For more information, please see full course syllabus of Math Analysis

For more information, please see full course syllabus of Math Analysis

### Midpoints, Distance, the Pythagorean Theorem, & Slope

- To find the midpoint in one dimension, we take the average of the two numbers involved:
a+b 2. - To find the midpoint in two dimensions (in the plane), we take the average location for each dimension on its own:
⎛

⎝x _{1}+ x_{2}2, y _{1}+ y_{2}2⎞

⎠. - To find the distance between two points in one dimension, we subtract one from the other. However, that could potentially cause a negative to pop up, so we deal with that by taking the absolute value of the result. Thus, the distance between any two numbers is
|a−b|. - The Pythagorean theorem says, "On a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two legs." In other words, if the two legs (the shorter sides) are a and b, while the hypotenuse (the longest side) is c, then we have
a ^{2}+ b^{2}= c^{2}. - The Pythagorean theorem allows us to find the distance between two points in the plane. We can plot the points, draw in a triangle, then figure out the lengths of the two legs. From there, we use the theorem to find the hypotenuse, which is the distance between them. This gives the distance formula
d =

√(x_{2}− x_{1})^{2}+ (y_{2}− y_{1})^{2} -
*Slope*is a way to discuss how "steep" a line is. Another way to interpret it is the__rate of change__: the rate the line increases (or decreases) for every "step" to the right. We symbolize slope with m, and it is defined as any of the following equivalent things:m = rise run= vertical change horizontal change= y _{2}− y_{1}x_{2}− x_{1}. - Slope tells us how much the value of a line will change for every "step" to the right. A slope of m=−3 means that if we go 1 unit right, the line will drop down by 3 units. It is the line's
__rate of change__. - The idea of slope is very important in math (especially in calculus), so it's useful to have an intuitive sense of how slope works. Keep these facts in mind when thinking about slope:
- Positive (+) slope ⇒ line
__rises__(when going right), - Negative (−) slope ⇒ line
__falls__(when going right), - Bigger number (+ or −) ⇒
__steeper__line, - m=1 ⇒ line rises at 45
^{°}angle, - m=0 ⇒ line is horizontal,
- m=−1 ⇒ line falls at 45
^{°}angle.

- Positive (+) slope ⇒ line

### Midpoints, Distance, the Pythagorean Theorem, & Slope

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
- Introduction
- Midpoint: One Dimension
- Example of Something More Complex
- Use the Idea of a Middle
- Find the Midpoint of Arbitrary Values a and b
- How They're Equivalent
- Official Midpoint Formula
- Midpoint: Two Dimensions
- The Midpoint Must Occur at the Horizontal Middle and the Vertical Middle
- Arbitrary Pair of Points Example
- Distance: One Dimension
- Absolute Value
- Distance: One Dimension, Formula
- Distance Between Arbitrary a and b
- Absolute Value Helps When the Distance is Negative
- Distance Formula
- The Pythagorean Theorem
- Distance: Two Dimensions
- Break Into Horizontal and Vertical Parts and then Use the Pythagorean Theorem
- Distance Between Arbitrary Points (x₁,y₁) and (x₂,y₂)
- Slope
- Interpreting Slope
- Example 1
- Example 2
- Example 3
- Example 4

- Intro 0:00
- Introduction 0:07
- Midpoint: One Dimension 2:09
- Example of Something More Complex
- Use the Idea of a Middle
- Find the Midpoint of Arbitrary Values a and b
- How They're Equivalent
- Official Midpoint Formula
- Midpoint: Two Dimensions 6:19
- The Midpoint Must Occur at the Horizontal Middle and the Vertical Middle
- Arbitrary Pair of Points Example
- Distance: One Dimension 9:26
- Absolute Value 10:54
- Idea of Forcing Positive
- Distance: One Dimension, Formula 11:47
- Distance Between Arbitrary a and b
- Absolute Value Helps When the Distance is Negative
- Distance Formula
- The Pythagorean Theorem 13:24
- a²+b²=c²
- Distance: Two Dimensions 14:59
- Break Into Horizontal and Vertical Parts and then Use the Pythagorean Theorem
- Distance Between Arbitrary Points (x₁,y₁) and (x₂,y₂)
- Slope 19:30
- Slope is the Rate of Change
- m = rise over run
- Slope Between Arbitrary Points (x₁,y₁) and (x₂,y₂)
- Interpreting Slope 24:12
- Positive Slope and Negative Slope
- m=1, m=0, m=-1
- Example 1 28:25
- Example 2 31:42
- Example 3 36:40
- Example 4 42:48

### Math Analysis Online

### Transcription: Midpoints, Distance, the Pythagorean Theorem, & Slope

*Hi; welcome back to Educator.com.*0000

*Today we are going to talk about midpoints, distance, the Pythagorean theorem, and slope.*0002

*We have a bunch of things to talk about.*0006

*The concepts in this lesson, like all the other introductory lessons, are all ideas you have seen in previous math classes.*0008

*None of this should be totally new to you, but we definitely want to review them.*0013

*There are some important concepts for the course in here.*0017

*We are going to be talking about these things later on; we are not going to directly talk about these other ideas.*0020

*We are not going to be teaching on them directly (other than this lesson, where we will be doing that directly).*0025

*But it is going to be assumed that you understand them all.*0030

*They are all going to repeatedly show up as we work on more complex things, so we really want to make sure*0034

*that all of these things are totally understood now, and that we really know what we are doing.*0038

*Not only that, but we really want to understand what we are doing.*0042

*We don't just want to be able to do these things; we want to understand how it works--how these formulas are operating:*0045

*not just how to use the formulas, but how they work--why they work--what they mean--what is causing them to be the way they are.*0053

*As we get into more advanced math, like this course right here, it is going to become more and more important*0061

*for you to understand the big picture--not just how you can do this one problem, but why doing the problem this way makes sense.*0066

*As we see more and more complex ideas, it is absolutely necessary for you to be able to make sense of why we are doing the things we are.*0073

*If you are just doing it because that is what you are told to, and that was the step that has to come next,*0080

*eventually things are going to fall apart, and you are not going to be able to see what the next step is going to be.*0085

*As you get older and older, you take more responsibility; as you get into more and more advanced subjects,*0089

*you are expected to understand what is going on and be able to take things on yourself.*0093

*Back in algebra, you were able to just take step-by-step formulas and apply them.*0097

