Lesson 1 - Real Numbers And Their Graphs (Algebra 1 Tutor)

hi and welcome to the algebra 1 tutor in this class we're going to embark on a journey that's going to take you from your knowledge of basic math which for this course more or less we're going to assume that you know how to add you know how to subtract you know how to multiply and divide you've worked with fractions in the past what we're gonna review it in this class you know you have some idea what a percent is all the things that that have to do with basic math that have no variables or anything like that we're gonna take you from your basic knowledge here at the beginning of this course and pull you up through the you know the end of algebra 1 there so you know we're going to expose you to variables and equations and inequalities and all of these things that for a lot of you guys out there I know terrify you okay that's just the way it is sometimes the title of algebra has this kind of a kind of cast to it that kind of makes you worry about it a little bit just the name of it sounds a little bit difficult but let me tell you if you take it baby steps one step at a time and take every little topic and break it down into little tiny understandable chunks then by the end of the section you'll find that you do understand it and more importantly once you begin to understand it you'll get some confidence and that is really the number one thing that holds people back in math you open up a textbook you see a bunch of equation it's a bunch of letters and exponents and things that that don't seem to make any sense because they're they're nothing to do with what you've learned in your basic math and so what we're going to do in this class is erase all of that and make it understandable so Jeff confidence and so that you'll you'll feel like you can do it and you can do it alright so what we're going to do in this class is start at the beginning I'm not gonna assume you know what it what an equation is I'm not gonna assume that you know what a variable is I'm not gonna assume you know anything other than what we just talked about and we're gonna go from there now here in the beginning of the class we're going to review some concepts that you typically might see in a pre-algebra course or some other reference that will give you a little primer in pre-algebra sort of in the beginning here we're gonna review some of those topics just to make sure that we don't leave anybody behind and then we're going to seamlessly move on into what most people would consider out for one alright now before we can get into algebra and all the things that you think about equations and all that stuff we need to start somewhere and so what we're going to do in this section which is called real numbers and they're graphs what we're going to do is talk about a few definitions just like any subject to study you need to study some definitions and understand them once you do that the terminology becomes easy and then you can just move on from there so not too much in the way of real math in this section there'll be some but not too much we're going to mainly learn some terminology most importantly you probably will be tested on this stuff and your whatever class you're taking so I want to make sure you understand the the first thing we're going to talk about in different types of numbers we all have an idea of what a number is obviously from when you're growing up but in math they assign labels to certain types of numbers and it's just something we do to categorize them so let's go ahead and go through those the first type of a number that you have experience with all your life is called the natural number so let me go ahead and and write that on the board and it it's written just the way it sounds natural number right now with a name like natural numbers you might think they have something to do with nature they might have something to do with you know everyday life and they do and so basically these things are essentially what you would if you had to just kind of boil it down to one little soundbite the natural numbers are essentially the positive numbers right now in math they don't like to give definitions like that they like to have you know definitions that have you know some meaning passed throughout history and things like that so we call them natural numbers so the set of natural numbers would be the numbers that your you know your you use all your life and that you used to count your jelly beans when you were three years old so they would be things like you know one two three four five dot dot dot the little dots mean you know the numbers keep going I'm going I going they're going on of course I can't keep writing them the little curly braces don't worry about that that's just you'll see it in your book that just means that it's it's describing the set so the way you would write this or read this is the set of numbers inside of these little curly braces describes what we call the natural numbers now the reason they're called natural really is because there are the everyday numbers that we work with you know they're the numbers that we use to count cotton balls and the numbers that we use to count our money and things like that the everyday numbers that you have on your fingers no crazy negative numbers you know no imaginary numbers nothing like that just a regular old natural numbers and that's why they're called that so congratulations you successfully understood your first concept here here in our journey of algebra now the next little guy that we're gonna learn about is the whole number and you'll see that there's a little bit of overlap really in a lot of these definitions so the whole numbers typically are going to look like this there'll be the set of numbers that will look very similar to what we just wrote down dot dot dot so again they go off on you know to infinity up higher and higher and higher the only real difference between the natural number and the whole number the set of those numbers is that the whole numbers include zero now why is it important to have a definition for whole numbers and why is it important to have a different definition for natural numbers I'm gonna be honest with you my background is engineering and also in physics but I'm mostly interested in solving problems myself so I am NOT a mathematician my concern isn't really the theory behind why this is is the case but those of you that are going off to study mathematics as an actual career there's a lot of theory behind that the theory of numbers but for everybody else that's just studying algebra and maybe they'll go into science field maybe they won't these are just labels that's really all they are and so you know a long time ago they decided to slap a label on on the numbers that are positive and when you