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Lecture Comments (5)

2 answers

Last reply by: Magesh Prasanna
Thu Sep 11, 2014 3:22 AM

Post by Magesh Prasanna on September 3, 2014

Hello sir! Awesome lecture...I am able to understand why negative times negative becomes positive by considering +ve number as money that I have and -ve number as money that I spend. Sir help me to find the formal proof.
Some of my teachers said that happens by Field operations.
What are the prerequisites to understand the depth of the field operations?
Thank You.

1 answer

Last reply by: Professor Selhorst-Jones
Fri Feb 21, 2014 9:24 AM

Post by Linda Volti on February 20, 2014

Great again, thank you!

Coordinate Systems

  • The real numbers (ℝ) have an inherent order to them. Large negatives are lowest, then small negatives, then 0, then small positives, and finally large positives.
  • We can show this order with the symbols   <  ('less than') and   >  ('greater than'). Examples: −7 < 2,  100 > 47.
  • If we want to indicate that the relationship between two numbers might be equal, we can use   ≤  ('less than or equal') and   ≥  ('greater than or equal'). Example: x ≤ 5 means that x can be any number up to and including 5, while y < 5 means that y must be strictly less than 5.
  • If we have a mathematical relationship based on one of the above, we call it an inequality because the two sides are not equal.
  • We can graphically represent this idea of order with the number line. We build out from 0 (the origin) to −∞ on the left and ∞ on the right.
  • If we want to talk about two numbers at the same time, we can create an ordered pair. We can represent these ordered pairs of numbers with the plane: two number lines crossed perpendicularly.
  • In the plane, we call the point of intersection the origin: (0,0). By convention, the first number in an ordered pair always goes by the horizontal, and the second by the vertical. While it changes, we often call the horizontal axis the x-axis, and the vertical axis the y-axis.
  • Sometimes we'll talk about which quadrant-quarters of the plane-a point is located in. We start with where both coordinates are positive: the top-right, then work counter-clockwise, counting off the four quadrants.
  • We can continue with the idea or ordered pairs by creating ordered triplets. These can be represented visually with another perpendicular number line to create a third dimension. We call this a (three-dimensional) space. This course won't explore much in three dimensions, but it's interesting to think about.

Coordinate Systems

Below, replace the question marks with the appropriate relation symbol:
3   ?   7               120    ?   −578               3.2   ?   3.2
  • A `relation symbol' is a symbol that tells the relationship between two things. Things like < (`less than'), > ('greater than'), or = ('equals').
  • When using < or > , remember that the wide part of the symbol faces the larger number.
  • A positive number is larger than a negative number, no matter how big the negative is.
3 < 7               120 > −578               3.2 = 3.2
What is the difference between x < 3 and x ≤ 3?
  • Because x is a variable, each relation gives a limitation on what values x can have.
  • If the symbol is < (or > ), the two things can not be equal. They can get very, very close, but they can never equal each other.
  • If the symbol is ≤ (or ≥ ), the two things can be equal.
x < 3 says that x can be any number less than 3, but x can not equal 3 or anything larger. (This is called exclusive: the relationship excludes 3.)
x ≤ 3 says that x can be any number less than 3 or equal to 3, but x can not be anything larger. (This is called inclusive: the relationship includes 3.)
The difference is that x < 3 excludes 3 as a possibility for x, while x ≤ 3 includes the possibility.
Place the numbers -4,  3, and 0.5 on the number line.
  • The number line has negatives go off to the left and positives go off to the right. It goes off infinitely in both directions (which we represent by drawing arrows on either end).
  • Each integer (whole number) is evenly spaced along the line.
  • Draw the number line, mark the location of each integer, then place the numbers that the problem gives.
Given the following diagram of the number line, order a, b, c, and d.
  • The number line has negatives go off to the left and positives go off to the right. It goes off infinitely in both directions (which we represent by drawing in arrows on either end).
  • Numbers that are more negative are considered lower (`less than').
  • Numbers that are more positive are considered higher (`greater than').
b < d < c < a
Plot the point (2, 4) on coordinate axes (the plane).
  • Begin by drawing the coordinate axes and applying some evenly-spaced tick marks to each axis.
  • The first number in the ordered pair is the location on the x-axis: the horizontal location. The positive direction on the x-axis is to the right.
  • The second number in the ordered pair is the location on the y-axis: the vertical location. The positive direction on the y-axis is up.
Plot the point (3,  −4) on coordinate axes (the plane).
  • Begin by drawing the coordinate axes and applying some evenly-spaced tick marks to each axis.
  • The first number in the ordered pair is the location on the x-axis: the horizontal location. The positive direction on the x-axis is to the right.
  • The second number in the ordered pair is the location on the y-axis: the vertical location. The positive direction on the y-axis is up, so the negative direction is down.
Plot the point (−5,  2.5) on coordinate axes (the plane).
  • Begin by drawing the coordinate axes and applying some evenly-spaced tick marks to each axis.
  • The first number in the ordered pair is the location on the x-axis: the horizontal location. The positive direction on the x-axis is to the right, so the negative direction is to the left.
  • The second number in the ordered pair is the location on the y-axis: the vertical location. The positive direction on the y-axis is up.
  • If you have a fractional or decimal number as one of the coordinates, just place it appropriately depending on how the tick marks are scaled. For example, if you put the tick marks as each being length 1, then 2.5 should be half-way between the `2' tick mark and the `3' tick mark.
Draw the coordinate axes and number each of the four quadrants.
  • The first quadrant (I) is the top-right portion of the coordinate axes. This is where both the x and y coordinates are positive.
  • The rest of the quadrants are assigned counter-clockwise from there.
Let m=−1 and n=−3. Plot the point (m,  n). Also name which quadrant the point is in.
  • Use substitution to see that the ordered pair (m,  n) = (−1,  −3).
  • Draw the coordinate axes and apply some evenly-spaced tick marks to each axis.
  • The first number in the ordered pair is the location on the x-axis: the horizontal location. The second number in the ordered pair is the location on the y-axis: the vertical location.
  • Refer to the previous question to see which quadrant it is in.

