WEBVTT mathematics/pre-calculus/selhorst-jones
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Hi--welcome back to Educator.com.
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For this lesson, we are going to talk about coordinate systems.
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The real numbers are great, because there is an inherent order in them.
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Whenever we think about numbers, we naturally get a sense of progression.
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There is a natural progression, at least in the positive numbers.
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If we consider the positive numbers, it seems fairly inherent to us, I think, that the larger a number is, the larger its quantity represented is.
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And so, the higher in the order it is--the number 1 is lower than 2, is lower than 3, is lower than 4.
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And if you are in between them, if you are, say, 1 and 1/2, then you would be between 1 and 2.
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You would be greater than 1, but you would be less than 2, if we are looking at the number 1 and 1/2.
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So, we have a pretty inherent sense of an order that fits to the real numbers.
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Now, what if we want to expand that to the negative numbers?
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If we want to consider the negatives--if we want to have our order be usable, not just on the positive portion
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(which makes obvious sense), but we also want to be able to use it on the negatives,
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then we want our negative and positive orders to agree.
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We want to be able to order the negative and positive numbers at the same time.
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So, to make sure that they agree, we make it so that small negatives, the negatives that are closer to 0,
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are higher in the order than large negatives, which makes a certain kind of sense.
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A small negative takes less away, so it seems more reasonable that it is a bigger thing, because it does less damage, in a way.
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That is not the best metaphor: but a smaller negative takes less away,
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so it is closer to being a positive than a big negative number, which takes even more away.
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So, it makes sense that really big negative numbers come lowest, then small negative numbers, then 0, then small positive numbers, then large positive numbers.
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So, this seems pretty reasonable, and it makes internal logic.
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There is an internal logic here; it doesn't contradict itself; so it seems like a good thing to run with.
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We take this idea, and now we will use symbols, so that we can denote it easily.
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We denote the order with two symbols: the "less than" symbol and the "greater than" symbol.
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So, -1 is less than 0; 0 is less than 1; 1 is less than 2; 5 is greater than -12; -12 is greater than -47.
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So, these properties are transitive--that is something we can notice.
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If a is less than b is less than c, then a is less than c, as well (and it is similar for greater than).
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This makes sense: if we have -12 in between them, we can just sort of cut out the -12, and we will get 5 > -47.
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And similarly, over here, -1 < 1; we have the order, and we don't have to have all of the elements in the middle for the order to still be there.
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If you have difficulty remembering which way the sign points (does it point to the big number,
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or does it point away from the big number?), I like this mnemonic that I was taught long ago, and it really helped me learn it.
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This sign--how do we know which way it goes?
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We imagine that it is an alligator; there is an alligator here, and the alligator is hungry.
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So, the alligator is hungry, and because he is hungry, he wants to eat the biggest thing possible.
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So, he says, "I am hungry; give me big food!"
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So, it makes sense that the alligator is going to point towards the bigger object.
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The bigger the number is in our scale...the alligator will want to eat that one in preference of the other one.
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So, the alligator is one way of thinking about it; the other one is just thinking that the wide part is always pointed at the bigger number.
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But I like the alligator mnemonic, and it worked well for me, so there it is for you, if you haven't heard it before.
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All right, less than/greater than...we can also talk about this in terms of variables.
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So, if we know that a variable is less than 2, but we don't know the precise value of the variable, we could say x < 2.
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That gives us the ability to use this order on a variable.
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We know that the variable is less than the number 2.
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We don't know what it is precisely, but we know whatever x is, it must hold true with this relationship.
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We call this kind of relationship an inequality, because the two sides are not just equal.
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An equality implies that there is an equal sign on the two expressions; inequality implies that the two expressions are not equal.
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And we know something about how they are unequal.
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So, an inequality is going to be less than or greater than, depending; and that is where we get inequalities.
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If we want to say that the relationship might be equal, we can use the signs "less than or equal" and "greater than or equal."
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These come from a merging of the less than (or the greater than) and the equals sign.
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We put these two things together, and together they put out things like this.
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That is where we are getting it: it is the less than or greater than sign up top, and the bottom half is 1/2 of an equals sign.
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So, that is where we are getting less than or equal and greater than or equal.
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So, 1 is less than 2; and technically, 2 is less than or equal to 2.
