If you're seeing this message, it means we're having trouble loading external resources on our website.

Хэрэв та вэб шүүлтүүртэй газар байгаа бол домэйн нэрийг ***.kastatic.org** and ***.kasandbox.org** блоклосон эсэхийг нягтална уу.

Үндсэн товъёог

Цаг: 0:00Нийт үргэлжлэх хугацаа:8:49

so we have a bunch of numbers listed up here and my goal in this video is to see if we can classify them into different types of number categories and let me draw the categories so this circle over here this we can this represents all of the numbers that can be represented as the fraction of two integers and of course the denominator can't be equal to zero because we don't know what it means to put a zero in the denominator so let's call these or the standard way of calling these things these things that can be represented as a fraction of two integers we call these rational numbers rational numbers and if something cannot be represented as the fraction as a fraction of two integers we call irrational numbers irrational numbers irrational numbers and the size of these circles don't show how large these sets are there is actually an infinite number of rational and an infinite number of irrational numbers so these are the irrational numbers irrational so these cannot be represented as a fraction of two integers and then within rational numbers you have integers themselves so I'll do that in do that in this blue color integers so integers are numbers that don't have to be represented as a fraction or a decimal so these are integers right over here integers and then a subset of integers are whole numbers so if you essentially say the non-negative integers you're then talking about whole numbers so let me do that subset right over here so these are going to be the whole numbers so whole whole numbers whole whole numbers right over here and actually let me just label it all these are rational we've done that same color rational numbers and of course irrational numbers irrational numbers irrational numbers an integer well if I could say literally that is an integer let's think about the integers but I would say let's just think about rationally let's talk about the rational numbers all right now that we have these categories in place let's categorize them and like always pause the video see if you can figure out what category these numbers are fall into where would you put them on this on this diagram so let's start off with three this is positive three it can be definitely represented as a fraction you can represent it as three over one but it doesn't have to be represented as a fraction a little rate could be just a three right over there but it's also non-negative so three is a whole number so three maybe I'll do it in the color of the category so three three is a whole number so it's a member of that set but if you're a whole number you're also an integer and you're also a rational number so three is a whole number it's an integer and it's a rational number now let's think about negative five now negative five once again it can be represented as a fraction but it doesn't have to be but it is negative so it's not going to be a whole number so negative five is going to sit right over here it's an integer and if you're an integer you're definitely going to be a rational number but it's not a whole number because it is negative now we have now we have 0.25 well this for sure can be represented as a fraction this is twenty five hundredths right over here so we can represent that as a fraction of two integers I should say it's twenty-five hundredths but there's no way to represent this except using a fraction of two integers so 0.25 is a rational number but it's not an integer and not a whole number now what about 22 over 7 well here it's clearly represented already as a fraction of two integers but I don't think I can represent this any other except as a fraction of two integers I can't somehow make this without using a fraction or some type of decimal that that may that might repeat so this right over here this would also be a rational rational number but it's also it's not an integer not a whole number now this over here zero point two seven one 0.271 three then the one three repeats this is anything 0.271 three one three one three that's what that line up there represents now you might not realize it yet but any number that repeats eventually this one does repeat eventually you have the point 1 3 1 through order you have the zero point two seven one three one three one three any number like this can be represented as a fraction for example and I'm going to do it here just for the sake of time but for example zero point three repeating that's the same thing as 1/3 it later on we're going to see techniques of how do you convert this to a fraction of two integers but for our sake we just know that this can be represented as a fraction of two integers just the way that zero point three repeating can be and so we would put this under rational numbers zero point two seven one three repeating but you have to represent it as a either as a decimal or a fraction of integers you if if you didn't have to then it could have been an integer but we're it up there in rational numbers now the square root of 10 square root of 10 this is interesting so any square root of a non-perfect square is going to be irrational so this is going to be irrational I'm not proving it to you here but you cannot represent this as the ratio of two integers or a fraction of with two integers with an integer in the numerator integer denominator you this will be if you were to represent it as a decimal it will not repeat it'll just keep being new new digits it will not repeat over time so this right over here is an irrational number it's not rational it cannot be represented as the ratio of two integers all right fourteen over seven this is the ratio of two integers four so this for sure is rational but if you think about it 14 over seven that's another way of saying fourteen over seven is the same thing as two these two things are equivalent so fourteen over seven is the same thing as two so this is actually a whole number it doesn't look like a whole number the remember a whole number is a non-negative number that doesn't need to be represented as the ratio of two integers and this one even though we did representative Rachel tutors it doesn't need to be represented as the ratio of two integers you could represent this as just two so that's going to be a whole number fourteen over seven which is the same thing as two that is a whole number now two pi now pi is an irrational pi is an irrational number so if we just take a multiple of Pi if we just take a an integer multiple of Pi like that this is also going to be an irrational number it's if you looked at its decimal representation it will never repeat so that's two pi right over there now what about let me do that same since I've been consistent relatively consistent with the colors so this is two pi right over there now what about the negative square root of 25 well 25 is a perfect square square root of that's it's going to be five so this thing is going to be this thing is going to be equivalent to negative five so this is just another representation of this right over here so it is an integer it's not a whole number because it's negative but it's an integer negative square root of 25 these two things are actually these two things are actually the same number just different ways of representing them and then you have a t of the square root of nine over square root of nine over seven well what's the principal root of nine this thing is going to be the same thing this thing is the same let me just in a different color this is this is the same thing as square root of nine is three it's the principal root of 9 so it's three sevenths so this is a ratio of two integers this is a rational number square root of nine over seven is the same thing as three sevens now let me just give you one more just for the road what about pi over pi what is that going to be well pi divided by pi is going to be equal to one so this is actually a whole number so I could write pi over pi right over there that's just a very fancy way of saying one