natural is a whole number because the natural number is a subset of the whole numbers so every natural number is a whole number every whole number is an integer and so transitively really every natural number is an integer and so let’s continue with this so natural numbers whole numbers integers what’s next what’s next is uh rational numbers K so rational numbers Eanes see they have a this has to work ratio in it so these really do involve ratios or proportions or division fractions things like that so rational numbers are also very very important so they get thrown notation as well a universal notation which is a beefy goofy-looking cue thing so you might see this in textbooks for things like that so the end the Z and the Q they will show up in textbooks and other types of mathematical literature so be sure that you’re able to identify those okay so with rational numbers we can’t really list them out the same way we listed all these out but what we can do is describe them using set-builder notation so we can say the set of all rational numbers that’s the set of all P divided by Q such that P and Q are in Z K so this elements so this means elements of so P and Q are elements of Z so this just means that P and Q are integers right summer busy represents the integers and this notation means element so if we talked about in earlier video and we

also require that Q is not 0 okay why do we require that because you’re not allowed to divide by 0 ever no matter what you’re doing in all of math ever you’re never ever ever allowed to divide by 0 this is not alone so that’s why we just require that Q is nonzero okay so nothing really special there okay so you know when we listed all these out it’s easy to see what’s a natural number what’s a whole number what’s an integer well what’s a rational number hey let’s see some examples of those so if we come over here basically any fraction you can write down where the on the bottom are both integers and the bottoms on zero that’s a rational number and so one two three over four five six okay so these aren’t guaranteed to be in lowest form but anyway one two three over four five six so that’s a rational number right it’s so this is our P over Q they’re both integers right one two three is an integer four five six is an integer and four five six is not zero okay so that’s an example of a rational number what about negative 52 over 117 that’s also a rational number how about 12 over negative 600 yeah 12 over negative 600 okay so of course that’s not reduced right now reduce but it doesn’t matter okay to be a rational number you don’t have to be reduced you just have to be a ratio of two integers or expressible as such anyway so 12 over negative 600 it can be reduced but still it’s a rational number because integer on top in nature on bottom if not zero that’s it yeah but also so to be a rational number you don’t have to be expressed in this form it only has to be possible to express you in that form so let’s say one point seven three two is also a rational number okay even though it doesn’t really exactly match this form but we could write this as 17 32 divided by 1000 okay so 17 32 divided by thousand that’s the same thing as one point seven three two okay so one point seven three two can be expressed in this form like this therefore one point seven three two is a rational number and so basically any decimal like this so any fraction or any decimal that doesn’t repeat infinitely with no pattern and so actually if you even if you repeat infinitely so let’s say if we have like one point seven three two like that remember this notation means the two repeats so it’s 1.73 tu-tu-tu-tu-tu-tu dot dot dot okay so the two just keeps repeating forever even things like that those are also rational numbers okay because it is possible to Express as a ratio of two integers okay it’s kind of weird and kind of tricky and kind of beyond the scope of what we want to talk about but it is possible okay but anyway so these are all examples of rational numbers so decimals that are that terminate that a finite number of decimal places any fraction any ratio of two integers like that where the bottom is not zero or any infinitely repeating decimals where you just repeating the same thing over and over okay so things like this or even things like this let’s say seventy one point eight six two or even if two digits repeat so that’s seventy one point eight six two six two six two six two odd and so even stuff like that that’s also an example of a rational number because it’s possible to express this as a ratio of two integers K and again it’s kind of goofy it’s kind of weird it’s kind of tricky to be able to do that there it is kind of tricky to actually do that but it is the point is it’s possible it can be done okay so again a rational number is any kind of fraction like this or ratio to integers and you kind of decimal like this that terminates after any number of digits here or any kind of infinitely repeating decimal where you have a part that just repeats over and over again okay so it’s important that if it’s an infinite number of decimal places it’s important that you have to have repeating here okay so if you don’t have the repeating and it goes on infinitely then it’s not it’s not rational that’s a different type of number we’ll talk about soon okay so those are all examples of rational numbers what else could be a rational number well hey uh integer / integer what if we say something like twelve over one is that rational yeah okay cuz its integer on top integer on bottom the bottom is not zero but what is 12 over one it’s 12 what kind of number is that it’s a natural number it’s a whole number it’s an integer okay what about negative 12 over 1 well it’s the same thing as negative 12 right so yeah that’s still that still counts right so the point is at the bottom here when you talk about a ratio of two integers the bottom could be if the bottom is one what do you have