Quadrant III
Let a=3 and b = [1/2]. Plot the point (3a−5,   −2b+1).
  • Use substitution to figure out the ordered pair (3a−5,   −2b+1). Once you substitute in a and b, simplify.
  • (3a−5,   −2b+1 )  = (3·(3)−5,   −2·([1/2])+1)  = (4,   0)
  • Draw the coordinate axes and apply some evenly-spaced tick marks to each axis.
  • The first number in the ordered pair is the location on the x-axis: the horizontal location. The second number in the ordered pair is the location on the y-axis: the vertical location.

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

Coordinate Systems

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Inherent Order in ℝ 0:05
    • Real Numbers Come with an Inherent Order
    • Positive Numbers
    • Negative Numbers
  • 'Less Than' and 'Greater Than' 2:04
    • Tip To Help You Remember the Signs
    • Inequality
    • Less Than or Equal and Greater Than or Equal
  • One Dimension: The Number Line 5:36
    • Graphically Represent ℝ on a Number Line
    • Note on Infinities
    • With the Number Line, We Can Directly See the Order We Put on ℝ
  • Ordered Pairs 7:22
    • Example
    • Allows Us to Talk About Two Numbers at the Same Time
    • Ordered Pairs of Real Numbers Cannot be Put Into an Order Like we Did with ℝ
  • Two Dimensions: The Plane 13:13
    • We Can Represent Ordered Pairs with the Plane
    • Intersection is known as the Origin
    • Plotting the Point
    • Plane = Coordinate Plane = Cartesian Plane = ℝ²
  • The Plane and Quadrants 18:50
    • Quadrant I
    • Quadrant II
    • Quadrant III
    • Quadrant IV
  • Three Dimensions: Space 21:02
    • Create Ordered Triplets
    • Visually Represent This
    • Three-Dimension = Space = ℝ³
  • Higher Dimensions 22:24
    • If We Have n Dimensions, We Call It n-Dimensional Space or ℝ to the nth Power
    • We Can Represent Places In This n-Dimensional Space As Ordered Groupings of n Numbers
    • Hard to Visualize Higher Dimensional Spaces
  • Example 1 25:07
  • Example 2 26:10
  • Example 3 28:58
  • Example 4 31:05