include zero and then called them whole numbers and then they have another set of numbers when you don't include zero those are called the natural numbers it's just something you're going to have to memorize now these two guys might seem trivial and they are but the next guy is one that you will see a lot all throughout algebra and will probably use the word over and over again integer integers right so you you probably even heard that number before take the algebra class are in now and if you have it and then you're definitely going to get to hear it over and over again integers are basically positive and negative numbers that's the way to really really write that down so if you were going to write down the numbers you would have something like negative 4 negative 3 negative 2 negative 1 0 not forget 0 1 2 3 dot dot that's a little curly brace right there so basically what you have is you know starting from 0 and include 0 the positive numbers that go on to positive infinity that way and the negative numbers going to negative infinity this way and the dot dot dots are the same thing and telling you that it's going off to negative infinity and this guy's going off to positive infinity integers are used so much in algebra the term is used so much in algebra because we're using negative and positive numbers all the time in algebra now I know that a lot of you out there probably most of you out there don't really know what a negative number really is and that's ok because we're gonna explore it in this course so negative number you know when you first hear about it you're not really sure is this a real thing is this something to have experience with when if I ever seen a negative number I don't know what that is that was my you know reaction when I first first heard of that that term a negative number is something you have experience with you just don't maybe realize it it's a convention basically in nature when you have phenomena you're always going to have to assign zero somewhere and a good example of that would be temperature so zero degrees Celsius right that is the temperature that water freezes so we know that if we take a cup of water and we cool it down to zero degrees Celsius at that moment that we get to zero a number that we've by the way kind of randomly assigned to the freezing point of water we just call it zero there's no reason at zero we just call it zero because it's convenient that's when water freezes we call it zero and we know that all of the temperatures greater than zero are gonna be positive so as we take that block of ice at zero and we warm it up we're gonna raise the temperature in a positive direction eventually you get up to room temperature you know if you keep raising the temperature but if you take that block of ice and put it a freezer or in a you know you know Antarctica or something then the temperature is gonna go lower than zero negative temperature below zero is what we call it so you see we've assigned a number zero to something and on the positive side we have numbers and then we have to have negative numbers to cover whatever's on the other side another example would be a sea level when you're flying an airplane you know the altitude of an airplane is above sea level when they say we're cruising a 30,000 feet or six miles or whatever it is that's above sea level which is the average water level really at sea level now there are some cities and the coastal areas that because you know near the coast you know if you have any kind of Hills or valleys in the coast you can actually dip down below sea level for some for some cities and so their altitude wouldn't be above sea level it would be below sea level if they had a little valley that went kind of below the average value of the sea level so they would have a negative you know negative altitude really so is that a crazy idea is that something that isn't real absolutely not it's completely real it's just that we're measuring it relative to zero I'll give you a one final thing and the reason I'm harping on it so much is because negative numbers are just they're used everywhere you're not gonna gonna watch another section of this course or turn a page in your book without looking at a negative number so I'm gonna spend a little time on the third thing that I want you to think about as an example of negative number is you know the concept of borrowing something so if I have if I have two markers in my hand then we say well I have two markers positive two that's over here right in my hand have positive two markers now if I give one marker away I now have one marker now if I give this marker away I now have zero marker so now I'm here so what happens if I start you know borrowing markers then I don't have anything else maybe I want some markers back so I borrow some markers from my friend I do have a marker in my hand but it's not really mine I don't really have my own anymore I borrowed it from somebody else so you could say that I actually have negative one markers at that point if I borrow money from the bank you can sort of say that I sort of have negative money yeah yeah I got money from them but it's not my money I actually owe them money I'm going to have to pay that back so I'm down in the hole down here I'm owing them money and that's what a negative number is so those are some ways in which to think about negative numbers they're just numbers below zero that's really all you have to know I just wanted to give you a few examples integers are the negative numbers the positive numbers with no decimals obviously and zero so no decimals here just the whole numbers positive negative and zero all right now the next thing I want to talk about is very important it's called the rational number rational number now let me ask you a question what do you think of when you see the word rational when someone is being rational what are they that means that they're cool-headed well thought out you know they're cool calm and collected they're they're rational you know you can talk to them so that's what these numbers really are it's kind of a descriptive word these numbers the rational numbers are the numbers that we can write as fractions and which actually is most numbers out there so we're gonna say that the rational numbers are the ones that we can write as a fraction so so let me give me an example let's say you know the number one point five is this number rational or is it not rational well the question you have to ask is can it be written written as a fraction because all rational numbers can be written as a fraction so it can be written as a fraction if you think about it hard enough you will realize that if you use the fraction of three halves if you put this in your calculator 3/2 three halves you're going to get one point five so that is