Introduction to the coordinate plane | Introduction to algebra | Algebra I | Khan Academy

The general idea behind the word 'abstract'

Watch the next lesson: https://www.khanacademy.org/math/algebra/introduction-to-algebra/overview_hist_alg/v/the-beauty-of-algebra?utm_source=YT&utm_medium=Desc&utm_campaign=AlgebraI

Missed the previous lesson?
https://www.khanacademy.org/math/algebra/introduction-to-algebra/overview_hist_alg/v/origins-of-algebra?utm_source=YT&utm_medium=Desc&utm_campaign=AlgebraI

Algebra I on Khan Academy: Algebra is the language through which we describe patterns. Think of it as a shorthand, of sorts. As opposed to having to do something over and over again, algebra gives you a simple way to express that repetitive process. It's also seen as a "gatekeeper" subject. Once you achieve an understanding of algebra, the higher-level math subjects become accessible to you. Without it, it's impossible to move forward. It's used by people with lots of different jobs, like carpentry, engineering, and fashion design. In these tutorials, we'll cover a lot of ground. Some of the topics include linear equations, linear inequalities, linear functions, systems of equations, factoring expressions, quadratic expressions, exponents, functions, and ratios.

About Khan Academy: Khan Academy offers practice exercises, instructional videos, and a personalized learning dashboard that empower learners to study at their own pace in and outside of the classroom. We tackle math, science, computer programming, history, art history, economics, and more. Our math missions guide learners from kindergarten to calculus using state-of-the-art, adaptive technology that identifies strengths and learning gaps. We've also partnered with institutions like NASA, The Museum of Modern Art, The California Academy of Sciences, and MIT to offer specialized content.

For free. For everyone. Forever. #YouCanLearnAnything

Subscribe to Khan Academy’s Algebra channel:
https://www.youtube.com/channel/UCYZrCV8PNENpJt36V0kd-4Q?sub_confirmation=1
Subscribe to Khan Academy: https://www.youtube.com/subscription_center?add_user=khanacademy

This right here is a
picture of Rene Descartes.

Once again, one
of the great minds

in both math and philosophy.

And I think you're seeing
a little bit of a trend

here, that the great
philosophers were also

great mathematicians
and vice versa.

And he was somewhat of a
contemporary of Galileo.

He was 32 years younger,
although he died shortly

after Galileo died.

This guy died at a
much younger age.

Galileo was well into his 70s.

Descartes died at what
is only 54 years old.

And he's probably most known in
popular culture for this quote

right over here, a
very philosophical

quote. "I think,
therefore I am."

But I also wanted to throw in,
and this isn't that related

to algebra, but I just thought
it was a really neat quote,

probably his least famous
quote, this one right over here.

And I like it just because
it's very practical,

and it makes you realize
that these great minds,

these pillars of
philosophy and mathematics,

at the end of the day, they
really were just human beings.

And he said, "You
just keep pushing.

You just keep pushing.

I made every mistake
that could be made.

But I just kept pushing."

Which I think is very,
very good life advice.

Now, he did many things in
philosophy and mathematics.

But the reason why
I'm including him

here as we build our
foundations of algebra

is that he is the
individual most responsible

for a very strong connection
between algebra and geometry.

So on the left over here, you
have the world of algebra,

and we've discussed
it a little bit.

You have equations
that deal with symbols,

and these symbols
are essentially

they can take on values.

So you could have something
like y is equal to 2x minus 1.

This gives us a relationship
between whatever x is

and whatever y is.

And we can even set up a table
here and pick values for x

and see what the
values of y would be.

And I could just pick
random values for x,

and then figure out what y is.

But I'll pick relatively
straightforward values

just so that the math
doesn't get too complicated.

So for example, if
x is negative 2,

then y is going to
be 2 times negative

2 minus 1, which is negative 4
minus 1, which is negative 5.

If x is negative
1, then y is going

to be 2 times negative
1 minus 1, which

is equal to-- this is negative
2 minus 1, which is negative 3.

If x is equal to 0, then y is
going to be 2 times 0 minus 1.

2 times 0 is 0 minus
1 is just negative 1.

I'll do a couple more.

And I could have
picked any values here.

I could have said,
well, what happens

if x is the negative
square root of 2,

or what happens if x is
negative 5/2, or positive 6/7?

But I'm just picking
these numbers

because it makes the
math a lot easier when

I try to figure out
what y is going to be.

But when x is 1, y is going
to be 2 times 1 minus 1.

2 times 1 is 2 minus 1 is 1.

And I'll do one more.

I'll do one more in a color
that I have not used yet.

Let's see, this purple.

If x is 2, then y is
going to be 2 times 2--

now our x is 2-- minus 1.

So that is 4 minus
1 is equal to 3.

So fair enough.

I just kind of sampled
this relationship.

I said, OK, this describes
the general relationship

between a variable
y and a variable x.

And then I just made it a
little bit more concrete.

I said, OK, well, then
for each of these values

of x, what would be the
corresponding value of y?

And what Descartes
realized is is

that you could visualize this.

One, you could visualize
these individual points,

but that could also
help you, in general,

to visualize this relationship.

And so what he
essentially did is

he bridged the worlds
of this kind of very

abstract, symbolic algebra
and that and geometry,

which was concerned with
shapes and sizes and angles.

So over here you have
the world of geometry.

And obviously, there are people
in history, maybe many people,

who history may have forgotten
who might have dabbled in this.

But before Descartes,
it's generally

viewed that geometry
was Euclidean geometry,

and that's essentially the
geometry that you studied

in a geometry class in
eighth or ninth grade

or 10th grade in a traditional
high school curriculum.

And that's the
geometry of studying

the relationships between
triangles and their angles,

and the relationships between
circles and you have radii,

and then you have triangles
inscribed in circles,

and all the rest.