Transcription: Coordinate Systems

Hi--welcome back to Educator.com.0000

For this lesson, we are going to talk about coordinate systems.0002

The real numbers are great, because there is an inherent order in them.0007

Whenever we think about numbers, we naturally get a sense of progression.0011

There is a natural progression, at least in the positive numbers.0016

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.0019

And so, the higher in the order it is--the number 1 is lower than 2, is lower than 3, is lower than 4.0030

And if you are in between them, if you are, say, 1 and 1/2, then you would be between 1 and 2.0038

You would be greater than 1, but you would be less than 2, if we are looking at the number 1 and 1/2.0045

So, we have a pretty inherent sense of an order that fits to the real numbers.0050

Now, what if we want to expand that to the negative numbers?0055

If we want to consider the negatives--if we want to have our order be usable, not just on the positive portion0059

(which makes obvious sense), but we also want to be able to use it on the negatives,0064

then we want our negative and positive orders to agree.0067

We want to be able to order the negative and positive numbers at the same time.0070

So, to make sure that they agree, we make it so that small negatives, the negatives that are closer to 0,0075

are higher in the order than large negatives, which makes a certain kind of sense.0080

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.0086

That is not the best metaphor: but a smaller negative takes less away,0095

so it is closer to being a positive than a big negative number, which takes even more away.0099

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.0106

So, this seems pretty reasonable, and it makes internal logic.0115

There is an internal logic here; it doesn't contradict itself; so it seems like a good thing to run with.0119

We take this idea, and now we will use symbols, so that we can denote it easily.0125

We denote the order with two symbols: the "less than" symbol and the "greater than" symbol.0129

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 -47.0135

So, these properties are transitive--that is something we can notice.0147

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).0150

This makes sense: if we have -12 in between them, we can just sort of cut out the -12, and we will get 5 > -47.0155

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.0166

If you have difficulty remembering which way the sign points (does it point to the big number,0176

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.0182

This sign--how do we know which way it goes?0189

We imagine that it is an alligator; there is an alligator here, and the alligator is hungry.0192

So, the alligator is hungry, and because he is hungry, he wants to eat the biggest thing possible.0203

So, he says, "I am hungry; give me big food!"0212

So, it makes sense that the alligator is going to point towards the bigger object.0220

The bigger the number is in our scale...the alligator will want to eat that one in preference of the other one.0227

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.0233

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.0239

All right, less than/greater than...we can also talk about this in terms of variables.0244

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 < 2.0250

That gives us the ability to use this order on a variable.0257

We know that the variable is less than the number 2.0261

We don't know what it is precisely, but we know whatever x is, it must hold true with this relationship.0264

We call this kind of relationship an inequality, because the two sides are not just equal.0271

An equality implies that there is an equal sign on the two expressions; inequality implies that the two expressions are not equal.0275

And we know something about how they are unequal.0282

So, an inequality is going to be less than or greater than, depending; and that is where we get inequalities.0285

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."0291

These come from a merging of the less than (or the greater than) and the equals sign.0298

We put these two things together, and together they put out things like this.0304

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.0312

So, that is where we are getting less than or equal and greater than or equal.0319

So, 1 is less than 2; and technically, 2 is less than or equal to 2.0322

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.0328

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.0334

The number line is a graphical representation of the order that the reals have inside of them.0343

We build this out from the origin, the 0, to the left with -∞ and to the right with positive ∞.0348

And remember, I talked about this previously, but infinity is not actually a number in the reals.0356

And it is not a number on the number line; it is just the idea of continuing on forever.0361

It is that arrow that says "just keep going in this direction."0365

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.0369

So, we never actually hit those values, because they aren't actually values.0374

It is just the idea of keeping going on forever.0377

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.0381

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.0390

So, this number line gives us a really easy visual representation of that order we were talking about.0395

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.0400

And we could talk about, say, 4.7 over here, and so on, and so on, and so on.0412

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.0418

The integers just make up landmarks; but the real numbers are that whole continuum, that fine spread of numbers.0426

They are tiny, tiny little numbers: 2.888 versus 2.8889--tiny, tiny differences, but still different numbers.0433

All right, ordered pairs: what if we want to talk about more than one number at a time?0442

