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hello welcome back so this lesson talks about signed numbers in other words we're going to start working with negatives I know the last couple of lessons have been a little shall we say intense today hopefully will be rather quick and easy as we learn a little bit about what negative numbers represent and then we'll start using them we've really gotten about as far as we can possibly get without using negative numbers in context so it's time we started working with them so here we go right with your guided notes if we were to look at a number line then the numbers to the right of zero while these are our positive numbers and of course the negative numbers are to the left of zero a lot of people think about negative numbers as being the opposite of a positive number and that's true opposites are numbers that are the same distance away from zero normally when people draw a number line to show negative numbers they draw one that shows what we call integers so let's get some space here scroll this up a little bit a number line that shows integers it looks like this so we have zero sitting here in the middle positive 1 positive to positive 3 positive 4 and so on on and on forever you remember that we talked about 1 2 3 & 4 being the natural numbers 0 1 2 3 & 4 being the whole numbers and if we work with the other half of the number line we have negative 1 negative 2 negative 3 negative 4 and continue on to the left now we're showing integers so integers are whole numbers and they're opposites we said that opposites we're numbers that were the same distance away from zero so for example here at positive 3 we have three units away from zero and here at negative 3 we have another three units away from zero so 3 and negative 3 are opposites of course we don't have to just put integers on our number line we could put all sorts of things maybe we could count by tenths so here we have 0.1 0.2 0.3 and so on and of course the negative numbers mirror this negative zero point 1 negative 0.2 and negative zero point three decimal points don't seem to want to stick today so opposites are numbers that have the same distance away from zero the opposite of 32 is exactly what you think it is negative 32 the opposite of 783 so when I go on the other side of zero is a positive 783 what's the opposite of zero well that's just itself zero is the only number that is its own opposite when you look at negative 5 we can read this in two different ways we can say negative 5 or we can read this negative sign here as the opposite of so we talk about negative 5 as being the opposite of 5 so the opposite of is another way to interpret a negative sign which way we do it depends upon context right we have these things all the time in the English language if you think about the word spell wi md maybe you read that as wind maybe you read that as whined it all depends on the stuff around the word and the same things true for math sometimes it's convenient to talk about negative 5 and other times it's convenient to talk about the opposite of 5 ok so your job now is to pause the recording and see what you can do with parts B&C of example 1 as soon as you and I finish with Part A so here we go part a I want to read this first negative here whoops sorry about that as the opposite of the opposite of and then I want to look at the stuff that's inside my parentheses and think of this as negative three so the opposite of negative three should be a positive 3 all right Utah try the next to pause and come back when you're done all right let's see how you did so the opposite of 7 is negative 7 oh my see what is this all right so let's see um we learned with order of operations to start with the innermost parentheses first so here we have eight and now we have the opposite of eight which is negative 8 but there's still a minus sign in front of that and another negative sign in front of that so let's look at these two here what is the opposite of negative 8 well that would be positive 8 but there's still another negative sign out there so now i need the opposite of positive 8 which is a negative 8 i know right about now at least some of you are saying can't you just count the negative signs and actually you can when you talk about the opposite of the opposite you get back where you started so every time we have two negative signs together it's as if we didn't change at all so the opposite of a negative is a positive and the opposite of a positive is a negative there we go negative 8 on to the next page do you remember with the symbols for less than and greater than look like I'll admit I have trouble with us because it's out of context however if you take your left hand and hold up your thumb and index finger you will see something that looks sort of like an L but is probably a little slanted it's also looks like a little less than symbol so I like to associate all the things that have L in them together less than with my left hand looks like an L the symbol for greater than goes the other direction alright so now what we want to do is compare some signed numbers I know you know that 2 is less than seven without looking up above can you put the right symbol in here a long time ago when I was like in grade school my teacher told me to imagine this it doesn't want to draw it as an alligator maybe a pacman eating the larger number you know whatever it works at any rate 7 is the larger number and 2 is less than seven if we were going to look at this on a number line we could plot two and seven we wouldn't need all of the hash marks in between we know that two is over here three four five six and maybe seven is right about there so 2 is less than seven let's try negative three and five you probably already know that negative 3 is less than five because you would rather have five dollars then be three dollars in debt that's certainly the better situation all right one two three four here's 5 negative 1 negative 2 negative 3 there is negative 3 negative 3 is less than five we're using the number line on purpose here because I want you to see something when we get done let's try comparing negative 2 and negative 6 so let's put zero way over here so we make some space negative 1 negative 2 negative 3 negative 4 negative 5 and over here is negative 6 all right so the question is what's the better situation would you rather be two dollars in debt or six dollars in debt yeah I'd rather be two dollars in debt that is the better one the larger amount so to speak I guess is dead but even so the real question is what is the number line telling us so what I want you to do is to take your pencil and come in and circle the smallest number for each pair so for example here with negative SR with two compared to 72 is the smaller number negative three compared to five which one was the smaller number negative three negative 6 compared to negative two which one was the smaller number hmm well 2 is greater than negative 6 that makes negative six smaller than negative 2 you can also think about it as the small end of the arrow points towards the small number