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It is true, although that would be one of those cases where it is not a very useful thing to say; but it is accurate.
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All right, moving on: now we can express this idea of our order that we have been just considering--this idea of order on the number line.
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The number line is a graphical representation of the order that the reals have inside of them.
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We build this out from the origin, the 0, to the left with -∞ and to the right with positive ∞.
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And remember, I talked about this previously, but infinity is not actually a number in the reals.
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And it is not a number on the number line; it is just the idea of continuing on forever.
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It is that arrow that says "just keep going in this direction."
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We never stop going to the right as we go to positive infinity; we never stop going to the left if we go to negative infinity.
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So, we never actually hit those values, because they aren't actually values.
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It is just the idea of keeping going on forever.
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All right, if a number is farther to the right, it is greater; the greater numbers go on the right; the less numbers go on the left.
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So, if you are to the right of a number, then you are greater than it; if you are to the left of a number, then you are less than it.
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So, this number line gives us a really easy visual representation of that order we were talking about.
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And we can put down any number we want: here is 1 and 1/2; here is 2 and 1/2; here, somewhere around here, is π.
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And we could talk about, say, 4.7 over here, and so on, and so on, and so on.
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The thing is that all of the real numbers, all of these very fine-grained numbers, fit in between the obvious landmarks of the whole numbers.
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The integers just make up landmarks; but the real numbers are that whole continuum, that fine spread of numbers.
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They are tiny, tiny little numbers: 2.888 versus 2.8889--tiny, tiny differences, but still different numbers.
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All right, ordered pairs: what if we want to talk about more than one number at a time?
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Say we want to talk about two numbers at once: consider this motivating example.
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We survey a number of households, and we ask how many dogs and how many cats each household has.
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We get these answers back: 0 dogs and 0 cats; 2 dogs and 0 cats; 1 dog and 2 cats; 0 dogs and 3 cats.
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Well, these are all of our answers, and we can write them out, as we just did.
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But I will be honest--I am lazy; I would like to find a way to be able to do this with less space--to be able to do this, having to write less.
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So, here is a useful place to bring up ordered numbers.
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We really only care about the numbers: 0 dogs, 0 cats...yes, OK, but really all I care about is that it was 0 and 0.
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So, as long as we know which number represents which animal, we can throw away the words.
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We can create ordered pairs, because we have to know what order it came in.
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Did it come in dog, then cat, or cat, then dog?
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So, we set up an arbitrary order: we set up that dogs go first and cats come second.
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And then, we can convert all of these words into just (0,0),(2,0),(1,2),(0,3).
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This takes much less space, much less writing, and the same information.
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So, if we want to talk about an ordered pair, some ordered grouping of numbers, it is (_,_), just numbers going in those blanks; and we close up the parentheses.
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Now, notice: there is some slight confusion with intervals in the reals--we talked about this before.
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If we had intervals in the reals, we might have shown it with parentheses.
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So, there is a possibility for a little bit of confusion when we are dealing with talking about ordered pairs and with talking about a point in two dimensions.
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But almost always, it is going to be obvious if we are talking about an interval, or if we are talking about a pair of numbers--if we are talking about a point.
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So, don't worry about getting these two things confused; it is almost always going to be totally obvious in the question which one is implied.
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It is very hard to get these two confused when we are actually working on problems,
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because it will be very clear, from the context, which one is meant.
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So, don't worry about that: even though they use the same notation, we will always know which one is actually being implied when we are working.
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So, this idea of an ordered pair allows us to talk about two numbers at the same time.
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Depending on the problem that we are working on, the relationship between the two numbers will change.
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In one problem, the relationship might be dogs and cats; in another problem, it might be the height of a ball and how many seconds we have gone in time.
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And in another problem, it might be the number of houses bought in a certain span of time, and the cost of all of those houses together.
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So, it is going to be totally different from problem to problem.
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And potentially, the two numbers could be completely unrelated.
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It could be the number of words in this lesson and the number of grains of rice that is currently sitting in a bowl in some restaurant in California.
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They are completely unrelated numbers, but we can just put them together, if we so desire.
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We won't want to do that in our problems, because it won't help us understand anything.
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But it is a possibility: the two numbers don't have to have anything to do with each other.