Say we want to talk about two numbers at once: consider this motivating example.0448

We survey a number of households, and we ask how many dogs and how many cats each household has.0453

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.0459

Well, these are all of our answers, and we can write them out, as we just did.0467

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.0470

So, here is a useful place to bring up ordered numbers.0481

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 0.0485

So, as long as we know which number represents which animal, we can throw away the words.0492

We can create ordered pairs, because we have to know what order it came in.0497

Did it come in dog, then cat, or cat, then dog?0500

So, we set up an arbitrary order: we set up that dogs go first and cats come second.0503

And then, we can convert all of these words into just (0,0),(2,0),(1,2),(0,3).0508

This takes much less space, much less writing, and the same information.0514

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.0518

Now, notice: there is some slight confusion with intervals in the reals--we talked about this before.0527

If we had intervals in the reals, we might have shown it with parentheses.0534

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.0538

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.0546

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.0556

It is very hard to get these two confused when we are actually working on problems,0563

because it will be very clear, from the context, which one is meant.0567

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.0572

So, this idea of an ordered pair allows us to talk about two numbers at the same time.0582

Depending on the problem that we are working on, the relationship between the two numbers will change.0586

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.0590

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.0599

So, it is going to be totally different from problem to problem.0608

And potentially, the two numbers could be completely unrelated.0611

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.0614

They are completely unrelated numbers, but we can just put them together, if we so desire.0624

We won't want to do that in our problems, because it won't help us understand anything.0628

But it is a possibility: the two numbers don't have to have anything to do with each other.0632

In all of our problems, though, they will be somehow connected.0636

And the problem will show us how they are connected.0639

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.0642

So, the reason they are called ordered pairs is because location in pair matters.0648

We care about what is first; we care about what is second.0660

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.0665

The location in the pair matters; but when we talk about an order, like in the real numbers,0672

I am talking about being able to say what place in line--what is closer to the front--what is farther ahead.0677

That is what I am talking about with this idea here.0688

So, this idea here is different than the ordered pair idea.0691

While they are called ordered pairs, they can't actually have an order.0696

We could compare the first values, and we could compare the second values.0700

But we can't actually say that an entire pair is greater or less than another pair.0704

Consider these three pairs: (-10,10), (5,-5), and (-3,3); and let's also consider (0,0).0708

None of these pairs is equal, because none of them are the same thing.0722

To be equal, they have to actually be the same thing.0727

(-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.0729

But we also can't put them in any order; who gets to be the biggest one--(-10,10)?0736

If (-10,10) is the biggest, what about (10,-10)? Would that be bigger or smaller?0742

And if it is bigger, why is it bigger than (-10,10)?0747

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.0751

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.0758

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.0767

That is just something to notice about them.0774

They are ordered, because we care about their first and second values.0776

We care what order the values come in, in the pair, but we can't put them in an order,0781

as in saying "this one goes here," and then followed by this one, and then followed by this one; that is not a possibility.0786

All right, this gives us the ability to talk about a two-dimensional surface, a plane where we can plot these ordered pairs.0792

We visually represented the reals with the number line; and now, we can represent our ordered pairs with the plane.0801

We call it the plane: to do this, we cross a horizontal number line with a perpendicular vertical number line.0807

They both cross at 0; so down here, at this little right angle, is 0 on both the horizontal axis and the y-axis.0815

I don't know if you can quite see that; that should be an arrow pointing down in there.0823

So, this gives us the ability to plot points, because now we can deal with both parts of our values.0828

Value, value: one of the values we can put on one axis, and the other value we put on the other axis.0835

And where they agree, we plot as a point.0844

That way, we can talk about (3,2) being different from (3,3), because that is (3,2), and here is (3,3).0849

So, we are able to talk about totally different locations by having this plane.0857

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.0863

We call the point in the middle, that point of intersection of the two number lines, the origin.0872

That is the origin; it is (0,0).0877

By convention, the first number in an ordered pair always goes by the horizontal.0880

So, if it is (first,second), then the horizontal location is always going to be based around the first value.0884

And the vertical location is always going to be based around the second value.0895

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).0900

So, this convention is an important convention to remember: the first thing always0915

gets placed in the horizontal; the second thing always gets placed in the vertical.0919