a rational number that is a rational number the number two is also a rational number because you can write it as a fraction 2 over one so you see all of these whole numbers that we've been talking about all of these integers you know negative 5 is also a rational number because we can write it as negative 5 divided by 1 so any whole number integer or anything like that it's automatically going to be rational in any fraction so let's say we have the fraction negative 45 divided by 9 that is also rational why because it's already written as a fraction so either the number that you write can be written as a fraction or you know of course if the numbers already a fraction it's it's a rational number as well so rational is cool calm and collected in the term of mathematics because it can be written as a fraction so those are nice well-behaved numbers if we're able to do that that's why they call them rational another example the final one 0.25 is that number rational well we can write that as 1/4 we can write it as 1/4 and so that guy's rational so you might say well if a number is rational and you can write it as a fraction then what would you call a number that you couldn't write as a fraction we'll give an example of a second what kind of numbers can you not write as fractions and what would you call them well the answer is you would call them an irrational number which is kind of a cool name actually because the word irrational means or you think about someone that's a little bit crazy they're not really well thought out they're not really stable they're doing their own thing kind of thing that's what irrational means but in the terms of the numbers it's it's not well behaved like these numbers and so the irrational numbers are the ones who cannot be written as a fraction so let's write that down we have irrational number numbers okay irrational number so the definition is you cannot write these numbers as a fraction now the most famous example of a number that you know is irrational you may not realize is high number high as you know that's a really special number you use it you know all throughout math you'll use it in algebra use it in geometry and trigonometry and calculus and on and on and on you'll use the number pi and we typically write the number pi is 3.14 that's typically the number that we use when we calculate you know the area of a circle or the circumference of a circle or something but actually pi is not 3 one four pi is 3.14159 dot dot dot so if you go in into a computer or even into a hand calculator and you put usually in the calculators nowadays there's a PI button you just press that button and look at look at what it puts spits out on the screen and it's gonna put 3.14159 it's gonna have a bunch of digits after that and if you go into a computer or a website that went calculate pi for you then you can go for weeks and look at the decimals that follow pi you can look to the million decimal place after pi and you'll never find a repeating pattern in this in this decimal you'll never find it and that's why we call it irrational because you cannot write this number as a fraction there is no fraction that you can write down that will equal pi exactly now some of you guys might have been taught incorrectly that the fraction 22 over 7 is equal to pi but actually that's not true if you put 22 over 7 into your calculator you'll get a number it'll be 3.1 something but it's not going to be 3.14159265 4 oh no no no no no not it's not going to be the exact right number it's an approximation so you cannot write pi as a fraction you just can't in fact there are people that do research into this stuff and they calculate pi to billions of decimal places trying to find a pattern it's really amazing that there is no pattern in this number and anytime you have a decimal with a what they you know with numbers after that have no pattern to it you can't write it as a as a fraction it's just not possible so that's irrational all right another example of an irrational number would be the square root of 2 if you put the square root of 2 in your calculator you're gonna get something like 1 point 4 1 doesn't blah blah blah you're gonna have a bunch of numbers that go on and on and on but there's not going to be any pattern in this decimal here so you're not going to be able to find any fraction at all that's going to exactly equal square root of 2 so these are both irrational numbers so you can sort of see that the the irrational numbers are the ones that are sort of special in a way that you're not able to write them as a fraction most of the numbers that you've dealt with in your everyday life certainly in money and things like that calendar time you know anything that you would use on a daily basis they're almost all gonna be rational and that's why we're calling that they're they're rational numbers because they're they're well-behaved now the next thing we want to talk about is in the title of this class we're gonna call this a real number a little bit real number what is a real number you know books are going to give you different definitions my job is not to be a textbook my job is to break down your textbook so that you understand what it's saying so go read your definition of a real number in your textbook I'm going to tell you that a real number is I'm gonna put little quotations here almost everything in terms of numbers it includes the integers it includes negative numbers positive numbers rational numbers irrational numbers basically everything the only thing that doesn't include it does not include does not include any type of imaginary number or complex number imaginary or complex now we're not gonna study imaginary or complex numbers too much in this course because it's not that necessary but I'm just trying to tell you that there are these numbers out there called imaginary numbers they do exist you will learn them as you get into algebra 2 and then if in some courses after that and you use them all the time when you get in to pass that into calculus and other classes but so they're in the big picture of the world there's the real numbers and then there's the imaginary numbers and so everything that you've ever learned up till now including everything on the board all the whole numbers the natural numbers to you and all these things that we're talking about those are all real because they're not imaginary so it's real or imaginary one other and so a real number is that's why I say it's almost everything the only thing it doesn't include is an imaginary number or a complex number which is really sort of like an imaginary number too so I don't want to get into the definitions of what these are because of August now but that's sort of what we're after here now some examples I don't really need to give too many