And we go into some depth in
that in the geometry playlist.

But Descartes
said, well, I think

I can represent this visually
the same way that you could

with studying these
triangles and these circles.

He said, well, if we
view a piece of paper,

if we think about a
two-dimensional plane,

you could view a
piece of paper as kind

of a section of a
two-dimensional plane.

And we call it two
dimensions because there's

two directions that
you could go in.

There's the up/down direction.

That's one direction.

So let me draw that.

I'll do it in blue because we're
starting to visualize things,

so I'll do it in
the geometry color.

So you have the
up/down direction.

And you have the
left/right direction.

That's why it's called
a two-dimensional plane.

If we're dealing in
three dimensions,

you would have an
in/out dimension.

And it's very easy to do
two dimensions on the screen

because the screen
is two dimensional.

And he, says, well, you know,
there are two variables here,

and they have this relationship.

So why don't I associate
each of these variables

with one of these
dimensions over here?

And by convention, let's make
the y variable, which is really

the dependent variable--
the way we did it,

it depends on what x is-- let's
put that on the vertical axis.

And let's put our
independent variable,

the one where I just
randomly picked values for it

to see what y
would become, let's

put that on the horizontal axis.

And it actually
was Descartes who

came up with the convention of
using x's and y's, and we'll

see later z's, in
algebra so extensively

as unknown variables
or the variables

that you're manipulating.

But he says, well, if we
think about it this way,

if we number these
dimensions-- so let's

say that in the x direction,
let's make this right over here

is negative 3.

Let's make this negative 2.

This is negative 1.

This is 0.

Now, I'm just numbering
the x direction,

the left/right direction.

Now this is positive 1.

This is positive 2.

This is positive 3.

And we could do the
same in the y direction.

So let's see, so this could be,
let's say this is negative 5,

negative 4, negative
3, negative-- actually,

let me do it a little
bit neater than that.

Let me clean this
up a little bit.

So let me erase this
and extend this down

a little bit so I can go all
the way down to negative 5

without making it
look too messy.

So let's go all
the way down here.

And so we can number it.

This is 1.

This is 2.

This is 3.

And then this could be
negative 1, negative 2.

And these are all
just conventions.

It could have been
labeled the other way.

We could've decided to put
the x there and the y there

and make this the
positive direction

and make this the
negative direction.

But this is just the
convention that people

adopted starting with Descartes.

Negative 2, negative 3,
negative 4, and negative 5.

And he says, well, I
think I can associate

each of these pairs of values
with a point in two dimensions.

I can take the x-coordinate, I
can take the x value right over

here, and I say, OK,
that's a negative 2.

That would be right over there
along the left/right direction.

I'm going to the left
because it's negative.

And that's associated
with negative 5

in the vertical direction.

So I say the y
value is negative 5,

and so if I go 2 to
the left and 5 down,

I get to this point
right over there.

So he says, these two values,
negative 2 and negative 5,

I can associate it with
this point in this plane

right over here, in this
two-dimensional plane.

So I'll say that point has
the coordinates, tells me

where to find that point,
negative 2, negative 5.

And these coordinates are
called Cartesian coordinates,

named for Rene
Descartes because he's

the guy that came up with these.

He's associating, all of a
sudden, these relationships

with points on a
coordinate plane.

And then he said, well,
OK, let's do another one.

There's this other
relationship, where

I have when x is
equal to negative 1,

y is equal to negative 3.

So x is negative
1, y is negative 3.

That's that point
right over there.

And the convention
is, once again,

when you list the
coordinates, you

list the x-coordinate,
then the y-coordinate.

And that's just what
people decided to do.

Negative 1, negative
3, that would

be that point right over there.

And then you have the point
when x is 0, y is negative 1.

When x is 0 right
over here, which

means I don't go to the left
or the right, y is negative 1,

which means I go 1 down.

So that's that point
right over there,

0, negative 1, right over there.

And I could keep doing this.

When x is 1, y is 1.

When x is 2, y is 3.

Actually, let me do it in
that same purple color.

When x is 2, y is 3, 2 comma 3.

And then this one right over
here in orange was 1 comma 1.

And this is neat by itself.

I essentially just
sampled possible x's.

But what he realized
is, not only do

you sample these possible
x's, but if you just

kept sampling x's, if you
tried sampling all the x's

in between, you would actually
end up plotting out a line.

So if you were to
do every possible x,

you would end up getting a
line that looks something

like that right over there.

And any relation, if you
pick any x and find any y,

it really represents
a point on this line.

Or another way to think about
it, any point on this line

represents a solution to this
equation right over here.

So if you have this
point right over here,

which looks like it's about
x is 1 and 1/2, y is 2.

So me write that, 1.5 comma 2.

That is a solution
to this equation.

When x is 1.5, 2 times
1.5 is 3 minus 1 is 2.

That is right over there.

So all of a sudden,
he was able to bridge

this gap or this relationship
between algebra and geometry.

We can now visualize
all of the x and y pairs

that satisfy this
equation right over here.

And so he is responsible
for making this bridge,

and that's why the
coordinates that we

use to specify these points are
called Cartesian coordinates.

And as we'll see, the first
type of equations we will study

are equations of
this form over here.

And in a traditional
algebra curriculum,

they're called linear equations.

And you might be saying,
well, OK, this is an equation.

I see that this
is equal to that.

But what's so linear about them?

What makes them
look like a line?

And to realize why
they are linear,

you have to make this jump
that Rene Descartes made,

because if you were to plot
this using Cartesian coordinates

on a Euclidean plane,
you will get a line.

And in the future,
we'll see that there's

other types of equations where
you won't get a line, where

you'll get a curve or something
kind of crazy or funky.