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In all of our problems, though, they will be somehow connected.
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And the problem will show us how they are connected.
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It should also be pointed out that ordered pairs of real numbers can't be put into an order, like we did with the reals.
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So, the reason they are called ordered pairs is because location in pair matters.
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We care about what is first; we care about what is second.
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It has a different meaning if we swap those two numbers--it is a different meaning if the second number comes first and the first number comes second.
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The location in the pair matters; but when we talk about an order, like in the real numbers,
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I am talking about being able to say what place in line--what is closer to the front--what is farther ahead.
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That is what I am talking about with this idea here.
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So, this idea here is different than the ordered pair idea.
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While they are called ordered pairs, they can't actually have an order.
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We could compare the first values, and we could compare the second values.
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But we can't actually say that an entire pair is greater or less than another pair.
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Consider these three pairs: (-10,10), (5,-5), and (-3,3); and let's also consider (0,0).
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None of these pairs is equal, because none of them are the same thing.
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To be equal, they have to actually be the same thing.
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(-10,10) is not the same as (5,-5), is not the same as (-3,3), is not the same as (0,0); none of these things are equal to each other.
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But we also can't put them in any order; who gets to be the biggest one--(-10,10)?
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If (-10,10) is the biggest, what about (10,-10)? Would that be bigger or smaller?
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And if it is bigger, why is it bigger than (-10,10)?
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And if it is not bigger, and it is not smaller, then it must be equal, if we are going to go with that idea of order that we have in the real numbers.
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So, any possibility of putting them in "this-comes-first, this-comes-second, this-comes-third"...that is not possible when we are talking about ordered pairs.
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We can give them out; we could give out a variety of them; but we can't really say anything about their location and where they came in.
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That is just something to notice about them.
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They are ordered, because we care about their first and second values.
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We care what order the values come in, in the pair, but we can't put them in an order,
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as in saying "this one goes here," and then followed by this one, and then followed by this one; that is not a possibility.
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All right, this gives us the ability to talk about a two-dimensional surface, a plane where we can plot these ordered pairs.
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We visually represented the reals with the number line; and now, we can represent our ordered pairs with the plane.
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We call it the plane: to do this, we cross a horizontal number line with a perpendicular vertical number line.
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They both cross at 0; so down here, at this little right angle, is 0 on both the horizontal axis and the y-axis.
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I don't know if you can quite see that; that should be an arrow pointing down in there.
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So, this gives us the ability to plot points, because now we can deal with both parts of our values.
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Value, value: one of the values we can put on one axis, and the other value we put on the other axis.
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And where they agree, we plot as a point.
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That way, we can talk about (3,2) being different from (3,3), because that is (3,2), and here is (3,3).
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So, we are able to talk about totally different locations by having this plane.
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We can put down both pieces of information from our ordered pair--both the first value and the second value--and that is really great for us.
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We call the point in the middle, that point of intersection of the two number lines, the **origin**.
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That is the origin; it is (0,0).
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By convention, the first number in an ordered pair always goes by the horizontal.
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So, if it is (first,second), then the horizontal location is always going to be based around the first value.
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And the vertical location is always going to be based around the second value.
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Once again, if I have something like (2,4), then the first value...we go here to 2, and then we rise up until we hit 4: (2,4).
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So, this convention is an important convention to remember: the first thing always
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gets placed in the horizontal; the second thing always gets placed in the vertical.
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And sometimes it will change: when we start working on functions, we will often call the vertical axis the f(x), or the value from the function.
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But normally, we are going to call the horizontal axis the x-axis, and the vertical axis the y-axis.
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Why do we do this--what is the reason for it?
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Well, often we talk about points (x,y), because they are coming from some equation y = ....involving the number x.
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All right, so we plug in some x; we will get some value here; and that will give us some value y, and then we will put them in.
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So, our y will be our second value; our x will be our first value.
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Often, we just associate x with being the horizontal, being that first value, and y with being that second value, being the vertical value.
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That is not always going to be the case; it could be something different; but that is normally what it is going to be.
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Also, if you have difficulty remembering what goes where (Is it x, then y? Is it y, then x? What is horizontal?
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What is not horizontal?), here is my mnemonic for you.
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Remember, it is going to be (x,y), because it is like the alphabet (w, x, y, z, so x, y--that is the order it comes in).