because like I told you includes almost everything but you know the number negative three it's an integer negative number that's a real number there's no imaginary there's no I here so it's it's real 25 over two that's a fraction that's fine it's a real number there's no I here there's no there's no I indicating that it's imaginary what about negative seventeen over nine that's a fraction it happens to be negative that's okay it's a real number because there's no I here there's no imaginary number the decimal zero point six nine that's also a real number again there's no there's no I indicating that it's imaginary because when you start to study imaginary numbers that you'll do in future courses you'll find that they always have a little I at the end if they're imaginary so that's why I say they almost include everything it's just that they don't include those imaginary numbers okay now what I want to do now is talk about something really really important kind of interesting and that is called the concept of a prime number and a lot of you may have heard about prime numbers even if you haven't really used them too much before but a prime number is a special number and the definition for a prime number is basically a number greater than the number one right a whole number greater than the number one that can only be divisible by itself and one can only be divisible by itself and one that's the definition you're gonna find in your book it's not terribly you know understandable first glance until you really start looking at it so let's let's do that now let me write down the definition first so it's a it's a whole number whole number greater than the number one so not number one but two three four so on that can be only that can only be divisible and when I say divisible I mean evenly divided by itself which is number you're talking about and the number one so we'll give a couple of examples obviously that's what we want to do we want to learn learn by doing so the number three the number three is a prime number and the reason is because you can divide 3 by itself and you'll get the number one right it divides evenly right you can also take the number three and divide it by one and then you're gonna get three so that divides evenly also right so it's a whole number that can be only be divisible by itself and one so here you're dividing by itself here are you dividing by the number one and you get a nice even number both tons but if you try to divide this number by anything else you're not gonna you're not gonna have a nice division 3 divided by 2 3 divided by 5 3 divided by 10 you're not gonna find another number that's going to divide evenly into it so 3 is prime right 3 is prime let me give you another example 7 right number 7 you can take the number 7 and you can divide it by itself and get a nice even number nice whole number I should say you can also take the number 7 and you can divide it by 1 which is what the definition says and you'll get a nice a nice number of 7 but if you take the number 7 and divide it by 20 1 by 4 or by 5 or by 3 or by anything else other than itself in 1 you're not gonna get a nice even division and so because of that 7 is prime just like 3 is prime just like 3 is prime so let's look at an example of a number that really isn't crime and I think you'll understand the difference so let's look at the number six number six is six prime that's the question so let's say six can be divided by six right it goes a nice even time with one six can be divided by one and that goes a nice nice even number of times you get the number six but you know you can also divide the number of six by three let's say and you get a nice and nice even number of two so you're able to divide this number like something other than itself and once and because you're able to do this six is not high 6 is not prime so most numbers really aren't from and it just it just depends on if you're able to divide it you know by anything other than itself in one so the list of prime numbers if you're just gonna write it down in the list the number two is prime the number three is probably the number five is prime the number of seven is probably the number 11 is prime and there's more numbers as you go you know like the number 13 is also prime but you know if you notice the number four is not listed here the number four is not prime because you can divide 4 by 2 the number 8 is not prime because you can divide 8 by 4 you can divide 8 by 2 also the number 10 is not prime not in that list because you can divide it by 5 you can divide it by 2 and so on so prime numbers also go on to infinity they keep going bigger and bigger and bigger it's just that you know you have to do a little bit of thinking to figure out if it's prime or not that's the basically the definition of a prime number it's it's a useful thing for the mathematicians you know they study these things they try to learn about them understand number theory for me it's just another label is a number prime or isn't it fun ok now the next thing I want to talk about is even and odd numbers and a lot of you may already know what even and odd numbers are but if you don't it's a super super easy concept to understand all right if you have if you're looking at even numbers even numbers basically these are the numbers that can be divisible by two so you're counting by two so it would be numbers like 2 4 6 8 10 you see you're counting by twos dot dot dot 12 14 16 on and on and on now it doesn't matter if it's negative or positive so negative 2 is also even negative 4 is also negative negative 6 it's also even not negative 8 negative 10 so anything that can can be divisible by 2 is an even number isn't even number no matter if it's positive or negative it's a very very simple concept of counting by twos now an odd number is everything else that's left over so things like 1 3 5 also the negative numbers that go along with this like you know negative 1 negative 3 negative 5 dah-dah-dah like that so it's everything leftover if it's not an even number now one thing I want to show you real quick you know look really quickly if you look at these okay this is a prime number 3 prime number because you have it listed here 5 is a prime number you have a listed here 7 is prime number you have it listed here 9 happens to be odd but it's not a prime number and that's because 9 can be divided by 3 it's can be divided by something other than itself and the number 1 so it's it's not a prime number so a number I'm playing it out only to show you that a number can have multiple labels this number is odd this number is odd but it also happens to be a prime number oh wait but it also happens to be an integer from our definition before and oh wait it also happens to be a natural number from our definition before so