For more information, please see full course syllabus of Pre Calculus
For more information, please see full course syllabus of Pre Calculus
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 xaxis, and the vertical axis the yaxis.
 Sometimes we'll talk about which quadrantquarters of the planea point is located in. We start with where both coordinates are positive: the topright, then work counterclockwise, 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 (threedimensional) space. This course won't explore much in three dimensions, but it's interesting to think about.
Coordinate Systems

 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.
 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 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.
 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.
 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').
 Begin by drawing the coordinate axes and applying some evenlyspaced tick marks to each axis.
 The first number in the ordered pair is the location on the xaxis: the horizontal location. The positive direction on the xaxis is to the right.
 The second number in the ordered pair is the location on the yaxis: the vertical location. The positive direction on the yaxis is up.
 Begin by drawing the coordinate axes and applying some evenlyspaced tick marks to each axis.
 The first number in the ordered pair is the location on the xaxis: the horizontal location. The positive direction on the xaxis is to the right.
 The second number in the ordered pair is the location on the yaxis: the vertical location. The positive direction on the yaxis is up, so the negative direction is down.
 Begin by drawing the coordinate axes and applying some evenlyspaced tick marks to each axis.
 The first number in the ordered pair is the location on the xaxis: the horizontal location. The positive direction on the xaxis is to the right, so the negative direction is to the left.
 The second number in the ordered pair is the location on the yaxis: the vertical location. The positive direction on the yaxis 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 halfway between the `2' tick mark and the `3' tick mark.
 The first quadrant (I) is the topright portion of the coordinate axes. This is where both the x and y coordinates are positive.
 The rest of the quadrants are assigned counterclockwise from there.
 Use substitution to see that the ordered pair (m, n) = (−1, −3).
 Draw the coordinate axes and apply some evenlyspaced tick marks to each axis.
 The first number in the ordered pair is the location on the xaxis: the horizontal location. The second number in the ordered pair is the location on the yaxis: the vertical location.
 Refer to the previous question to see which quadrant it is in.
Quadrant III
 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 evenlyspaced tick marks to each axis.
 The first number in the ordered pair is the location on the xaxis: the horizontal location. The second number in the ordered pair is the location on the yaxis: 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 screencaptured 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
 Inherent Order in ℝ
 'Less Than' and 'Greater Than'
 One Dimension: The Number Line
 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
 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
 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
 Three Dimensions: Space
 Higher Dimensions
 If We Have n Dimensions, We Call It nDimensional Space or ℝ to the nth Power
 We Can Represent Places In This nDimensional Space As Ordered Groupings of n Numbers
 Hard to Visualize Higher Dimensional Spaces
 Example 1
 Example 2
 Example 3
 Example 4
 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
 ThreeDimension = Space = ℝ³
 Higher Dimensions 22:24
 If We Have n Dimensions, We Call It nDimensional Space or ℝ to the nth Power
 We Can Represent Places In This nDimensional 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
Precalculus with Limits Online Course
Transcription: Coordinate Systems
Hiwelcome 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 isthe 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 negativesif 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 transitivethat 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 signhow 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 consideringthis 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 finegrained 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 2.8889tiny, 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 honestI am lazy; I would like to find a way to be able to do this with less spaceto 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 realswe 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 numbersif 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 numbersit 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 linewhat is closer to the frontwhat 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 "thiscomesfirst, thiscomessecond, thiscomesthird"...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 twodimensional 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 yaxis.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 pairboth the first value and the second valueand 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 xaxis, and the vertical axis the yaxis.0931
Why do we do thiswhat 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, ythat 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 Frenchsorry, 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 himthe 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 planefour 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 xvalue is greater than 0, and the yvalue 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 itsorry.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 issorry.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 yaxis; we have gone over the place that is 0 on the xaxis.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 xaxis), and now we have made it onto the negative side1208
of the yaxis, 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 xaxis, but we are still in the negative part of the yaxis.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 plane1239
it has to actually be completely inside of the quadrant to be considered in a quadrant.1244
If it was on a midground 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 threedimensional object with a twodimensional 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 notsogreat way.1301
We call this threedimensional space: space is the word we use for it, because it is just like the space we live in.1307
We live in a threedimensional 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 ndimensional 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 ndimensional 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 higherdimensional spaces, like this.1398
We live in and are adapted to exist in a threedimensional 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 twodimensional beings1432
coming to live in a threedimensional 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 onthe 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 pointswe 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 scaleit 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 horizontallywe 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 numberswe 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 > 0what 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 xcoordinate, its horizontal coordinate, is negative; it is on the left side of the vertical line.2033
And we know that its vertical coordinate, its ycoordinate, is positivethat 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 latergoodbye!2100
0 answers
Post by Macy Li on July 24 at 01:41:06 PM
Is it possible to increase the speed of this video, so I can just glimpse it for a review? Thanks.
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 SelhorstJones
Fri Feb 21, 2014 9:24 AM
Post by Linda Volti on February 20, 2014
Great again, thank you!