you have p / one p / once gives you P what’s P P is an integer okay so basically what the point of trying to make here is that every integer is a rational number okay every integer is just a special kind of rational number where the denominator is one okay so every integer really is a rational number or in other words everything in here everything in here is also in here everything in here is also in here what does that mean that means Z is a subset of key so we can come over here and we could say okay Z is a subset of Q okay okay so natural numbers whole numbers integers rational numbers every natural number is a whole number every whole number is an integer every integer is a rational number okay but again the whole numbers aren’t really as important so we don’t talk about them as much so we could just get that say every natural number is an integer every integer is a rational number and so on okay so two other types we’re really three technically with two we want to talk about in detail here okay so rational numbers the next two times we want to talk about together so there’s rational numbers now there’s also irrational numbers let’s talk about real numbers though so we have irrational numbers and real numbers let’s talk about real numbers first so real numbers and irrational numbers so irrational that’s kind of the counterpart to rational okay so real numbers they’re also extremely extremely important in all of math so the real numbers they also get their own notation and that’s kind of this beefy looking R okay so some people like to connect it over here doesn’t really matter though I don’t really do that but just to have it I guess so anyway B if you look an R type thing R of course for real and it’s it’s harder to describe what these are so basically it’s the real number is the set of real numbers the rational numbers together with the irrational numbers so what’s an irrational number first of all they’re not really important enough to get their own notation so just like with the whole numbers we just gave them one for the video so let’s call these guys I for irrational so be careful don’t confuse that with the integer you know I for irrational but at Z for integer that comes from the German words L and okay okay so irrational numbers this is things like things like the square root of 2 the square root of 3 the square root of 17 pi so know this number PI from you might’ve talked about that a geometry class you know this relates to circles like the area of a circle is pi times radius squared things like that ok so that’s what irrational numbers are also roots 33 hey things like that’s it’s basically the square root of anything that’s not a perfect square so like the square root of 9 is 3 right so that’s not irrational it’s actually an integer which makes it rational also and it’s it’s also a natural number okay so it’s actually every type of nut we talked about so far natural whole number engineering rational so anyway but like the square root of so the square root of 9 is 3 so that’s not irrational but the square root of 3 is irrational ok for example so irrational numbers would be something like if you tried to express them as a decimal it would be infinitely it would go out infinitely far but it wouldn’t repeat okay so remember rational numbers you could have decimals like 1.73 2 where the two repeats that’s rational but if you just have like a non repeating pattern and say like one point seven three two five six five three one eight nine four seven six Don Todd and this just keeps going on forever without ever repeating okay so if you have infinitely many decimals or infinitely many play is after the decimal point and no repetition there’s no repeating anywhere then that’s an irrational number okay and all of these numbers actually have that property of square root of 2 square root of 3 root 17 PI root 33 they all have that property that they repeat or sorry they don’t they don’t repeat they have infinitely many decimal points or sorry infinitely many places after the decimal point and they never repeat ok so that’s a very important distinction there if you have the repetition it’s rational without the repetition and if it’s still infinite then it’s irrational okay but if it terminates after a finite number of spots like like we just had this if it stops here even though that’s a lot of decimal places that’s still rational okay this is rational because we can express it as a ratio of two integers

right but again if we repeat our sorry if we keep going on forever and ever without repeating if you just have infinitely many places after the decimal without repeating then it’s irrational okay so very important distinction there so that’s what an irrational number is so so the irrational is the set of all things like you know that so it’s really not a really good way to describe this set but there’s it’s it’s harder to describe the irrational numbers and the Reta and the real numbers but basically the real number is it’s all of the rational numbers so let’s actually keep the colors consistent at least so all of the real numbers is all of the rational numbers union with the irrational numbers okay so the real andros that’s all of the rationals union with the Irrational’s so a real number is really almost any kind of number you can write down or think of now there is something more complicated called a complex number but we don’t really want to talk about that too much yet so it’s sometimes depending on where you are sometimes it does show up in an intermediate algebra course sometimes they hold off on they hold off until college algebra precalculus or something like that anyway a complex number is you know it’s a little more complicated than a real number but anyway I just want to point out that something does exist but we’ll talk more about that in a much later video okay so I just want to point out that there are other types of numbers out there for example complex numbers but we don’t want to get too crazy nothing too complicated in this video but anyway as far as just real numbers go okay well how does this fit in here so how does the real numbers how do they fit in here okay well first of all how do we get the real numbers we take all the rational numbers Union them with irrational numbers okay so basically what does that tell us well it tells us the real numbers it’s made up of rationals and Irrational’s okay so the real numbers actually if you have a rational number that is a real number okay because every rational number is a real number every irrational number is a real number and so basically that tells us that the rational numbers that’s a subset of the real numbers okay that’s we can actually get that just directly from here okay this right here tells us that every rational is a real number every irrational is a real number because we have to Union these two to get the real numbers okay so anything in here anything in queue is automatically an R just from this Union here anything in AI is automatically an R from the Union here okay so Q is a subset of R because everything in Q is automatically in R which is exactly what this tells us here okay so Q is a subset of R oops maybe make that a little better Q is a subset of R okay now the irrational numbers don’t really fit into this and so the irrational numbers don’t fit into the scheme here but we could also say that the irrational numbers are a subset of R but really who cares the irrational numbers they’re they’re just kind of separate so the irrational numbers really are separate from everything okay but here the point is every natural number is a whole number but even that who cares you know the whole numbers they aren’t really that important so when you go into later math classes after intermediate algebra things like that the whole numbers really aren’t talked about that much it’s just the natural numbers that are important the integers that are important the rational numbers and those are important the Irrational’s it not really as much but you know we um we could still talk about them in terms of reals and rationals but anyway so if we want to really summarize everything then every natural number is a whole number every whole number is an integer every integer is a rational number every rational number is a real number and also every irrational is a real number and so those are common sets of numbers and how they all relate to each other so in the next video we’ll do an example of identifying types of these numbers here okay so that’s a natural numbers whole numbers integers rational numbers real numbers and Irrational’s and natural numbers integers rationals and reals by far the most important types and they have these notations here that go along with them so you’ll want to make sure that you’re able to identify those in case you come across them in a textbook or something so that’s common sense of numbers and an example in the next video