these definition of numbers are just multiple labels to define the buckets that you can put number ten and most numbers are gonna have more than one label because by the way this is also a rational and a rational number because it can be written as a fraction yeah you can write that as 7 over 1 you can write that as 9 over 1 so there's there's more than one thing to think about when you're looking at these things and so on your tests a lot of times what'll happen is they'll give you a number and they'll say tell me if it's rational tell me if it's prod list everything you can about this number so you'll have to do that you'll have to do that all right now the next thing I want to talk about will switch gears here the next thing I want to talk about sort of switching gears is the symbols that you're going to find symbols you know there are a lot of symbols in algebra that you're going to learn parenthesis we're going to get into and all these things that look foreign and that's why it scares a lot of people but we're breaking down one step at a time so let's go with the super basics the thing that you already understand let's talk about the equal sign you know I like to talk about things that everybody understands all right so that's and it's a something you have to understand so the equal sign there you go it's a symbol that you've used all your life how would you use an equal sign then you know the answer to this but you could say something silly like 2 is equal to 2 you know that's your first equation written right there on the board it's a silly equation it doesn't really mean anything all it says is if I have two jellybeans over here that's exactly equivalent to you know this other bucket with two jellybeans so if I have two jellybeans here and two jellybeans here those are equivalent number of beans that's that's what it's saying everything on this side of the equal sign has to be equal to what's on the other side of the equal sign so that's really simple let's talk about something that you may not have seen you can also say something like 3 is not equal to 2 now this symbol is an equal sign with a slash through it and that just simply means that the two sides are not equal it's exactly what you might expect the three jellybeans over here is not equal to the two that I have all right and then finally we need to spend a little bit of time talking about greater than and less man which I know you've seen before but I'm going to hopefully make it really easy to understand we can write something down we're gonna draw a little line to kind of divide off here we can write something down like this the number two is less than than number three and you know all right you all know that the number two is less than three but we need to use our symbols to write that mathematically because the math is basically describing what we know to be true and as you build your skills you'll be able to solve problems that you can't do in your head so we have to make sure we understand everything two is less than three how do you read this this this little symbol here you probably see it in your book it's called the less than symbol last name symbol and then if you flip it around and we will hear in a second you'll call it the the greater than symbol so let's do that right now let's say the number five is greater than the number one which I know that you all know to be true and we say that this symbol is the greater than symbol it's the greater than something which is exactly true there's nothing wrong with that but here's the deal I want you to not worry too much about these guys as far as being separate symbols you know a lot of books will say here's the less than symbol and here's the greater than symbol here's you know you need to use them whenever you need to use them I prefer to teach you that whenever you're comparing two things just take this arrow and point it always to the smaller number I'm going to say that two more times to make sure you understand always point that arrow to the smaller number always point that arrow to the smaller number to the smaller number because that's really what the symbol means really it's an arrow and so the smaller side of this arrow the side that comes to a tiny little point should always be next to the smaller thing the bigger part of the arrow the arrow at the part of the arrow that's big in fact should always be next to the bigger thing so you see the arrow is always pointing to the smaller thing here it's pointing to two because it's smaller and here the arrow is pointing to one which is smaller so yes you could say they're separate symbols because they point different ways but really it's just one symbol it's an arrow and the arrow points always to the smaller number so in this case we have the arrow pointing to the smaller number as well and so try not to think of these things as two separate symbols that you have to memorize them try to think of them as just an arrow that points to the smaller number and then so when you do that the bigger part of that arrow is gonna point to the the bigger number or be next to the bigger number and when you read it you just read it just like you would in the sentence five is greater than one two is less than three and that's how it that's how it comes about now you can also do some interesting things with these things that you're going to need to understand and so it's sort of building upon greater-than less-than so we're gonna do that now what I'm going to do talked in the beginning of the course that the word variable scares a lot of people all right so what I need to do now is sort of introduce that just to sort of teach you a little bit more about how to graph these these inequalities here on the number line and introduce the number line which is half the reason we're teaching this section here so think of a if you don't know the answer to a problem you know when you're younger and you're adding up jelly beans you know you're you're just basically counting them and you arrive at the answer when you get up into algebra and beyond when you don't know the answer or something you assign it a letter and that letter is just something you don't know the answer to it it's called a variable it's just a letter that just represents whatever the answer happens to be and that's a variable so when you solve your equations later on in the class you're just gonna solve for that that the answer to that problem it's going to be the answer to that variable now here I'm going to introduce a variable X you're going to see the letter X used a lot because it just it just happens that algebra uses the letter x to represent the variable most of the time it's a letter of the alphabet you've seen the letter X all your life don't be worried about it it's just an unknown value of something