the wide large end is by the large number either way the negative 6 is over here and so what I want you to notice is that the smaller numbers are all on the left there we go all of the left stands are on the left so if we had a number line here with some mystery numbers here's B and here's a at first you might think that we couldn't tell which one is larger but we can we don't even need to know where zero is what we know is that for a number line the farther left we go the smaller the number gets in this picture B is the smaller number so here's our big deal maybe take your highlighter or circle this put some stars by it something like that big deal big deal less in here if X is to the left of Y on the number line then x is less than or greater than Y which do you think should be less than all right so your job here is to pause the recording and see if you can fill in the lengths with greater than less than or equals whichever one it makes the statement true pause and come back alright let's see how you did comparing negative eight thousand to negative 30 again it's probably easier to think about debt which is the better situation I would rather only be in debt thirty dollars then I would be in debt eight thousand dollars but if I imagined a number line negative eight thousand would be much farther to the left then negative 30 is this one might be a little more challenging negative 11.2 compared to negative 11.3 maybe we need a number line here's negative 11 here's negative 11.1 here's negative 11.2 and way over here is negative 11.3 there now we can clearly see it negative 11.3 is to the left of negative 11.20 compared to negative 57 having no money is always preferable to being a debt and now this one hmm well let's remember the opposite of that's hella translate this first negative sign the opposite of and then inside the parentheses negative 12 the opposite of negative 12 is a positive 12 so the left-hand side of this is worth positive 12 the right-hand side is worth negative 12 and now we know which way the inequality goes alright the next thing we want to talk about is absolute value absolute value of a number measures its distance from zero not distance in terms of feet but distance in terms of units when we want to talk about the absolute value we have vertical lines so vertical lines like this the absolute value of and then whatever we're talking about goes on the inside be careful to make these nice and tall we don't want to mistake the lines being used in absolute value for negative for ones that looks kind of sloppy so I'm going to fix this up there we go with that so if we were looking at the absolute value of 3 we're saying how far away is 3 from 0 and 3 is 3 units away from 0 if we wanted to say the absolute value of negative 1 point2 then we want to know how far away is negative 1 point2 from 0 and it's one point two units away from zero right when you throw a ball and you tell somebody how far you through it you don't think about the direction in which you through it you just measure the distance and so you report that as something that is not negative even if you threw the ball backwards you say you threw the ball backwards a certain amount so absolute value is never negative and that's because it measures distance or it describes distance alright so now it's your turn to try again pause the recording and see what you can do with the next three examples okay so now you're back let's check and see what you did the absolute value of negative 62.4 is 62.4 the absolute value of 273 is 273 the absolute value of 0 is 0 because 0 is right on top of 0 it is no distance away okay so true or false the absolute value of a number is always the opposite of the number look back up at what we just did an example 3 and it's pretty easy to see that this is false some people say that the absolute value of a number is always positive and that's false also when we can see that here in C right for be we didn't change 273 to negative 273 it stayed the same so what we're really doing is saying that the absolute value is just never negative all right let's try something else here see if we can combine what we've talked about so far again we want to fill in these blanks with greater than less than or equals to make something true pause the recording and see how you do okay let's compare your answers with mine let's see the absolute value of negative 8 that's worth 8 so when I compare 828 these are equal the absolute value of negative 24 is 24 the opposite of negative 24 is also 24 those are equal the absolute value of negative 14 is 14 and when we compare 14 to 12 we use greater than the opposite of the absolute value of negative 6 whoo okay let's just work inside the absolute value first let's just do this as much the absolute value of negative 6 is 6 bring this negative sign right down there we go negative 6 compared to 0 less than so when might you see a negative number well I suppose there's the obvious there's always debt which I hope you never do see but um you know it's possible right debt certainly a case where we might see negative numbers we also have seen in the winter time temperatures below zero we can talk about sea level elevations that are below sea level are talked about with negative numbers for all of you football fans you've heard negative numbers when they talk about loss of yardage things more related to heating and air conditioning let's see in an AC circuit right this alternating current we have negative voltage if we are rotating something and you go this direction counterclockwise this is actually a positive rotation but if you go the other way this is a negative rotation you might also see a negative exponent when we start talking about very very small quantities will use them to use the negatives to write them in what we call scientific notation and you'll see a negative exponent there you may also see a negative sign used to describe the slope of a pipe coming away from a house a lot of different times when you can see negatives other times we create negatives because they're helpful whenever you have a benchmark that gets set amounts below this benchmark it represented with negative numbers so rather than saying temperatures below zero we can talk about temperatures below normal whatever normal happens to be maybe you've got your house set for 72 degrees Fahrenheit and then you start recording the temperature differences right then we would have positive three degrees will be three degrees above normal negative three degrees would be three degrees below normal those of you who are golfers there's a benchmark called par right and if you are better than par then you have a negative score a certain number of strokes below par we can also talk about the time before something happens that you might say negative two seconds if you've ever watched a rocket being launched you've heard them say ah t minus two and Counting something like that talking about the time before that magic liftoff moment all right well that's probably enough for right now go see what you can do with the homework and we'll talk to you soon take care