*But now, you have to understand why those step-by-step formulas work,*0101

*because you have to understand why that works, so you can now tackle more complex ideas.*0104

*So, don't just understand how you can use these things, but understand what is going on on a deeper level.*0109

*That is what I really want you to get here, and what I want you to get out of the entire course at large.*0113

*That should be the goal of your education at this point: being able to understand what it is doing--*0118

*why it works--not just going through it so you can get the next grade.*0123

*All right, let's get started.*0127

*Let's say we want to find the point that is halfway between 0 and 4.*0129

*It seems pretty easy, right? Halfway between 0 and 4...well, there is 0; click over; click over; look, it is 2.*0133

*It is just half of 4: 4 over 2 equals 2.*0140

*Well, what if I want something a little bit more complex--like I want to figure out the midpoint between -5 and 17?*0144

*What point is halfway between those two?*0149

*There are two ways we could approach this idea.*0151

*Say we want to find the midpoint between -5 and 17: there are two ways to look at this question.*0153

*First, we can look at it through the idea of distance.*0158

*The distance between -5 and 17: how far would we have to travel to get from -5 to 17?*0161

*We would have to travel 22: - 5 to 0--we travel 5; 0 on to 17--we travel another 17.*0166

*We can also look at it from the point of view of 17 minus the one before it (-5); we get 22 either way we do it.*0173

*Technically, we haven't formally defined what distance means; we will in just a little bit.*0180

*But this makes sense: we can see that that should be 22.*0184

*So, to find the midpoint, what we can do is start at -5, and then we will work halfway up.*0187

*We will go up 22/2: we start at -5, and then we add 22/2.*0194

*22/2 is 11, so -5 + 11...we get 6: 6 is our answer; 6 is our midpoint.*0199

*Let's look at another way to do it, though: what if, instead, we wanted to find it through the idea of a middle?*0206

*We are looking for the midpoint, so it makes sense that the midpoint has to be halfway between them.*0212

*What is going to be halfway between them? Well, it would be the average of the two numbers.*0217

*What would be (if we could combine those two and figure it out) the most common place you have between the two?*0221

*What is going to be the middle between the two? It would be the average.*0226

*So, we take 17; we take -5; and we don't have to worry about the distance; we just realize, "Look, the midpoint is going to be halfway."*0229

*To be halfway, you have to be at the average of the two values.*0236

*So, we take -5 + 17, and we divide it by 2; that gets us 12/2, which is 6, the exact same thing we figured out last time.*0239

*That is great: it agrees with our previous work, and that is definitely what we want.*0247

*Since both of these things made logical sense to us, they had better both work; otherwise there is going to be a mistake somewhere in there.*0250

*All right, at this point, we could maybe look at this in general.*0256

*Let's say somebody hands us two points, A and B, and guarantees that A comes before B--that A is less than or equal to B.*0260

*We have this order that A will come before B, or maybe on top of B.*0268

*We don't know what they are, but we still want to be able to talk about the midpoint.*0271

*From our previous work, we have two ways to find this.*0275

*The distance from A to B is going to be B - A, so if we move up half of that distance, it is going to be (B - A)/2.*0277

*So, we start at A; and then we would add (B - A)/2; that is our half of the distance idea.*0288

*But we can also think, "Look, I know I am going to be here, and I know I am here; so I am looking for the place that is halfway between them."*0293

*So, we get the middle: that is (A + B)/2.*0300

*Now, from what we saw before...we saw both of these ways worked; as we had hoped, they give the same value.*0304

*We can go with either A + (B - A)/2 or (A + B)/2; they both are the same thing.*0311

*Let's prove that they are actually equivalent: if we start with our half-distance formula right here,*0318

*and we have A + (B - A)/2, well, let's try to put them on the same fraction.*0323

*We make it 2A/2; we can now combine fractions, and we get (2A + B - A)/2.*0327

*At this point, 2A - A...we get (A + B)/2; so sure enough, our half distance is equal to our middle; they are just two different ways of saying the same thing.*0333

*Since we have two ways to find the midpoint, and they are really just equivalent ways to get the same answer,*0347

*we will just, out of laziness, make one of them the official one.*0352

*A good motivator to do anything is because it is the easier way, as opposed to having to memorize two different things,*0356

*or go with the slightly longer one, let's go with the slightly shorter one; so we make our midpoint formula (A + B)/2; there we are.*0361

*Whatever the two points are that we are trying to find the middle between them, it is just (A + B)/2,*0369

*because we are just looking for the middle place, and the middle place must be the average of our two locations.*0374

*What if we wanted to do this in two dimensions, though?*0380

*What if we wanted to find the midpoint between (0,0) and (6,2).*0382

*Now, notice: we could look at this, as opposed to trying to figure out what is the midpoint on the line that connects the two of them*0385

*(I wish I had made that slightly more perfect)--instead of trying to figure out, "What is the middle going to be here?"*0393

*we can say, "Well, we know that there is going to be some distance vertical and some distance horizontal."*0397

*It must be that it splits our distance halfway horizontally, and it splits our distance halfway vertically.*0406

*So, the two of them come together, and that is our midpoint.*0413

*It is going to have to be the horizontal middle: our horizontal distance was 6, so 6/2 is where we are going.*0419

*Our vertical distance was 2, so it is 2/2, so our location (that is going to be halfway between them)*0425

*will be half of our horizontal distance and half of our vertical distance: (6/2,2/2), or (3,1).*0431

*The midpoint is going to occur at the horizontal middle and the vertical middle, put together into a single point.*0439

*What if we are doing this for some arbitrary pair of points, (x _{1},y_{1}) and (x_{2},y_{2})?*0446

*Well, the same basic idea: we can think, "What is the vertical, and what is the horizontal?"*0450

*What are going to be the midpoints of those two things?*0457

*We think with that idea, and we are able to come up with the same logic that is going to occur at the horizontal and vertical middles.*0459

*This point is going to be the same vertical height, because we never changed height as we went along.*0466

*So, horizontally, we are going to have changed to a new horizontal location; but vertically, this thing right here is going to be the same thing here.*0473

*The same sort of idea here: y _{2} is going to change when we switch down, as we go down.*0484

*But horizontally, we didn't change there; so we have fixed things here, so the point that we are meeting the two at--*0488

*if we drop a perpendicular and throw out a horizontal, we are going to meet up at (x _{2},y_{1}).*0498

*So, the midpoint horizontally is going to be (x _{2} + x_{1})/2,*0505

*which will get us the middle location, because it is the average of our two horizontal locations.*0516