so if I had something like X is greater than or equal to 6 I would write it like that now this symbol looks very very similar to this symbol but there's a difference this one has a line underneath it this one has a line underneath and that line is basically telling you that whatever X is it's got to be greater than six but it can also be equal to six that's what the line entered there means if there's no line then it's just if you're if you're just looking at like that it's just X is greater than six so seven eight nine ten eleven twelve and so on that what that's what X could be but if you make it greater than or equal to six then X can be everything including six so in this case X is greater than or equal to six is how you read it and the answer would be that X is equal to six seven eight nine ten Dada Dada on and on and on we're talking about integers here so X is greater than or equal to six so it's basically including six because of the equal sign that's why we list it and then every number bigger than that because we're talking about numbers bigger than six now just to give you another little example if you have Y is less than or equal to three right that's the way you read it less than or equal to that's why the way you write that then Y is going to be equal to three two one zero negative one negative two dot dot because basically it's equal to 3y can be equal to three but it can also be equal to two two one zero negative one you keep going less and less and less and so when you think about it if you get less than zero you're gonna be going into the negative into the realm of the negative numbers which is what we talked about so that is the symbols I wanted to show you we have the simpl equal sign we have when two things are not equal and we have greater than less thing which are exactly the same thing you just point the arrow to the smaller number and then we have greater than equal to less than or equal to and in these cases you're basically trying to express a whole range of numbers either bigger or smaller than the number you have and if there's a if there's a bar under it it includes that number and if there's no bar listed here well then you wouldn't include that number that's really all that that means so let's go ahead and erase the board and start actually graph numbers on the number line okay now we've done a lot in this section we've talked about numbers we've talked about categories of numbers labels for these numbers we've talked about symbols greater than less than equal to and we've kind of danced around the issue of a number line but we've never actually done that and drawn that so that's what we're going to do now imagine all the numbers that you've ever known stretched out on the line and so you know numbers can go on forever so the number line goes on forever and it goes off into the positive direction because the positive numbers go on to infinity it also goes on to the negative direction because the negative numbers also go to negative infinity and so this line really is double-arrow line and it goes in both directions so what we're going to talk about now is the number line and you know you need to get comfortable with it because I'm going to tell you later on when we're gonna start adding and subtracting negative and positive numbers the number line is going to help you do that and help you visualize it the number line is a very simple concept just like we talked about it's on line the little arrow means it goes on forever and the other little arrow means it goes on forever that way now in the middle of the line just like we said you always have to have a reference point you always have zero in the middle and so over on going this direction you have your positive numbers one two three you know then you have four five six seven eight nine ten they go on forever I'm only gonna draw these and then I'm going this direction you have negative one negative two negative three and then keep going you would have negative four negative five and then we go on to negative infinity there now if I were going to graph a number on this number line it would be a simple matter of putting a dot wherever I want the dot to go so if I were going to plot the point negative one on this number line I would put my pencil right on top of this mark or negative one and put a dot there this would be the graph of the number negative one on this number line and you immediately see how it relates to all the other numbers so if I wanted to put you know plot the number negative three I would put a big fat dot on negative three now notice that even though the numbers on the number line are you know they're whole numbers you can plot points in between those numbers so if I were going to plot the point two point five or two-and-a-half then I would go right between two and three and I put a dot right there and I would say this one is two point five and you know this number line you don't want to put too many numbers here so you just put the whole numbers and then anything in between the apply you just you know put it put it as best you can so that is a concept of a number line it's not a complicated concept it's a line with numbers and you use it to represent and plot the points on there it's going to help you visualize negative numbers and positive numbers and that's going to be really useful now let's say that I wanted to do a little bit more with that let me draw another number line all right so I'm gonna go do the same thing before 0 1 2 3 negative 1 negative 2 negative 3 now what I want to do here what if I wanted to graph the inequality X is greater than negative 2 and when I say inequality by the way when something is equal you know that that would be something that was you have an equal sign in there when something is in equal you call it an inequality these greater than or less than symbols they're just called inequalities so if you hear me say the word inequality it just means it has one of these symbols there that's it so what this number means or what this what this thing means is that X which is some some number that you're just trying to represent here on the graph has got to be greater than negative 2 got to be greater than negative 2 so what we're going to do is we're gonna put a circle right on top of negative 2 because that's a number here and an X is every number bigger than that so we're going to actually shade the number line and you do this on your paper you literally get your pencil out and you would literally shade it like this so this graph is telling you that X is greater than negative 2 you start at negative 2 and every number bigger than that is what's representative now notice very very important this circle here around negative 2 it's an open circle when you have an open circle when you're plotting