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And then, when we read, you read left to right, which is to say horizontally;
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and then you read up and down; you start high and you go low, which is to say vertically.
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So, when we do reading, just like normal reading, we start reading horizontally (at least in English).
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We start reading horizontally, left to right, and then after we have done that, we do up/down; we do vertical motion.
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So, it makes sense that (x,y) is like the alphabet; the alphabet goes like that.
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And then, if we are also continuing to talk about the alphabet, left to right is how we read first, and then up/down (vertical).
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So, x will go with the left/right, and y will go with the up/down.
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That is the mnemonic I am going to give you for this.
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That is maybe not the perfect mnemonic; but you really have to understand this one,
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because you have to be ready to see these things over and over and over.
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The plane has many different names: sometimes the plane is called the coordinate plane,
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because we call these values, the first value and the second value, the coordinates.
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Sometimes they will also be called the x coordinate and the y coordinate, the horizontal coordinate and the vertical coordinate.
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We also call it the Cartesian plane; why do we call it the Cartesian plane?
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It is because Rene Descartes (I am not very good with my French--sorry, Rene Descartes) was a French philosopher and mathematician
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in the early 1600s who did a lot of work with talking about things in the plane.
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He did a lot of really great math, and so it is named in honor of him--the Cartesian plane, from his name, Descartes.
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So, "Cartesian plane" is just coming from his name, Descartes.
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So, that is another name for it: coordinate plane, Cartesian plane, and one more way you can call it,
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which you probably don't see until you get into college much, but you will see it now and then
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if you get into advanced math in college: R2.
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We will talk about R2, because what we have is one real line crossed with another real line; so it is R and R, or 2 R's put together, R2.
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We can also talk about quadrants within the plane--four quarters of the plane.
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We want to be able to talk about a point being in one of the quadrants, the four quadrants of the plane.
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We need to know where each quadrant occurs.
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So, we start with where both coordinates are positive: that is quadrant 1, where the x-value is greater than 0, and the y-value is greater than 0.
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Both values are positive: positive and positive.
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Then, from there, we work our way counterclockwise.
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Why do we work our way counterclockwise? There is no good reason.
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We just chose one, because humans had to choose one at some point, and it just became the way we do it--sorry.
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If you would rather it was clockwise, then yes, it is a little confusing.
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But maybe clockwise would be just as confusing as counterclockwise; it is just the way it is--sorry.
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We go counterclockwise from here; we start (I) in the positive location, positive and positive; and then we go to II.
00:19:49.100 --> 00:19:56.000
At this point, we have crossed over the y-axis; we have gone over the place that is 0 on the x-axis.
00:19:56.000 --> 00:20:02.200
So now, we are in "negative x land"; so it is going to be negative on the horizontal, but still positive there.
00:20:02.200 --> 00:20:07.800
And that is quadrant II; after that, we move on to quadrant III; now, it is going to be negative
00:20:07.800 --> 00:20:13.400
(because we are still on the negative side of the x-axis), and now we have made it onto the negative side
00:20:13.400 --> 00:20:16.900
of the y-axis, because we have dropped below the horizontal axis.
00:20:16.900 --> 00:20:23.700
So, here it is going to be negative and negative; and then, from there, we finally go on to quadrant IV, finishing things up.
00:20:23.700 --> 00:20:30.900
And now we have managed to flip over to being on the positive side of the x-axis, but we are still in the negative part of the y-axis.
00:20:30.900 --> 00:20:33.600
So, it is positive here and negative here.
00:20:33.600 --> 00:20:39.500
If a point is on one of the coordinate axes or both of the coordinate axes, it is not in a quadrant.
00:20:39.500 --> 00:20:44.100
It isn't in any quadrant; it actually has to be not on the lines building our plane--
00:20:44.100 --> 00:20:48.200
it has to actually be completely inside of the quadrant to be considered in a quadrant.
00:20:48.200 --> 00:20:52.000
If it was on a mid-ground between quadrants, we wouldn't really have a good way to talk about it.
00:20:52.000 --> 00:20:58.900
And we could say it is between quadrant I and quadrant II; but instead, we just say it doesn't have any quadrant,
00:20:58.900 --> 00:21:03.500
unless it is actually completely inside of a quadrant.