*The average of our two vertical locations is going to be (y _{2} + y_{1})/2.*0521

*We bring these two things together, and we get where our middle is--we get our midpoint that way.*0528

*So, from the midpoint in one dimension, we can figure out what it is horizontally.*0538

*The horizontal motion was x _{1} + x_{2}, so its midpoint is (x_{1} + x_{2})/2.*0542

*And vertically, our locations were y _{1} and y_{2}, so the middle of our vertical locations will be (y_{1} + y_{2})/2; great.*0548

*Our midpoint formula is just (x _{1} + x_{2})/2, and (y_{1} + y_{2})/2; awesome.*0559

*The next idea: distance--what if we want to find the distance between 2 and 7.*0567

*That is easy: 7 minus 2 equals 5--done, right?*0571

*Well, we could make a mistake, though; we are not perfect; what if we accidentally put it in in the wrong order.*0575

*We put in 2 - 7 = -5: well, that doesn't really make sense, because distance has to be a positive length.*0581

*There is no such thing as a negative length, if we are measuring something; you can't say, "Oh, that man is -2 meters tall."*0589

*It doesn't make sense; we can't talk about his distance, his length, as being -2.*0595

*So, -5 doesn't really work; but notice, 5 and -5...they are very different in one way; but in another way, they are very similar.*0599

*One of them is the same thing, just with a negative sign; the other one is the same thing, but with a positive sign.*0610

*So, they are the same number, but with different signs on them.*0616

*In one way, we can think of 5 and -5 as being very different numbers; they are opposites, after all.*0620

*But in another way, we can think of them being the same number, but with different signs.*0624

*They are the same distance from 0; so what we really want is some way of being able to force "positive-ness."*0628

*5 and -5 are pretty close to both being the same thing; it is just that one of them is the wrong sign.*0637

*So, if we could force it to be positive, it wouldn't matter if we did 7 - 2 or 2 - 7,*0643

*because, since we are forcing positive, it will always give us the same thing.*0648

*We would always find that distance, even if we put it in the wrong way.*0651

*This is where the idea of absolute value comes in.*0654

*We call this idea of forcing a positive, making something always come out as positive, absolute value.*0657

*It is represented by vertical bars on either side.*0664

*So, whatever we want to take the absolute value of, we just put inside of two vertical bars.*0666

*We could have |x - 5|, and whatever comes out after we plug in x, we would take its absolute value--we would force positive-ness.*0670

*It is going to take negatives, and it will make them positive; and it will take positives and not do anything; also with 0, it won't do anything.*0677

*If you are positive, you stay positive; if you are 0, you stay 0; if you are negative, you flip to being the positive version.*0683

*You hit it with another negative: so -5 would become 5, but positive 5 would just become positive 5, as it already started.*0689

*-47 would become 47; 47 would just stay as 47; great.*0697

*With this idea of absolute value, we can now tackle how we talk about distance in one dimension.*0703

*So, if we want to talk about it just arbitrarily, if we have two points, A and B, and we don't know which comes first--*0707

*we don't know if it is going to be A then B, or B then A--we have no idea which comes first (if it is A first or B first)--*0714

*but we still want to be able to talk about what the distance is between them,*0726

*well, our previous logic can tell us that one of these two is going to be right: A - B or B - A.*0729

*But the other one is going to be wrong, although almost correct, because it will be the negative version.*0735

*So, we have A - B versus B - A: we want some way of being able to say, "Let's just get rid of the negative signs," right?*0741

*Let's force everything to be positive; then, it doesn't matter what order we put it in, because it is going to be the same distance,*0750

*because it is just a negative version or a positive version;*0756

*it doesn't matter, because we will flip everything to positive; we will always get the distance.*0758

*So, we toss some absolute values on there: absolute value to the rescue!*0761

*We wrap them in absolute values, and they both become the same positive, correct distance.*0764

*So, the absolute value of A - B is the same thing as the absolute value of B - A,*0770

*because the only difference would be whether it is negative or positive.*0774

*And now they are both forced to be positive, so |A - B| is equal to |B - A|,*0776

*which is just going to be the distance between A and B, which is the distance between B and A.*0782

*For ease, we will just make the first one official: so the absolute value of A - B is the distance between those two locations.*0787

*We just take the absolute value of the difference, and that gives us how far the two things are apart*0794

*when it is in one dimension--when we are just on the number line.*0799

*What if we are in more dimensions, though? Let's take a look at the Pythagorean theorem, because we will need that to discuss two dimensions.*0803

*To discuss distance in two dimensions, we need to understand the Pythagorean theorem.*0809

*You have probably learned this before; if it isn't really something that you know well, you are going to want to go back and relearn it.*0813

*Make sure you have this idea, because it is going to show up all sorts of places in precalculus and in calculus.*0817

*And it is definitely going to show up a whole bunch in the trigonometry portion of this course.*0823

*So, definitely make sure that you go and re-study it if you don't remember it.*0826

*What it was: we have a right triangle (a right angle in the corner): the square of the hypotenuse*0829

*(that is the long side, the side that is opposite our right angle) is equal to the sum of the squares of the other two legs.*0835

*So, we square each of the other two, smaller legs.*0846

*And when we add them together, a ^{2} + b^{2}...each of the smaller legs squared,*0849

*then added together...that is going to be equal to our hypotenuse, squared.*0854

*a ^{2} + b^{2} = c^{2}: leg 1 squared, plus leg 2 squared, equals hypotenuse squared.*0859

*That is the idea of the Pythagorean theorem.*0866

*So, any time we see a right triangle showing up, anything we have showing up with perpendiculars--*0868

*it is a good idea to think, "Oh, I wonder if I could use the Pythagorean theorem here."*0873

*It will be very, very useful in a whole bunch of situations.*0877

*If you are not really comfortable with using it at this point, definitely go back and review this idea.*0880

*Either search for it on the Internet, or just try to do a couple of exercises and make sure you have practiced on it.*0883

*Or go review it on Educator.com: listen to the lecture, and then practice some exercises.*0888

*But you want to make sure that you are definitely comfortable with the Pythagorean theorem,*0891

*because it is going to show up a whole lot for the rest of the time you are doing math.*0894

*All right, on to distance in two dimensions: what if we wanted to find the distance between (0,0) and (6,8)?*0898

*Well, we can't just subtract and take absolute values, because we have two dimensions that we are running in.*0904