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.0923

But normally, we are going to call the horizontal axis the x-axis, and the vertical axis the y-axis.0931

Why do we do this--what is the reason for it?0937

Well, often we talk about points (x,y), because they are coming from some equation y = ....involving the number x.0939

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.0948

So, our y will be our second value; our x will be our first value.0954

Often, we just associate x with being the horizontal, being that first value, and y with being that second value, being the vertical value.0958

That is not always going to be the case; it could be something different; but that is normally what it is going to be.0965

Also, if you have difficulty remembering what goes where (Is it x, then y? Is it y, then x? What is horizontal?0970

What is not horizontal?), here is my mnemonic for you.0976

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).0979

And then, when we read, you read left to right, which is to say horizontally;1000

and then you read up and down; you start high and you go low, which is to say vertically.1015

So, when we do reading, just like normal reading, we start reading horizontally (at least in English).1024

We start reading horizontally, left to right, and then after we have done that, we do up/down; we do vertical motion.1030

So, it makes sense that (x,y) is like the alphabet; the alphabet goes like that.1037

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).1042

So, x will go with the left/right, and y will go with the up/down.1048

That is the mnemonic I am going to give you for this.1055

That is maybe not the perfect mnemonic; but you really have to understand this one,1058

because you have to be ready to see these things over and over and over.1062

The plane has many different names: sometimes the plane is called the coordinate plane,1066

because we call these values, the first value and the second value, the coordinates.1071

Sometimes they will also be called the x coordinate and the y coordinate, the horizontal coordinate and the vertical coordinate.1074

We also call it the Cartesian plane; why do we call it the Cartesian plane?1079

It is because Rene Descartes (I am not very good with my French--sorry, Rene Descartes) was a French philosopher and mathematician1083

in the early 1600s who did a lot of work with talking about things in the plane.1092

He did a lot of really great math, and so it is named in honor of him--the Cartesian plane, from his name, Descartes.1097

So, "Cartesian plane" is just coming from his name, Descartes.1104

So, that is another name for it: coordinate plane, Cartesian plane, and one more way you can call it,1107

which you probably don't see until you get into college much, but you will see it now and then1112

if you get into advanced math in college: R2.1116

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, R2.1119

We can also talk about quadrants within the plane--four quarters of the plane.1131

We want to be able to talk about a point being in one of the quadrants, the four quadrants of the plane.1136

We need to know where each quadrant occurs.1141

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 0.1144

Both values are positive: positive and positive.1156

Then, from there, we work our way counterclockwise.1161

Why do we work our way counterclockwise? There is no good reason.1164

We just chose one, because humans had to choose one at some point, and it just became the way we do it--sorry.1166

If you would rather it was clockwise, then yes, it is a little confusing.1173

But maybe clockwise would be just as confusing as counterclockwise; it is just the way it is--sorry.1176

We go counterclockwise from here; we start (I) in the positive location, positive and positive; and then we go to II.1183

At this point, we have crossed over the y-axis; we have gone over the place that is 0 on the x-axis.1189

So now, we are in "negative x land"; so it is going to be negative on the horizontal, but still positive there.1196

And that is quadrant II; after that, we move on to quadrant III; now, it is going to be negative1202

(because we are still on the negative side of the x-axis), and now we have made it onto the negative side1208

of the y-axis, because we have dropped below the horizontal axis.1213

So, here it is going to be negative and negative; and then, from there, we finally go on to quadrant IV, finishing things up.1217

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.1224

So, it is positive here and negative here.1231

If a point is on one of the coordinate axes or both of the coordinate axes, it is not in a quadrant.1233

It isn't in any quadrant; it actually has to be not on the lines building our plane--1239

it has to actually be completely inside of the quadrant to be considered in a quadrant.1244

If it was on a mid-ground between quadrants, we wouldn't really have a good way to talk about it.1248

And we could say it is between quadrant I and quadrant II; but instead, we just say it doesn't have any quadrant,1252

unless it is actually completely inside of a quadrant.1259

We can continue this idea to an even larger level.1263

We can take these ideas and start running with them.1266

If we want, we can create ordered triplets.1269

Before, in two dimensions, we had (x,y); now we can go to three dimensions, and we can have (x,y,z).1271