00:21:03.500 --> 00:21:06.300
We can continue this idea to an even larger level.
00:21:06.300 --> 00:21:09.100
We can take these ideas and start running with them.
00:21:09.100 --> 00:21:11.400
If we want, we can create ordered triplets.
00:21:11.400 --> 00:21:17.200
Before, in two dimensions, we had (x,y); now we can go to three dimensions, and we can have (x,y,z).
00:21:17.200 --> 00:21:21.400
To visually represent this, we have our same perpendicular thing.
00:21:21.400 --> 00:21:26.700
We have that same sheet that we used to have here; that same plane is back here.
00:21:26.700 --> 00:21:30.700
But then, in addition to that, we create another vertical axis.
00:21:30.700 --> 00:21:36.000
It is a little hard to see, because we are trying to represent a three-dimensional object with a two-dimensional thing.
00:21:36.000 --> 00:21:40.800
But we have one line, one line, and then a third one coming out of them.
00:21:40.800 --> 00:21:46.700
All right, so we can sort of see it from my fingers in this not-so-great way.
00:21:46.700 --> 00:21:53.500
We call this three-dimensional space: **space** is the word we use for it, because it is just like the space we live in.
00:21:53.500 --> 00:21:58.700
We live in a three-dimensional world: you can go forward, backward, left, right, and up and down.
00:21:58.700 --> 00:22:03.400
This would be the combination of those three major directions, which turns into 6 if we include
00:22:03.400 --> 00:22:08.400
the positive direction and the negative direction that we are able to move through the world we exist in.
00:22:08.400 --> 00:22:17.100
So, it is just like the space we exist in: since it is 3 real lines put together, we call it R3.
00:22:17.100 --> 00:22:21.300
This course won't explore much in three dimensions, but it is an interesting thing to think about.
00:22:21.300 --> 00:22:23.600
And we will have a little bit of stuff on it.
00:22:23.600 --> 00:22:26.400
If we want to, we can take these up to even higher dimensions.
00:22:26.400 --> 00:22:30.100
We can continue this idea and run up to as many dimensions as we want to have.
00:22:30.100 --> 00:22:41.200
If we have n dimensions, we call it n-dimensional space, which we might also refer to as Rn, because it is R...the real line, put n times together.
00:22:41.200 --> 00:22:45.200
We can represent places in this n-dimensional space as ordered groupings of n numbers.
00:22:45.200 --> 00:22:53.000
If we are in two dimensions, we have (x,y); if we are in three dimensions, we have (x,y,z).
00:22:53.000 --> 00:22:59.900
If we are in four dimensions, we just put in another one to that grouping: (x,y,z,w), or some other symbol.
00:22:59.900 --> 00:23:03.400
And so, we can keep running this up to as many symbols as we want.
00:23:03.400 --> 00:23:08.900
We can have as many different coordinate locations as we want for whatever our Rn is.
00:23:08.900 --> 00:23:18.500
You give me an n, and I can make a coordinate that has that many, n, slots in it to give us a coordinate system.
00:23:18.500 --> 00:23:22.600
However, there is no good way to visualize higher-dimensional spaces, like this.
00:23:22.600 --> 00:23:27.500
We live in and are adapted to exist in a three-dimensional world.
00:23:27.500 --> 00:23:33.400
All right, it is very hard, if not completely impossible (perhaps) to represent anything higher than three dimensions
00:23:33.400 --> 00:23:38.100
in a way that we can really see and intuitively grasp in a single picture.
00:23:38.100 --> 00:23:42.900
So, this course isn't going to discuss higher dimensions; but I think this stuff is really, really fascinating.
00:23:42.900 --> 00:23:44.800
And it is an interesting thing to ponder.
00:23:44.800 --> 00:23:48.300
If you think this is really interesting, and you are thinking, "Wow, I actually really want to think about this more,"
00:23:48.300 --> 00:23:52.300
there is a book called *Flatland* that is a pretty fun book.
00:23:52.300 --> 00:23:58.000
I actually haven't read it, but I know about it: *Flatland* is a book about two-dimensional beings
00:23:58.000 --> 00:24:02.700
coming to live in a three-dimensional world, and what their experience is like, and various things like that.