*We have to deal with both of these things at once.*0910

*What we do is say, "Let's turn this into a triangle."*0912

*We drop a perpendicular from (6,8); we now have this right angle in the corner.*0916

*And with this right angle in the corner, we can use the Pythagorean theorem.*0921

*When we did midpoint, we dropped down perpendiculars; we drew out horizontals and perpendiculars.*0926

*And we were able to get a right triangle going on, which will help us to find middle locations for horizontal and vertical.*0931

*Now, we are allowing us to find horizontal lengths and vertical lengths.*0937

*We break it into horizontal and vertical parts.*0940

*So, if we are at (6,8) up here, then the distance that we traveled horizontally is 6.*0943

*The distance that we traveled vertically is 8: remember, 6 is because that is the horizontal portion; 8 is because that is the vertical portion.*0949

*So, we use the Pythagorean theorem: we know that d ^{2}, the diagonal, the hypotenuse,*0957

*is going to be equal to 6 ^{2} + 8^{2} = 36 + 64 = 100.*0961

*So, for our diagonal, our distance, d, equals 10.*0968

*So, we can figure out that this has to be 10, up here on that side, because we can turn it into a right triangle, which allows us to apply the Pythagorean theorem.*0972

*What if we look at this in a more general way, where we just get two arbitrary points,*0982

*where we don't know what they are--(x _{1},y_{1}), our first point, and (x_{2},y_{2}).*0985

*Now, I didn't talk about this explicitly the first time, but when I say x _{1}, I am not saying x times 1; I am just saying our first x.*0990

*x the first, y the first, x the second, y the second--(x _{1},y_{1}),(x_{2},y_{2})--*0996

*that is what you should interpret when you see those little subscripts, those little numbers on the bottom right.*1003

*So, (x _{1},y_{1}), (x_{2},y_{2}): they are just two arbitrary points, sitting out in a plane.*1008

*We can continue with this idea: we will make a triangle.*1013

*We will toss out a horizontal from this one; we will go straight with a horizontal out.*1016

*And we will drop directly down with a vertical, like this; and that will guarantee us that we have a right triangle that we can now work with.*1023

*And now, we have a way of being able to talk about the distance of that.*1034

*So, we draw that in, and we can say, "Oh, what is the horizontal length?"*1037

*Well, since we ended up at x _{2} (because it is going to have the same horizontal location as our second point),*1040

*we went from x _{1} to x_{2}; our distance is the absolute value of (x_{2} - x_{1}).*1047

*The horizontal length is going to be the absolute value of (x _{2} - x_{1}).*1053

*What is the vertical length--what is the vertical leg of our triangle?*1056

*Well, we end up at y _{2}; and what is the location that we are starting on this triangle?*1059

*It is going to be y _{1}, because it is going to be the same as over here.*1064

*So, we take y _{2} - y_{1}; the absolute value of that is going to be our vertical length.*1067

*The length of the vertical leg of the triangle is the absolute value of (y _{2} - y_{1}).*1074

*So, if we want to know what the distance of the diagonal is, it is going to be d ^{2} = (|x_{2} - x_{1}|)^{2} + (|y_{2} - y_{1}|)^{2}.*1078

*Now, there is a little thing that we can notice at this point.*1096

*OK, if I have |x _{2} - x_{1}|, and then I square it, well, if I just take*1099

*x _{2} - x_{1}, and I square that, that is going to be the same thing.*1104

*Remember, if I have (-7) squared, that comes out to be 49, which is the same thing as 7 squared.*1108

*So, we don't have to take an absolute value to begin with, because, when it is the number times itself, if it has a negative,*1116

*if it multiplies by itself with another negative, those negatives are going to cancel each other out.*1123

*But if we start on a positive, we are going to have no negatives anyway.*1127

*So, the absolute value of x _{2} - x_{1}, squared, is equal to the quantity*1130

*(x _{2} - x_{1}), squared, because they are both going to come out to be positive, in any case.*1134

*So, this is also the same for y; so we can actually drop our absolute values--we don't have to worry about absolute values when we are doing this.*1139

*And the distance will be equal to the square root, because it is d ^{2} equals this thing squared, plus this thing squared*1144

*(horizontal length squared, plus vertical length squared); so d equals the square root--take the square root of both sides, so we get just d.*1151

*It is the square root of ((x _{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}).*1157

*So, it is the difference in our two horizontal locations, squared, plus the difference in our two vertical locations, squared; great.*1163

*That is distance in two dimensions.*1168

*All right, slope: slope is a way to discuss how steep a line is--how quickly it is going up--how much it is changing one way or the other.*1171

*Another way to interpret is the rate of change--the rate that the line increases or decreases for every step to the right.*1181

*So, if we take one step over to the right, it tells us how much up we should go or how much down we should go,*1190

*depending on if it is a positive slope or a negative slope.*1195

*So, it is one step over, and we change by the slope.*1198

*Either way we look at slope--whether we look at it as how steep it is (the angle we are working at)*1202

*or if we look at it as the rate of change (how much we are changing for every step we take on the line), we define it the same way.*1206

*We take some arbitrary portion of the line--any chunk of the line--and we see how much it "rises" and how much it "runs."*1213

*So, rise is the vertical amount of change, and run is the horizontal amount of change.*1219

*So, if we had some chunk of line, like this, then what we would do is just set two arbitrary points, here and here.*1224

*And then, we would say, "OK, how much did we move vertically?" (that is our run) and "How much did we move horizontally?"...*1233

*Oops, sorry, not our run--I said the wrong thing there: our vertical change is our rise--you rise vertically.*1243

*And our horizontal change is our run, because you run along the ground; you run along (generally) horizontal things.*1251

*So, our rise, compared to our run--we divide the rise by the run.*1258

*We symbolize it with m: why do we symbolize it with m? Because, clearly, the first letter of slope starts with m, right?--it makes sense.*1262

*I am kidding; there is actually no good reason, and nobody knows why we use m.*1270

*Anyway, m equals rise over run; that is how we symbolize it--the amount that we rise by, divided by the amount that we run.*1274

*We can also talk about this, since rise and run are just other words for vertical change and horizontal change.*1281

*The slope is equal to the amount of vertical change in our line, divided by the amount of horizontal change in our line.*1286

*Now, keep in mind: vertical change could go down, at which point we would have a negative rise (I keep accidentally swapping them).*1292

