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Okay, in this video clip, we're going to do a little bit of a math review.

The purpose here is not to actually teach the math involved, but

hopefully remind those of you who need a refresher of a few things here.

So, we're not going to start from first principles or anything like that.

But, just to remind you about a view things, especially with respect to

algebra and plotting that will be useful later on in our course.

So certainly, if you are doing a more quantitative approach to the course,

you'll want to be up on these things.

But even if you want to do the more qualitative approach or

just taking auditing approach,

there will be times you want to just sort of follow along with the video.

And it will be helpful to remember these certain things.

So, when we talk about algebra of course we use letters, and

then we can replace, plug in numbers for them.

So we have things like this where we say okay, equations a + b = c.

So we have two numbers represented by a and

b, when we add them together we get number c, etc., etc.

Remember a few things with exponents.

If we have an actual number 2 squared,

that is the same as 2 times 2 which is 4 of course.

Or 2 cubed = 2 times 2 times 2, or 4 cubed,

4 times 4 times 4, so it's 2 multiplied three times.

So we can also try about a squared, certainly any number squared, a cubed.

I don't think we really actually get into anything more than squared in here,

if I remember correctly.

So, but we'll see a lots of squares, a squared, b squared, x squared,

y squared things like that.

So just remember how that works.

Another thing along with this, we have negative exponents.

So if we have say 3 to the -2 power,

that's defined to be 1 over 3 squared, okay.

So 1 over 3 squared, 3 to the -2.

Or if we use a, a to the -2 power = 1 over a squared.

Remember also that a to the 1st power, a to the 1, simply = a.

So a to the -1 is the same as 1 over a.

We can also, of course, talk about things like the square root of 3, where if

you take the square root of 3 times the square root of 3, you'll get 3 back again.

So really if we have the square root of a times the square root of a,

that's going to give us a back again.

In exponent notation we write instead of using the square root

radical sign there, we sometimes write this as a to the one-half.

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And the reason I just did a reminder over here is,

why would we call this a to the one-half square root of a?

Well square root of a times square root of a equals a, back again.

Square root of 3 times square root of 3 equals 3.

We write this as a to the one-half

times a to the one-half.

Bases are the same, the as are the same.

Like bases, so we add the exponents.

This becomes a to the one-half + one-half, which of course, is just 1.

So, we get a to the 1, which is just a.

So, what we've just shown in different notations,

square of a times square of a = a.

So, that's why we use a to the one-half, that one-half exponent as a symbol for

the square root of a, or it's equivalent to the square root of a.

So that's a little bit with exponents, a few reminders about that.

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Remember how we did that?

Because if we have this, we can get this back because we just go a times 1,

so that's going to equal a.

And I've got a times b over a.

Remember how this works if it's really been a while for you?

So this is the same thing as a times b over a.

And the a here in the numerator essentially, and

in the denominator cancel, and we just get a+b again.

So essentially we can factor out an a in this case.

Or we could have done a similar thing with b.

Why is that important?

Because we will see later on something like this.

We'll have not just a + b,

we'll have a squared + b squared.

That form comes into the theory in a number of cases.

And for various reasons we'll want to write it like this.

We'll write this as (a squared)

times (1 + b squared / a squared), okay?

because these are equivalent because I take a squared times 1,

that's just a squared.

A squared times b squared over a squared, the a squareds cancel,

I'm just left with b squared.

Even though this looks, why would we want to do this when this is simpler?

Well, it turns out in certain cases that will give us a nicer form to see

what's going on with the math.

Okay, so

that's factoring something out to just write something in a different form there.

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Another thing, just as a reminder,

if we have a/b + c/d, so maybe we have something like,

what should we do, we just, well maybe

two-thirds plus,, five-halfs, okay?

Oops, just let me change that, that's true to five-fourths.

There, just so that a, b, c and d are all different there, okay?

Obviously, it could be the same, but

two-thirds plus five-fourths, we want to combine those two fractions together.

So remember we need to get a common denominator.

We want the bottom part to be the same.

The way we do that is we do a little trick, and we multiply in the form of one.

Remember so we have two-thirds times,

look at the other denominator the four, so four fourths, that's just one.

I haven't changed the two thirds, two thirds times one is one.

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because later on the course, we'll do some things with spacetime diagrams,

which can be a very powerful tool for analyzing certain situations.

And it's useful if you haven't plotted things for a long time,

it's useful to remind ourselves How we do that.

So plotting, and

actually as we go along the course there may be few other things that come up that

we won't have mentioned here but we'll mention them along the way as we go just

as reminders of how things work and won't try to jump too far ahead.

Remember make haste slowly as it were So plotting, remember just how that works?

If you've got, I'm going to use x an y now.

Some x an y value.

So maybe the values for x, often we choose these, maybe -2, -1, 0, 1, 2, 3.

And given x, we can find a value for y.

Maybe they're just given, maybe there's some experimental data and

we know the x and y values and we just want to plot them.

In this case, we'll use this.

We'll say x is negative two, y is four.

One, zero, one, four, nine.