To visually represent this, we have our same perpendicular thing.1277

We have that same sheet that we used to have here; that same plane is back here.1281

But then, in addition to that, we create another vertical axis.1287

It is a little hard to see, because we are trying to represent a three-dimensional object with a two-dimensional thing.1291

But we have one line, one line, and then a third one coming out of them.1296

All right, so we can sort of see it from my fingers in this not-so-great way.1301

We call this three-dimensional space: space is the word we use for it, because it is just like the space we live in.1307

We live in a three-dimensional world: you can go forward, backward, left, right, and up and down.1313

This would be the combination of those three major directions, which turns into 6 if we include1319

the positive direction and the negative direction that we are able to move through the world we exist in.1323

So, it is just like the space we exist in: since it is 3 real lines put together, we call it R3.1328

This course won't explore much in three dimensions, but it is an interesting thing to think about.1337

And we will have a little bit of stuff on it.1341

If we want to, we can take these up to even higher dimensions.1343

We can continue this idea and run up to as many dimensions as we want to have.1346

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.1350

We can represent places in this n-dimensional space as ordered groupings of n numbers.1361

If we are in two dimensions, we have (x,y); if we are in three dimensions, we have (x,y,z).1365

If we are in four dimensions, we just put in another one to that grouping: (x,y,z,w), or some other symbol.1373

And so, we can keep running this up to as many symbols as we want.1380

We can have as many different coordinate locations as we want for whatever our Rn is.1383

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.1389

However, there is no good way to visualize higher-dimensional spaces, like this.1398

We live in and are adapted to exist in a three-dimensional world.1402

All right, it is very hard, if not completely impossible (perhaps) to represent anything higher than three dimensions1407

in a way that we can really see and intuitively grasp in a single picture.1413

So, this course isn't going to discuss higher dimensions; but I think this stuff is really, really fascinating.1418

And it is an interesting thing to ponder.1423

If you think this is really interesting, and you are thinking, "Wow, I actually really want to think about this more,"1425

there is a book called Flatland that is a pretty fun book.1428

I actually haven't read it, but I know about it: Flatland is a book about two-dimensional beings1432

coming to live in a three-dimensional world, and what their experience is like, and various things like that.1438

So, if you think these ideas are really cool, go and check out this book, Flatland; it is pretty cool stuff.1443

Example 1: if we want to order 5, 18, and -7, how do we order it?1448

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."1453

There is our answer; but that is not the best way to approach it.1465

Instead, it might be useful to be able to say, "Well, let's see if we can see it visually first."1470

So, instead, we make a number line, and we won't be very careful about giving it a scale.1477

But we can still get a sense of where these numbers are.1487

Well, here is -7, somewhere over here on the left side; and then 5 is kind of closer to 0.1491

And then, 18 is way out farther to the right.1496

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.1499

So, for this kind of problem, where we are just ordering three numbers that we can actually see, it is not that useful.1507

But it becomes really handy when we are working with numbers that we can't actually lay hands on.1511

We don't know what the value of the number is.1516

For example, if we know that a is greater than 0, and we want to order a, 2a, and 3a;1518

it becomes really handy to think of it in terms of this number line.1527

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.1530

So, it is somewhere over here: well, if a is over to the right, then 2a would just be adding on another a.1538

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.1546

Now, we see what the order is: it goes a to 2a, and then 2a to 3a; now we have our order.1551

We can see, visually, what might have been difficult to talk about in a really analytical way, with just symbols.1557

By being able to make a picture, it becomes easier for us to understand; great.1564

The next example: now we are going to really use this idea of using a number line to understand what is going on.1569

If b is less than 0, we want to order b, 2b, 3b, -b, -5b, and 0.1574

So, what we do here is set up that same number line; and let's arbitrarily place a 0 somewhere.1582

Now, the first thing we need to do is, since everything (with the exception of the 0 right here) is in reference to b:1591

we want to be able to say, "Well, where is b?"--we don't know its precise location.1597

But we know which side it must fall on, because we know b is less than 0.1601

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.1605

Now, if we take 2b, well, 2b is going to go in the same direction as the original one.1612