*We would have a negative rise if we ended up dropping down.*1300

*All right, let's go back really quickly and address this asterisk.*1305

*Why is it that we can look at any arbitrary portion of the line?*1308

*Why doesn't it matter which section of the line we look at?*1312

*Shouldn't it matter if we look at a big section or a small section, or if we look at a high section or a low section?*1315

*No, because the line never changes slope: that is what it means to be a line.*1320

*Every section of it is going along at the same steepness; every section of it is going along at the same rate of change.*1324

*If we put a bunch of sections together, they will all agree on their slope.*1331

*If we look at just one tiny section in a very different place, it is still going to have the same slope.*1334

*Whatever part of the line we look at, it will always have the same slope.*1338

*So, we get the same value for slope, no matter where we look on the line.*1341

*So, that is why we don't have to worry about what portion of the line we are considering.*1345

*We just look somewhere, and that is our slope; great.*1349

*So, if we have two points, they can define a line; so we want to find the slope between two arbitrary points, (x _{1},y_{1}) and (x_{2},y_{2}).*1353

*Well, what we have to do is say, "All right, how much did I rise in that chunk?"*1361

*And I compare it to how much I ran in that chunk.*1368

*We figure out both of those, and we will be able to get what our slope has to be.*1372

*So, we build a right triangle to help us find these distances.*1376

*Since this here is going to be the same as the horizontal of our second point, that matches up there.*1379

*And it is going to be the same as the vertical, since vertical doesn't change as we go horizontally.*1385

*Those match up there, so the amount of run that we have is (x _{2} - x_{1}).*1388

*That is how much we changed as we went from left to right.*1395

*And the amount vertically that we changed is (y _{2} - y_{1}), because we went up from y_{1} to y_{2}).*1398

*We went up from y _{1} to some y_{2}; so it is y_{2} - y_{1}.*1405

*So, our rise is y _{2} - y_{1}, and our run is x_{2} - x_{1}.*1411

*Our slope is equal to the rise divided by the run, which is our vertical change divided by our horizontal change.*1417

*That gets us (y _{2} - y_{1})/(x_{2} - x_{1}).*1426

*We have been using this formula for years, but now...hopefully, you actually understood it before...*1429

*but even if you didn't understand it before, why this formula was slope,*1434

*hopefully now you are thinking, "Oh, now I see why slope is what it is!"*1437

*It is because it is just coming from rise over run; that is how we defined it; so we get (y _{2} - y_{1})/(x_{2} - x_{1}).*1440

*because (y _{2} - y_{1} is how much we rose, and (x_{2} - x_{1}) is how much we ran.*1446

*Being able to understand what we are doing with slope, though, requires being able to interpret it on an intuitive basis.*1452

*We want to know what to immediately imagine when we are talking about something that has a slope of 50.*1457

*So, slope tells us how much the value of a line changes for every step to the right.*1464

*If we have a slope of m = 2, then that means, if we take one step to the right, then we will go up 2 steps.*1469

*Our line will end up looking like this.*1480

*If we had a slope of, say, -3, then it would be that for one step to the right, we take 3 steps down, so our line would look like that.*1487

*What we have here is a way of being able to talk about the line's rate of change--how much you change for one click.*1508

*You click over, and you change by your slope.*1514

*So, we can think of slope as how steep it is (bigger numbers will make it steeper, because it means more steps to be made in our rate of change).*1517

*All right, there is one step, but how many times we go down or how many times we go up is the number of our slope.*1524

*So, a line's rate of change is its slope: it is a way of talking about how fast this line is changing as we slide along it.*1530

*You want to keep these facts in mind as we think about slope.*1539

*If we have a positive slope, it means that the line is rising; we are always thinking about it as we go from left to right.*1543

*That is how we are always reading how our slope works: it is always what happens as we go from left to right.*1552

*So, positive slope means we rise as we go to the right; a negative slope means that we fall when we go to the right.*1558

*We either go up by positive, or we go down because it is negative.*1565

*A bigger number, whether it is positive or it is negative, means a steeper line.*1568

*The steeper the line is, the bigger the slope has to be; the bigger the slope is, the steeper the line is.*1573

*A big slope, like, say, m = 50, is going to be really, really super steep.*1581

*It is going to go up really, really fast, because for every step it takes to the right, it has to take 50 steps up.*1587

*Similarly, in m = -50, it is going to be very similar; but for every step it takes to the right, it takes 50 steps down; so it is super steep going down.*1592

*A big number, whether it is a big positive number or a big negative number--that is going to imply a steep line.*1603

*Some specific locations to keep in mind: if m is equal to 1, then that means our line rises at a 45-degree angle,*1609

*because for every step to the right, we take one step up.*1615

*So, it means that we have a nice, even-sided triangle: 45...these two have to be the same.*1618

*If we have m = -1, then for every step we take over, we take a step down.*1626

*So, we have the same idea; but instead, we are going down now; so these two angles have to be the same.*1633

*We have a nice 45-degree angle in that triangle as well--what makes up the line.*1639

*So, we are either rising at 45 degrees (if we have a positive one) or we are falling at 45 degrees (if we have a negative one).*1644

*And if m = 0, then we take one step over; we take no steps up; we just continue taking steps over forever and ever.*1650

*So, m = 0 means our line is horizontal; m = 1 means our line rises at 45 degrees; m = -1 means the line falls at 45 degrees.*1657

*This also means that everything between positive 45 and -45 is all going to happen in fractions--things that are between -1 and 1.*1664

*If we want to get really steep lines, that is as we approach either positive infinity, or as we approach negative infinity.*1676

*We can never be perfectly vertical with a slope, because that will require either positive infinity or negative infinity.*1681

*And we are not able to actually call those out, because they are not really numbers.*1689

*But as we go from 1 and click up more and more and more, and approach infinity more and more and more,*1692

*we will need larger and larger numbers to become steeper and steeper and steeper.*1696

*All right, there are lots of ideas that we have covered here; now we are ready to start talking about some examples.*1700

*First, the idea of midpoints: if we have a midpoint, and we are looking between -3 and 37, then remember,*1706

*our formula for midpoints was just (A + B)/2; it is the average of the two.*1712

*So, the average of -3 and 37...put those two together: we get 34/2, which equals 17; so, 17 is our answer.*1718

*If we want to find (6,2) to (1,-12), then we do it on each of the components, because we look at the horizontal average, and we look at the vertical average.*1731