these guys it means that you're representing every point bigger but you're not including non negative 2 itself in other words negative 2 is not represented here in this little equality because we said X is greater than negative two we didn't say it was greater than or equal to we didn't say was equal to negative two we just said had to be greater than so if something is greater than one it can't be one it has to be you know two three four or five or whatever if something's greater than negative two you cannot include negative two in the answer because we say this greater than negative two so we put an open circle to remind us that negative two is not really part of this answer but everything bigger is all right now if we change the problem if we change this so that we said it was X greater than or equal to negative two that would be something different and in order to represent that the way we would have graphed it instead of drawing it again I'm just gonna go right back up here everything is exactly the same except we have to make this a solid circle because this solid circle is telling us that yes negative two is included in that and it's negative two and it's everything bigger than that so that's very very important it's a great place to make a mistake on a test if you ever see any open circles then you know that number and that point is not included in the inequality and if you see those closed circles then you know that it is so it's very important for you to know all right let's do a couple of additional examples let's say let me draw another number line so we have 0 1 2 3 negative 1 negative 2 negative 3 now let's say I wanted to represent X is less than or equal to negative 1 X is less than or equal to negative 1 well the first thing you have to say is well known negative 1 is what I'm looking at so I have to put a circle here and it has to be a closed circle and the reason it has to be closed is because it's less than or equal to negative 1 so negative 1 is included in the answer and since we're talking about numbers less we're gonna shade this direction so every number way down here below negative 1 all the way to negative infinity is included in this and what this is trying to represent including the number negative 1 and that's why we have a pot you know a solid circle there to remind us of that so don't get too confused and definitely don't make any mistakes with the open the clothes you have to pay attention if it's less than or equal to or if it's not equal to as well and that's why that's important now let's go on and do something similar but in this case we're gonna do a little bit different I'm going to draw something on the number line zero one two three negative one negative two negative three and what I'm going to do is I'm going to first draw it and then we're going to write down the inequality that it represents so let's put an open circle here and let's shade this direction okay and let's go ahead and put an open circle here and let's shade this direction okay so we have two open circles and they're both going opposite directions now the question is how do you write that down what would be the inequalities that you would use to represent that well it looks really confusing but you need to break it down into two parts you have this part and you also have this part so let's write that down this part is going to be X is less than negative three and the reason it is because of course it's less than negative three because we've shaded that it's an open circle which means we're not gonna have any equal sign here because negative three is not included in you know in the graph we also have a second part and this is gonna be X is greater than one again it's open circle so you know you don't have an equal sign here and it's greater than the one so we're gonna have have it like that so this really is the answer it's two separate little inequalities and and that's what's representing the graph but you can combine them and a lot of times they'll do this on your test and it confuses people for no good reason and so let me show you the way you combine these things is write your X in the middle so what we do is we write X in the middle and we say X is greater than one which is exactly what we have here so far we haven't done anything other than rewrite down what we have but we also say that X is less than negative three and so here we've combined the two things but in order to read it properly in two to not get too confused by it because he does look confusing I will admit to you you have to start in the middle a lot of people try to start reading it over here and it gets really confusing read in the middle X don't forget about this X is greater than 1 okay great X is greater than 1 it's not including 1 so there's no underline here it's just greater than 1 so that's what happens if you covered this part now if you cover this part of it up then if you read it properly starting at X X is less than negative 3 X is less than negative 3 because the arrow is pointing to the smaller thing so X is the smaller thing X is less than negative 3 reading it from the inside out and X is less than negative 3 and that's why we have it written that way so if you try to start reading in here you can you can do it I'm not telling you that you can't do it but for me it's always easier if you start in the middle X is greater than 1 okay that's what I have X is less than negative 3 ok that's true this is what you'll see a lot of times on your tests sometimes they'll give you something like this and they'll say graph it the first thing people do is freak out a little bit and panic just cover up half of it and graph that and cover up the other half and graph that and you'll be a really good shape let's do one more similar to that it's an important skill to have it's an important skill to have so we'll have 0 1 2 3 negative 1 negative 2 negative 3 and let's say the graph that we're after is going to be just change colors a little bit let's say negative - we're gonna have a dot on and positive 3 we're gonna have a dog on and we're gonna shade everything in between these two dots question is what inequality or inequalities could be written to to represent that and so again you have to you know you have to you know take it one step at a time and and put them together without trying to you know go to the straight to the final answer you need to work in steps so what you need to first look at here and say ok X is greater than or equal to negative 2 because here's negative 2 here's a solid dot so it's greater than or equal to negative 2 because the shading goes on to the bigger numbers those greater than equal to negative 2 however it doesn't go on forever and ever so we also know that X is less than or equal to positive 3 X is less than or equal