It is not going to be that b is below 0, and then 2b pops up to 0.1618

We are going to continue to go backwards by another b; so now we will be at 2b.1622

We do that again, and we get to 3b; there is b; there is 2b; there is 3b.1627

We have ordered, first, all of the negative numbers.1632

Now, what happens if we look at -b? Well, -b is going to take this same distance here,1635

and it is going to flip it here; so what had been here to get to b will instead flip to -b.1641

If we take 2 and we put a negative on it, we get to -2.1648

We flip to the opposite side of 0, but that same distance away: b now flips to -b.1652

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.1658

So now, we see what our order is: 3b < 2b < b < 0 < -b < -5b.1665

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,1676

trying to think purely in terms of the numbers going on, becomes a lot easier with a visual representation.1682

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."1687

We could plug in b = -1 and try that out.1694

We try out b = -1, and sure enough, b < 0...that fits with all the requirements that we have for b so far.1698

So, it is a reasonable hypothetical number to choose.1707

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 5.1710

So, we get that same ordering going on--the exact same thing that is going to happen if we try out a hypothetical number.1726

But I really like the idea of being able to see this visually, so that works out really well for this sort of thing.1730

We get a good understanding of what is going on.1735

Third example: Plot these points--we get all of these points; to plot them, we will need a plane to start with.1738

So, we draw a vertical line; we draw a horizontal line; we get our horizontal axis and our vertical axis now.1745

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, -6.1754

We have a scale going with it now: my scale is not perfect, but it is pretty good.1776

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.1781

So, plot the points: (0,5)...remember, the first goes to the horizontal; the second goes to the vertical.1788

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, 5.1797

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).1813

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, -3.1829

And we have all of our points plotted.1857

Remember, the first value always goes to the horizontal; the second value always goes to the vertical.1859

Final example: Let's say x is 2 and y is -1; now, we want to plot the points (x,y) and (2x,-3y).1865

And we also want to say what quadrants they are in.1873

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).1875

So, first, let's do plotting the (x,y) and (2x,-3y).1884

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.1891

We just swap out the numbers--we substitute: x is 2, so we get 2; y is -1, and there is (x,y).1896

If we want to figure out what (2x,-3y) is, then we substitute for the values, and we will get 2(2)...1905

and let me move it down to the next line...so (2(2),-3(-1)), which is the same thing as (4,3).1918

So, there are our two points that we are looking to plot.1932

We draw our coordinate axes, and quickly put on a scale for it to have, so we have places to plot.1935

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).1950

So, (4,3) is in the first quadrant; and we count counterclockwise, 1 to 2, 2 to 3, 3 to 4.1963

And so, this is in the fourth quadrant: fourth quadrant, first quadrant, and there they are plotted.1973

What if we wanted to figure out what a < 0, b > 0--what quadrant it would be in?1980

Well, we don't actually know what a is; we don't actually know what b is.1985

But we have enough information to figure out what quadrant it is in.1990

So, if a is less than 0, it is a negative number; and if b is greater than 0, it is a positive number.1993

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.1998

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.2004

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.2009

So, it goes up; and so, b is going to be somewhere here.2017

Who knows where it is specifically, but we are going to have (a,b) somewhere in this area.2021

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,2028

because we know that its x-coordinate, its horizontal coordinate, is negative; it is on the left side of the vertical line.2033

And we know that its vertical coordinate, its y-coordinate, is positive--that it is on the top side of the horizontal line.2039

So, we know that we are somewhere in this quadrant, which is quadrant II; we are somewhere in quadrant II.2045

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.2051

If b is here, then it must be the case that -b is down here.2059

So, we put the two together: and (-a,-b)...who knows if it is going to be at that specific point,2064

but we know from this logic that, since it was previously negative horizontally, it is going to be positive horizontally;2070

since it was previously positive vertically, it is going to be negative vertically.2076

That drops us into this quadrant down here; we must be in quadrant IV.2083

We get quadrant II and quadrant IV from the two points for this.2089

All right, I hope you learned a bunch; I hope everything is clear to you, and you are remembering everything that you need,2093

so you can really do precalculus and get a great understanding of what is going on here.2097

We will see you at Educator.com later--goodbye!2100