*So, (6 + 1)/2, the average of our horizontal components, and (2 + -12), the average of our vertical components, will get us 7/2 and -10/2.*1743

*7/2...we can't simplify that anymore, so that will lock in; but -10/2...we can simplify that, so we get -5.*1761

*(7/2,-5): that is our midpoint for this one right here.*1769

*And our last one: what if somebody handed us things that weren't numbers--they hand us 2a, 3b, 6k, -7b?*1775

*They are numbers, in that a represents some number--it is a placeholder--it is a variable.*1782

*b represents a number; k represents a number; they all represent numbers.*1786

*But we can't actually solve and get numbers, like we did with these previous two ideas, these previous two questions.*1789

*But we can still use the numbers--we can still use the variables.*1796

*We just put them into the formula, just the same: we are still looking for what is the average of our horizontals--what is the average of 2a and 6k.*1799

*What is the average of our horizontal locations?*1808

*And what is the average of our vertical locations, 3b + -7b?*1811

*We are still looking for the same sort of average ideas, horizontal and vertical.*1817

*It just is that we can't combine 2a and 6k, because a and k are speaking totally different languages.*1822

*And b and b, we can combine, because they are speaking the same language.*1828

*So, 2a + 6k--we can't combine that, but we can have our fraction, the denominator,*1831

*go onto both of them: 2a/2 + 6k/2; we have the denominator split onto both of them.*1835

*And 3b + -7b; that begins -4b/2; so 2a/2 becomes just a; 6k/2 becomes 3k; no comma--they are combined together through addition.*1842

*But they can't do anything more: a and k don't speak the same language, so they can't combine.*1856

*But we have a + 3k; we know that is what our horizontal location is.*1861

*So, if we were given a and k later, we can easily get what the midpoint is, in terms of actual numbers.*1864

*-4b/2: that is -2b; there we are--we don't have numbers in the terms of 53 or something,*1869

*but we have answers that are still pretty good.*1879

*If we get what these variables are later--if we somehow get them because we solve for them,*1882

*or somebody hands them to us--we will be able to immediately find out what actual numbers would be.*1886

*And this gives us a great idea of where the midpoint is, based on variables.*1890

*We don't have to be working with numbers to be able to solve for these things; we can also just put in variables,*1894

*and just follow the exact same rules that the numbers would follow.*1898

*The next one: let's talk about distance--what is the distance between -7 and 8?*1903

*Remember, we do this based off of the difference between the two numbers, its absolute value.*1907

*So, we could take |-7 - 8| or |8 - -7|; either way we do this, -7 - 8 will be -15; 8 - -7 will turn that into positive 15.*1914

*Either way, they both equal 15; so the answer is 15.*1929

*The distance between -7 and 8 is 15, which makes sense, because -7 clicks up to 0 by going 7, and 0 clicks up to 8 by going 8; so 7 + 8 is 15.*1933

*Great, that makes a lot of sense.*1943

*What if we want to figure out what the distance is between (3,7) and (9,-1)?*1945

*Well, remember: now we are working off of what we figured out before, with the Pythagorean theorem and how that applied to distance.*1949

*So, it is going to be the square root of the difference between our horizontals, squared, plus our difference between our verticals, squared.*1955

*It is the square root of all of those things.*1965

*So, formulaically, it is d = √[(x _{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}]--the square root of all of that.*1967

*So, in this case, let's arbitrarily set this as our second one; and we will set this as our first one.*1981

*The distance is equal to the square root of (x _{2} - x_{1})^{2}...that would be...*1987

*the second x is 9, minus...the first x is 3, squared; plus our y portion: the second y is -1, minus the first y (is 7), squared.*1995

*So, distance is equal to the square root of 6 squared plus -8 squared; distance equals √(36 + 64),*2009

*which means distance is equal to √100, which equals 10.*2024

*So, the distance between those two points is 10.*2028

*The final one: what if we get some things that don't turn out nicely?*2031

*We have these ugly decimal numbers: we still follow the exact same idea.*2035

*The distance is equal to the square root of...arbitrarily, we will just make this one the first one, and this one the second one...*2039

*the answer would turn out the same, because of all of the things that we talked about before.*2045

*It doesn't matter which gets turned into the second and which gets turned into the first; the distance is going to be the same between them.*2048

*If that doesn't really make sense, go back to when we talked about how we figured out that this is the formula.*2053

*And notice that (x _{2} - x_{1})^{2} is the same as (x_{1} - x_{2})^{2}.*2057

*It comes up with the ideas where we were talking about absolute value before.*2063

*So, the first x is -0.2, minus the second x (is 2.5), squared, plus...the first y is 1, minus the second y (is 1.7), squared.*2066

*So, distance equals the square root of...-0.2 - 2.5 becomes -2.7, squared; plus...1 - 1.7 becomes -0.7, squared.*2084

*We use a calculator to figure this out; or we could do it by hand, but I used the calculator when I figured it out.*2100

*And I figured out before that (-2.7) ^{2} becomes 7.29, plus...(-0.7)^{2} becomes 0.49,*2105

*because those negatives just end up canceling with each other when they square against themselves.*2115

*So, this is 7.29 + 0.49; so our distance is equal to the square root of 7.78 (what we get when we combine these two numbers--7.78).*2119

*That is the distance between those two things.*2129

*Of course, the square root of 7.78 is kind of hard to actually use, if we had to measure something and cut something,*2132

*if we were making, say, a bookshelf (who knows?); so we could approximate this.*2139

*We could take the square root of 7.78 with our calculator, and that would be approximately equal to 2.79.*2144

*In reality, the decimals actually keep going, and they keep shifting around forever, because 7.78 is not a perfect square.*2149

*So, we can't just come up with a nice, easy number; but 2.79...we can just cut it off, and we will say, "Yes, 2.79 is pretty good."*2155

*So, we get an approximate value; the true answer, though, is √7.78; that is what the answer really is.*2164

*But if we need to be able to work with something that we can actually know what the number is pretty close to,*2172

*because we are more concerned with having a sense of where it is than knowing precisely the right answer, then we could also get 2.79.*2176

*Most teachers would probably accept both; technically, √7.78 is a better answer than 2.79,*2183

*because we have lost a tiny bit of accuracy when we take the square root, and then we round it.*2190

*But we will be able to use both; they are both very good answers.*2195

*All right, the next one: what would be the distance between the origin and (3,12,4)?*2199