00:24:02.700 --> 00:24:08.400
So, if you think these ideas are really cool, go and check out this book, *Flatland*; it is pretty cool stuff.
00:24:08.400 --> 00:24:13.000
Example 1: if we want to order 5, 18, and -7, how do we order it?
00:24:13.000 --> 00:24:25.600
Well, first, we can just say, "OK, well, that is pretty easy, right? 5 < 18, and since negative is less than positive, it must be -7 < 5 < 18."
00:24:25.600 --> 00:24:29.900
There is our answer; but that is not the best way to approach it.
00:24:29.900 --> 00:24:37.000
Instead, it might be useful to be able to say, "Well, let's see if we can see it visually first."
00:24:37.000 --> 00:24:47.200
So, instead, we make a number line, and we won't be very careful about giving it a scale.
00:24:47.200 --> 00:24:50.800
But we can still get a sense of where these numbers are.
00:24:50.800 --> 00:24:56.500
Well, here is -7, somewhere over here on the left side; and then 5 is kind of closer to 0.
00:24:56.500 --> 00:24:59.300
And then, 18 is way out farther to the right.
00:24:59.300 --> 00:25:06.900
And so, we see this in its order: it goes -7 to 5, and then 5 to 18, which is exactly what we have right here.
00:25:06.900 --> 00:25:11.300
So, for this kind of problem, where we are just ordering three numbers that we can actually see, it is not that useful.
00:25:11.300 --> 00:25:16.200
But it becomes really handy when we are working with numbers that we can't actually lay hands on.
00:25:16.200 --> 00:25:18.200
We don't know what the value of the number is.
00:25:18.200 --> 00:25:27.200
For example, if we know that a is greater than 0, and we want to order a, 2a, and 3a;
00:25:27.200 --> 00:25:29.800
it becomes really handy to think of it in terms of this number line.
00:25:29.800 --> 00:25:38.400
We don't know where a is, but we know it is somewhere to the right of 0, because it is a positive number: a > 0 implies that a is a positive number.
00:25:38.400 --> 00:25:46.100
So, it is somewhere over here: well, if a is over to the right, then 2a would just be adding on another a.
00:25:46.100 --> 00:25:51.500
So, we get to 2a, because that would be a up, and then, if we want to get to 3a, we just add up another a.
00:25:51.500 --> 00:25:57.500
Now, we see what the order is: it goes a to 2a, and then 2a to 3a; now we have our order.
00:25:57.500 --> 00:26:04.500
We can see, visually, what might have been difficult to talk about in a really analytical way, with just symbols.
00:26:04.500 --> 00:26:09.200
By being able to make a picture, it becomes easier for us to understand; great.
00:26:09.200 --> 00:26:14.600
The next example: now we are going to really use this idea of using a number line to understand what is going on.
00:26:14.600 --> 00:26:22.100
If b is less than 0, we want to order b, 2b, 3b, -b, -5b, and 0.
00:26:22.100 --> 00:26:30.900
So, what we do here is set up that same number line; and let's arbitrarily place a 0 somewhere.
00:26:30.900 --> 00:26:37.400
Now, the first thing we need to do is, since everything (with the exception of the 0 right here) is in reference to b:
00:26:37.400 --> 00:26:41.600
we want to be able to say, "Well, where is b?"--we don't know its precise location.
00:26:41.600 --> 00:26:45.500
But we know which side it must fall on, because we know b is less than 0.
00:26:45.500 --> 00:26:52.400
Since b is less than 0, that is the same thing as being negative; so let's just put it here; b is less than 0 right now.
00:26:52.400 --> 00:26:57.700
Now, if we take 2b, well, 2b is going to go in the same direction as the original one.
00:26:57.700 --> 00:27:02.400
It is not going to be that b is below 0, and then 2b pops up to 0.
00:27:02.400 --> 00:27:06.700
We are going to continue to go backwards by another b; so now we will be at 2b.
00:27:06.700 --> 00:27:12.600
We do that again, and we get to 3b; there is b; there is 2b; there is 3b.
00:27:12.600 --> 00:27:15.100
We have ordered, first, all of the negative numbers.
00:27:15.100 --> 00:27:20.700
Now, what happens if we look at -b? Well, -b is going to take this same distance here,
00:27:20.700 --> 00:27:28.500
and it is going to flip it here; so what had been here to get to b will instead flip to -b.