to positive 3 it's has a little equal sign there as well because you have another solid dot okay so this by itself would be solid dot going on and on to infinity this by itself would be solid dot going on and on to negative infinity but together when you put them together it just represents the range of numbers here because when you put them together it's the way you would write that you put X in the middle and do exactly what I said you would start with one of them greater than or equal to negative two just rewriting this and also less than or equal to three and if you look at it this part is it's just totally rewritten over here and then all the other stuff on this side right here is just this but flipped around a little bit so the arrows pointing to X here the arrows point to X big part is pointing to three big parts pointing to three so this is really exactly the same thing as this it's just written with X in the center so the way you read it is you start in the middle and you say X is greater than negative two check and X is less a less than or equal to three so X is greater than or equal to negative two and X is less than or equal to three and you put those two things together it's representing the region of points between those two numbers and including those two numbers because of the solid dots there because that's what that represents so this is you know there's probably a fancy word for it but basically it's a double inequality it's when you're describing a range or you're describing two parts of the number line and using both of these together you can rewrite it so don't be afraid to look at this and pick it apart to graph it and go in reverse or if you're just looking at a graph write them down individually and then combine them into something like this all right now the next thing we're going to talk about which is actually going to be the last thing in the section it's also very important and that's gonna be the topic of absolute value absolute value is one of those things is pretty easy but has a big word as a big name so a lot of people get worried about it basically the absolute value of a number is just telling you how far away that number is from zero how many units of distance is that number away from zero so the easiest way to do it is just to draw a number line and take a few examples so here's a number line here zero one two three course goes on forever negative one negative two negative three and goes on forever that direction so if I were gonna ask you what is the absolute value of the number three just regular all three all you have to do is take the number on the inside and write it down that's the answer the absolute value of that is three because the number of distance units the number three is away from zero is simply one two three three distance units alright so the absolute value of five is fine for the same reason because if you have you know four out here and five out here the distance that the number five is away from zero is just simply five so you say well this is silly why we why we even learning this well the absolute value of negative six is equal to what it's equal to six the absolute value of negative two is equal to what it's equal to two and the reason is because if you look at negative two here which is right here how many distance units is that away from zero it's one distance unit two distance units away from zero so yeah the negative numbers do go on the other side of zero true but did the absolute value the value that you have independent of its sign the distance that that number is away from zero is always going to be a positive number even if the number itself is negative and so you're just counting units that's all you're doing practically speaking all you have to do if you have an absolute value of ten thousand three hundred and forty seven negative is just gonna be ten thousand three hundred and forty seven whatever you have in the inside is what you write down as the absolute value no matter if it's a positive number the absolute value is positive if it's a negative number the absolute value is positive so that concludes our section in real numbers and their graphs you see we've we've we've covered a tremendous amount of material we've talked about numbers we've talked about you know integers and all these categories of numbers we on numbers rational numbers all these things are just kind of can seem to be a little bit of mumbo-jumbo but you have to learn the number one because you're going to be tested on them but number two it's sort of the language of math if somebody says hey here you know here's an integer in this equation you need to know what the word integer means because if you don't you're just gonna get scared by it and the number one thing you need to realize is that if you understand something you build your confidence up so when you do that you know that you can conquer everything else that follows I guarantee every one of you guys when you were all babies you know when I was a baby too we learned how to speak in the English language you know we didn't we didn't have a textbook we didn't have a teacher we just listened to people and somehow out of all of that came the ability to to use words and then sentences and then paragraphs and then we can write things down and make long you know stories if we want to it's an incredible thing that a child can do to learn how to speak without any books or anything they're just listening by examples they're listening to what people are saying and they're putting it together in their brains putting it together that is how I plan to teach this entire course and all of my courses is by examples so rather than put a bunch of definitions on the board really lengthy ones like you've seen a book we're gonna do it by working problems we're going to do it by example so that if you see enough of these things you'll realize yeah yeah I do kind of understand how to add negative numbers together I really do understand how to multiply fractions I know how to solve equations because I've seen it done so many times building one skill at a time I'm Jason I hope you got something out of this lesson stick with me on this journey I guarantee you that you'll be comfortable in algebra but not only that you'll be really really good

Channel: Math and Science

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Keywords: real numbers, rational number, irrational numbers, algebra 1, math, mathematics, algebra, learn algebra, number line, integers, rational numbers, natural numbers, whole numbers, real numbers and their properties, real numbers and the number line, graph, rational numbers on a number line, rational numbers examples, rational numbers and integers

Id: Vuj5CZDy-Pc

Length: 53min 37sec (3217 seconds)

Published: Wed Feb 03 2016