*So, we talked about things in two dimensions before: what if we had to deal with three dimensions, though?*2206

*This problem is going to happen in three dimensions.*2211

*We talked briefly about three-dimensional things before, where we have three different axes.*2213

*They are each perpendicular: so here is our x-axis; here is our y-axis; here is our z-axis; so it goes x, y, z, like this.*2221

*If we were to plot this point, just to try to get a sense of what is going on, then we would count out 3 on x;*2234

*we would count out 12 on y; and we would count out 4 on z.*2242

*So, our point would be out 3; from 3 out to a distance of 12; and then, from there, up 4.*2248

*I am starting to accidentally encroach into my words.*2258

*So, our line is going to look like this.*2262

*Now, notice, though: we could also think about a cross-section here.*2266

*This is getting a little complex to see: let me see if I can get it a little bit more sensible with my hands.*2269

*Imagine that here is our x; here is our y; here is our z.*2275

*So, our number is go x forward; it is going to go y 12, and it is going to go up 4 (forward by 3).*2279

*We go out like this: we are going to start here, and we are going to go in all of these, all at once.*2286

*We go out by 3; we go over by 12; and we go up by 4.*2292

*We could also think of this as being a cross-section: we could take a cross-section, and we could make a triangle in here.*2298

*We can say, "What happens in the x,y plane?": here is x; here is y.*2307

*We go over 3; we go up 12; and we can talk about how we got here--we can do that portion of our trip.*2316

*We go and travel x and y; then we go up.*2327

*We travel x and y; and we could figure out, "Look, the x and y plane is perpendicular to the z portion of our axis."*2330

*So, that is going to be perpendicular there, as well; so we can figure out what this length here is.*2337

*And then, we already know what this is; it has to be 4, because our height was 4.*2342

*If we could figure out what this portion right here is, what this length for our cross-section, the base of our triangle, is, we would be good to go.*2347

*We look at the x,y plane--we look at this portion here--and we can turn it into a nice, flat object that we can see--a nice planar object.*2356

*So, we do the same things that we have been doing before.*2365

*Here, 3 is our horizontal; 12 is our vertical; so our distance is going to be the square root of 12 squared, plus 3 squared.*2368

*Well, let's swap that around; that was exactly correct, but just to keep doing the exact same way we have been doing it precisely before.*2379

*We will have 3 ^{2}, because we had horizontal before first; and then 12^{2}, because we are always following with vertical.*2387

*The other way was just as right: 3 ^{2} + 12^{2} is the same thing as 12^{2} + 3^{2}, after all.*2393

*But that way, we are just following our nice pattern from before.*2398

*So, we can figure out what the length of this portion right here is.*2402

*We have it right here: we now have that--now we just bring in this thing, and we just do another one.*2408

*For d of our triangle here, let's use a different color; we will go with green for the distance in our three-dimensional object.*2414

*In our three-dimensional object, it is going to be the square root of what it was in the x,y plane--that distance,*2424

*squared (what was it in the x,y plane? It was the square root of 3 squared plus 12 squared; that was its distance before;*2430

*but we have to square it), plus...what was the jump that it had up--what was its vertical leap in the z?--that was 4, so plus 4 squared.*2438

*Now, notice: when we take a square root and square it, like we have in here, d =...the square root of 3 ^{2} plus*2447

*12 ^{2}, squared, is just going to crack it open, and we will get 3^{2} + 12^{2}, and then plus 4^{2}.*2453

*We simplify this, and we get d = √(9 + 144 + 16); simplify that some more; we get √169, which equals 13; so our distance is 13.*2460

*What we have done is: we are able to look at how it changed on the first plane.*2476

*We sort of take a cut, so that we can look at how it changed in the x,y plane; and then, we put on the z.*2481

*Now, you might be getting a sense of "Oh, maybe there is something we could do in general."*2489

*And I didn't tell you this before, because it is not really going to come up much in this course.*2494

*But we can actually get a distance for three dimensions, as well.*2497

*It is going to be the distance of the square root of a whole bunch of stuff, now...of (x _{2} - x_{1})^{2},*2500

*the square in our horizontal, plus (y _{2} - y_{1})^{2}, the square in our vertical,*2510

*plus (z _{2} - z_{1})^{2}, the square in our coming out of the x,y plane.*2517

*That comes out, because what we do is clear out the x,y plane first.*2525

*It is going to have a square root around it; but then, when we put in that z--when we toss out that coming out perpendicularly--*2528

*we are going to square root again, because now we are doing another right triangle.*2536

*And so, it will simplify to just each one of these differences, squared.*2539

*This might be a slightly complex idea for you, so don't worry if this didn't make sense.*2542

*Just take it out of your head; throw it away; it is not really going to come up.*2546

*It is just a really cool thing that...if you are thinking, "Oh, there is something interesting going on here"...you are right!*2549

*This is the interesting thing that is going on; we can actually generalize this to three dimensions.*2553

*And if we wanted, we could even keep going to 4, 5, 6...any number of dimensions we want.*2558

*And you might have some idea of what is going to happen as we go on to four dimensions.*2562

*See if you can figure out what goes on in four dimensions--it is kind of cool.*2565

*All right, Example 4: What is the slope between (-1,8) and (1,14)?*2568

*So remember, we figured out that slope is rise over run; the amount that we rise is our change in our vertical,*2573

*(y _{2} - y_{1})--our two vertical locations--the change--and our run is our two horizontal locations--their change.*2583

*So, in this case, arbitrarily, let's make this one the second one, and this one the first one.*2589

*There is no particular reason; it is just because that one came first, and that one came second.*2595

*So, let's look at it that way: it also makes sense, because if we were to draw a picture of it, we would have something like this:*2600

*(-1,8), and then (1,14); so it makes sense that they give this one as the second one and this one as the first one.*2606

*But as we will see in a little bit, it actually doesn't matter which one we choose first.*2612

*So, we want to find out the slope between (-1,8) and (1,14).*2616

*So then, m = (y _{2} - y_{1})/(x_{2} - x_{1}).*2621

*Our second y is 14; so 14 minus our first y (is 8), divided by our second x (is 1); our first x (is -1), so minus -1; that equals 6...*2632

*1 minus -1...those negatives cancel; we get 6/2 = 3, so our slope is 3.*2648

*What is the slope going to be if we switch our first and second points?*2655

*Instead, we make this one 1, and we make this one 2; well, we do the same thing: m =...*2659