00:27:28.500 --> 00:27:31.700
If we take 2 and we put a negative on it, we get to -2.
00:27:31.700 --> 00:27:37.900
We flip to the opposite side of 0, but that same distance away: b now flips to -b.
00:27:37.900 --> 00:27:45.000
If we want to look at -5b, then it is going to be a total of 5 b's up from 0, so we will be at -5b here.
00:27:45.000 --> 00:27:56.400
So now, we see what our order is: 3b < 2b < b < 0 < -b < -5b.
00:27:56.400 --> 00:28:01.700
So, what would otherwise be a very difficult problem for us to solve, if we were just trying to do it all in our head,
00:28:01.700 --> 00:28:07.300
trying to think purely in terms of the numbers going on, becomes a lot easier with a visual representation.
00:28:07.300 --> 00:28:14.300
One other way, if you have real difficulty with this, is to say, "OK, I don't know what b is, but we could use a hypothetical number."
00:28:14.300 --> 00:28:18.200
We could plug in b = -1 and try that out.
00:28:18.200 --> 00:28:27.100
We try out b = -1, and sure enough, b < 0...that fits with all the requirements that we have for b so far.
00:28:27.100 --> 00:28:30.500
So, it is a reasonable hypothetical number to choose.
00:28:30.500 --> 00:28:45.700
If that is the case, then b = -1; 2b would equal -2; 3b would equal -3; -b would equal -(-1), so positive 1; -5b would equal -5(-1), so positive 5.
00:28:45.700 --> 00:28:50.600
So, we get that same ordering going on--the exact same thing that is going to happen if we try out a hypothetical number.
00:28:50.600 --> 00:28:55.500
But I really like the idea of being able to see this visually, so that works out really well for this sort of thing.
00:28:55.500 --> 00:28:58.600
We get a good understanding of what is going on.
00:28:58.600 --> 00:29:04.900
Third example: Plot these points--we get all of these points; to plot them, we will need a plane to start with.
00:29:04.900 --> 00:29:14.100
So, we draw a vertical line; we draw a horizontal line; we get our horizontal axis and our vertical axis now.
00:29:14.100 --> 00:29:35.900
Let's mark off some sort of scale: 1, 2, 3, 4, 5, -1, -2, -3, -4, -5; 1, 2, 3, 4, 5, -1, -2, -3, -4, -5, -6.
00:29:35.900 --> 00:29:40.800
We have a scale going with it now: my scale is not perfect, but it is pretty good.
00:29:40.800 --> 00:29:47.900
I am not absolutely perfect in drawing on this thing; but it is not a terrible scale--it is good enough for us to get a good idea of where these would show up.
00:29:47.900 --> 00:29:57.600
So, plot the points: (0,5)...remember, the first goes to the horizontal; the second goes to the vertical.
00:29:57.600 --> 00:30:13.000
So, (0,5) is going to be 0 horizontally and up 5 vertically; 0 horizontally--we are right here; and then up 5: 1, 2, 3, 4, 5.
00:30:13.000 --> 00:30:29.400
Here we are at (0,5); (5,0) is forward 1, 2, 3, 4, 5; and we go up 0 because it is (5,0), so here we are at (5,0).
00:30:29.400 --> 00:30:57.000
If we want to do (-1,3), we go -1 horizontally and up 3 vertically: 1, 2, 3; and (4,-3)--we go over 4, 1, 2, 3, 4; down 3 because it is -3: -1, -2, -3.
00:30:57.000 --> 00:30:58.700
And we have all of our points plotted.
00:30:58.700 --> 00:31:05.300
Remember, the first value always goes to the horizontal; the second value always goes to the vertical.
00:31:05.300 --> 00:31:13.400
Final example: Let's say x is 2 and y is -1; now, we want to plot the points (x,y) and (2x,-3y).
00:31:13.400 --> 00:31:15.500
And we also want to say what quadrants they are in.
00:31:15.500 --> 00:31:23.900
Then, after we do that, let's start by saying a < 0 and b > 0, and then we need to say the quadrants of (a,b) and (-a,-b).
00:31:23.900 --> 00:31:30.800
So, first, let's do plotting the (x,y) and (2x,-3y).