In fact, you may recognize this as y equals x squared.

In other words, take the value of x, square it, and that gives us y.

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Put some marks on here.

One, two, three four.

One, two, three four.

Something like that.

So a negative 3, negative 2, negative 1, 0.

1, 2, 3, 4.

Of course we could be going this way to one, two, three, four, five etc.

Right?

And so we plot.

So first point, negative two four.

When x is negative two, y is four.

So Negative 2, up 4.

So there is our point, in fact let's use red.

A little more colorful, negative 2, 4.

We got that so it's about right there.

Negative 1, 1, right?

So negative 1, 1.

0,0 so a point right thee.

When x is 1, y is 1.

X is 1, y is 1 so about right there.

When x is 2, y is 4.

So again about right there.

And when x is 3 y is 9 so it'd be way up here some place.

And [INAUDIBLE] and so then we connect the dots.

So if we actually plotted a whole bunch more points

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So negative 3, negative 2, negative 1, 1, 2, 3, 4, 1,

2, 3, 4, negative 1, negative 2, and so on and so forth.

So our points are negative 2, negative 4, so

we'll actually need to go a little bit farther down here,

down to Negative 4 so negative 2 for X, negative 4 for Y,

got the red out, red marker so there is that one.

Negative 1, negative 2, so hopefully my scaling is more or

less correct here so we get a nice line out of it.

Zero, zero is this point right here.

One, two, so over one, up two Doing okay.

2, 4, over 2. Up 4.

So like that.

And then 3, 6.

Over 3. And up 6, would be up here, someplace.

And you see if we connect the dots, we get, I don't know if I can do this very

well, but we'll try here, if I can Not too bad there.

A little quicken.

But that's essentially the equation of a line.

This is a line, 2 times x, and again, if you dread your mathematical memory,

if it's been a while, you may remember, this number right here.

In other words, A is the slope, Of the line.

The slope.

And this is something actually will be important to us to remember,

it's not crucially important but it's useful information in that

if we look at this number here, it tells us how steep the line is.

The bigger this number in front there, the a value, two in this case,

It tells us it's a slope of two, has a steepness, in a sense, of two.

For those of you again who remember the exact of slope or for a line rise,

it's rise over runs.

So we measure the rise over the run here.

It goes over one For the run and up two and obviously if it went over,

went up three, it would be a steeper slope.

This would be actually a value three here.

So the bigger the number here, the steeper the slope of the line.

The smaller the number, assume we were talking about a positive number,

the slope of the line would be more like this.

If it was one half there,

then we would be seeing something more like Line sort of like that.

That'd be a slope of 1/2 if we plugged in new numbers for

that and plotted that line.

A slope of 1 is going to be at a 45 degree angle,

assuming our scaling on each access is identical.

So slope of 1 if it was just y equals x Plotting 1, 1.

When x = 2, y = 2, and so on, so forth.

And so we get a line right down the center there, right at 45 degree angles,

splitting the center of those other two.

So, that's the slope, of a line.

One other thing you might say, well what about B here?

What's the deal with B?

Well if you did an example where we did say y equals two x, plus one.

All right, so you're doing y equals two x, we'll do two x plus one.

So slope will still be two, the steepness will be two.

What's the one mean?

Well you may remember in dredging your memory.

That it's called the y-intercept.

And if you just think about that name a minute,

it literally means it's where the line intercepts the y-axis.

So, here, when y equals just 2x and B equals 0, the line,

all three of our lines here, intersected the y-axis.

Here's the y-axis.

At zero in this case y equals 2 x plus 1.

If we changed our numbers here right, the x values would be the same.

But now we have new y values because we changed the equation a little bit.

Then we'd find that we still have a line of slope 1 but

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The y values over change.

When x is negative two,

remember we didn't review this in our little math review a minute ago, but

a negative number times another negative number gives you a positive number.

A negative number times a positive number gives you a negative number.

A positive number times a negative number gives you a negative number.

Two negative numbers multiplied together, the negatives cancel out as it were and

gives you a positive number.

So in this case.

X is negative two times the negative two that gives me a positive four.

So this actually becomes a plus four there.

Same thing here, negative one times negative two gives me a plus there.

Zero times zero is still zero, but if x is one I get negative two.

So that becomes a negative two.

This is a negative four.

That's a negative six It just the other way around.

When x is negative, now y is positive.

When x is positive, now y is negative.

And if you actually plotted this,

you're going to get something that looks like this.

We'll do it like this.

Now we're at 1, negative 2 So down here, two and negative four.

Down there so something that looks like this.

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Okay?

That is a line with slope equals negative two.

So when we have negative slope, it slopes downwards from left to right.

Pause a slope, slopes upwards from left to right.

That will be another thing to remember.

It won't be a crucial point if we start hazy on it but

just something to refresh in your memory bank as it were.

Pause a slope, slope upwards.

The bigger the number the steeper it is Negative slopes slope downward.

The bigger the negative number, the steeper the negative slope here.

So if it was negative 3, it'd be more steep going that way.

If it was negative one-half,

it'd be a less steep line going downward from left to right.