*it is going to be...our new second one is 8, minus our new first one (is 14), divided by x _{2},*2667

*our new second one (is -1), minus our new first one (is 1).*2681

*8 - 14 is -6: -1 minus 1 is -2; will you look at that--these cancel, and we get the exact same thing.*2688

*If we switch our first and second points, which we arbitrarily decided to make second and first, does it affect what the slope comes out to be? No.*2699

*It doesn't--why? Because of the negatives: it introduces negatives on both the top and the bottom; they cancel out.*2708

*So, if we have negatives showing up because of the switch, they are going to show up on both the top and the bottom, so we will always see cancellation.*2713

*So, it doesn't matter if we plug in our one as the first one, or if we plug in that one as the second one.*2719

*It doesn't matter which one gets to be called first and second, as long as we match up our seconds and our firsts.*2725

*They have to match up vertically: if we have one point be the second point on the top, the second y-coordinate,*2731

*then it has to also be the second x-coordinate; it has to come first on the bottom, as well.*2737

*So, we have to make sure that the points match up vertically.*2741

*8 and -1; (-1,8); 14 and 1; (1,14); they match up there.*2745

*Let's prove this, though: if we want to prove that this always comes out to be the case,*2753

*to prove this, what we want to show is that it doesn't matter if it is (y _{2} - y_{1})/(x_{2} - x_{1}),*2759

*versus (y _{1} - y_{2})/(x_{1} - x_{2}); if we swap the location of which gets to come first,*2771

*which gets to be more on the left in the fraction, it doesn't matter which gets to be more on the left and which is more on the right.*2781

*That is what we want to show; so how do we prove it?*2787

*Well, let's start with this one here: we will have (y _{2} - y_{1})/(x_{2} - x_{1}).*2789

*Now, we want to be able to get that to start looking like this thing.*2800

*And we say, "Well, (y _{2} - y_{1})...that is pretty much the same thing, but it has a negative introduced to it."*2804

*So, how could we introduce some negatives here?*2811

*Well, let's write it again: (y _{2} - y_{1})/(x_{2} - x_{1}).*2814

*We could multiply it by 1, right...wait, wait, what?--yes, 1, right?--I can multiply anything by 1, any time I want.*2818

*You can't stop me from multiplying by 1; I can take any number and multiply it by 1, and it has no effect.*2826

*So, everything is equal to just itself times 1.*2832

*Now, the cool thing about math is that there are a lot of ways to say the number 1.*2835

*I can say 1 as 1, but I can also say it as 1/1; or I could say it as 5/5, or I could say it as -1/-1.*2840

*And that is how we introduce our negatives; and this is also the idea that is coming along when we change denominators.*2849

*We introduce by multiplying the same thing on the top and the bottom; we multiply by -1 on the top and -1 on the bottom.*2855

*Now, notice: since we are multiplying the top, we are not just multiplying the first part of the top.*2861

*We are multiplying the whole top; because it is multiplication, it is going to apply to this fraction as if it started in parentheses.*2865

*So, times -1, over -1; (y _{2} - y_{1}) times -1 becomes -y_{2} + y_{1},*2872

*over -x _{2} + x_{1}; and this thing right here is just the exact same thing as this thing right here.*2881

*We have just swapped the location; instead of -y _{2} + y_{1},*2891

*it becomes y _{1} - y_{2}, what we are a little more used to seeing.*2895

*So, we have managed to prove that it doesn't matter what order we put it into, using this (y _{2} - y_{1})/(x_{2} - x_{1}) formula.*2899

*It doesn't matter, because it is going to end up giving out the same answers.*2905

*But the really key idea to think about, when we are talking about slope, is that it is the rise over the run.*2908

*It is the rate of change--how quickly the line is changing.*2912

*All right, I hope you learned a bunch here; I hope it has been a great refresher, and everything is really understandable,*2915

*because we will be using these things a whole bunch, later on.*2919

*All right, see you at Educator.com later--goodbye!*2922

1 answer

Last reply by: Professor Selhorst-Jones

Tue Nov 4, 2014 11:29 AM

Post by Jamal Tischler on November 4, 2014

How can the Pythagorean theorem pe proved ? I saw an explaination with some triangles and squares bonded, but I didn't realy understand it.

1 answer

Last reply by: Professor Selhorst-Jones

Sun Sep 21, 2014 9:46 PM

Post by Magesh Prasanna on September 20, 2014

Hello sir! asusual superb lecture...By Definition Slope =rise/run. i.e no.of rises per run. Are we only concerned about rises per run why we aren't for runs per rise?..I'm unable to imagine how line runs per rise. The rise/run of a straight line is proportinal to the angle of the line. Let me know how the value of rise/run is related to the angle of the line?

1 answer

Last reply by: Professor Selhorst-Jones

Sat Jul 5, 2014 3:56 PM

Post by Thuy Nguyen on July 4, 2014

In computer science, when implementing a binary search, using the shorter formula for finding a midpoint is wrong because it could cause an overflow of integers. I like the concise formula for midpoint, but a good reason for using the longer version: a + (b-a)/2, would be in programming a stable algorithm.

2 answers

Last reply by: Linda Volti

Fri Feb 21, 2014 6:00 PM

Post by Linda Volti on February 21, 2014

Totally agree with the first three posts: absolutely fantastic! Even though I knew most of these things, I'm now learning them at a completely different level. I wish I had a teacher like you when I was at school many years ago now!

1 answer

Last reply by: Ian Henderson

Mon Aug 12, 2013 10:45 PM

Post by Ian Henderson on August 12, 2013

Sorry I may be a bit confused here, but when we're looking for M the slope, would that not be the equivelant of looking a2+b2 = c2? The pythagorean theorem? Is C not usually the slope?

1 answer

Last reply by: Professor Selhorst-Jones

Thu Jul 11, 2013 12:16 PM

Post by Jonathan Traynor on June 26, 2013

What a perfect way to recap old material. IO love the way you appeal to intuition and then explain it in maths terms. Outstanding!!!

2 answers

Last reply by: thelma clarke

Mon Mar 16, 2015 7:16 AM

Post by Montgomery Childs on June 25, 2013

Great refresher. Love the way you break it down - 4th. dimension = t(time)?

1 answer

Last reply by: Professor Selhorst-Jones

Thu Jun 13, 2013 8:17 PM

Post by Sarawut Chaiyadech on June 13, 2013

thank you very much you make maths visible :) cheers ]