00:31:30.800 --> 00:31:36.600
We know what the value of x is; we know what the value of y is; we can actually figure out what (x,y) is.
00:31:36.600 --> 00:31:44.900
We just swap out the numbers--we substitute: x is 2, so we get 2; y is -1, and there is (x,y).
00:31:44.900 --> 00:31:57.700
If we want to figure out what (2x,-3y) is, then we substitute for the values, and we will get 2(2)...
00:31:57.700 --> 00:32:12.600
and let me move it down to the next line...so (2(2),-3(-1)), which is the same thing as (4,3).
00:32:12.600 --> 00:32:15.000
So, there are our two points that we are looking to plot.
00:32:15.000 --> 00:32:30.100
We draw our coordinate axes, and quickly put on a scale for it to have, so we have places to plot.
00:32:30.100 --> 00:32:43.500
We go over 2, down 1 to -1, and there is (2,-1); we go over 4, 1, 2, 3, 4; up 3, 1, 2, 3; and there is (4,3).
00:32:43.500 --> 00:32:53.600
So, (4,3) is in the first quadrant; and we count counterclockwise, 1 to 2, 2 to 3, 3 to 4.
00:32:53.600 --> 00:33:00.000
And so, this is in the fourth quadrant: fourth quadrant, first quadrant, and there they are plotted.
00:33:00.000 --> 00:33:05.300
What if we wanted to figure out what a < 0, b > 0--what quadrant it would be in?
00:33:05.300 --> 00:33:10.500
Well, we don't actually know what a is; we don't actually know what b is.
00:33:10.500 --> 00:33:13.100
But we have enough information to figure out what quadrant it is in.
00:33:13.100 --> 00:33:17.800
So, if a is less than 0, it is a negative number; and if b is greater than 0, it is a positive number.
00:33:17.800 --> 00:33:24.500
If we want to figure out where (a,b) is at, well, if a is a negative number, then it is going to be somewhere over here.
00:33:24.500 --> 00:33:29.100
We don't know what the precise value is; but we are just being rough, so we can get a sense, visually, of where it goes.
00:33:29.100 --> 00:33:37.200
And b is going to be a positive number, so it is going up; remember, positive is this way; negative is this way; positive this way, negative this way.
00:33:37.200 --> 00:33:41.300
So, it goes up; and so, b is going to be somewhere here.
00:33:41.300 --> 00:33:48.400
Who knows where it is specifically, but we are going to have (a,b) somewhere in this area.
00:33:48.400 --> 00:33:53.100
We have no idea what the specific values of a and b are, but we know that it is going to have to fall in there,
00:33:53.100 --> 00:33:58.800
because we know that its x-coordinate, its horizontal coordinate, is negative; it is on the left side of the vertical line.
00:33:58.800 --> 00:34:05.600
And we know that its vertical coordinate, its y-coordinate, is positive--that it is on the top side of the horizontal line.
00:34:05.600 --> 00:34:11.600
So, we know that we are somewhere in this quadrant, which is quadrant II; we are somewhere in quadrant II.
00:34:11.600 --> 00:34:19.100
If we want to figure out where (-a,-b) is, well, if a is here, then it must be the case that -a is over here.
00:34:19.100 --> 00:34:24.300
If b is here, then it must be the case that -b is down here.
00:34:24.300 --> 00:34:30.100
So, we put the two together: and (-a,-b)...who knows if it is going to be at that specific point,
00:34:30.100 --> 00:34:36.500
but we know from this logic that, since it was previously negative horizontally, it is going to be positive horizontally;
00:34:36.500 --> 00:34:42.800
since it was previously positive vertically, it is going to be negative vertically.
00:34:42.800 --> 00:34:48.800
That drops us into this quadrant down here; we must be in quadrant IV.
00:34:48.800 --> 00:34:52.800
We get quadrant II and quadrant IV from the two points for this.
00:34:52.800 --> 00:34:57.100
All right, I hope you learned a bunch; I hope everything is clear to you, and you are remembering everything that you need,
00:34:57.100 --> 00:35:00.600
so you can really do precalculus and get a great understanding of what is going on here.
00:35:00.600 --> 00:35:02.000
We will see you at Educator.com later--goodbye!