Hello. I'm Professor Von Schmohawk and welcome to Why U.
We have seen that sets of real numbers can be visualized using the number line.
A line is a 1-dimensional object
and the number line represents real numbers as points in this 1-dimensional space.
Every real number corresponds to a single point in this space and vice versa.
In the previous lecture we discussed the operation of creating a "Cartesian product".
In this lecture, we will see how the Cartesian product
allows us to construct mathematical objects
which correspond to points in two or more dimensions.
As we saw in the last lecture, the Cartesian product of two sets A and B
is formed by pairing one element from each set to form ordered pairs.
The collection of ordered pairs formed by the Cartesian product forms a new set.
This set contains ordered pairs representing every possible combination of elements
where the first element is from set A
and the second element is from set B.
The Cartesian product is written using a symbol which looks like a multiplication symbol.
We can display the ordered pairs of this Cartesian product in a grid
to form a simple 2-dimensional coordinate system.
By specifying the colors of each element
we can locate any horizontal and vertical position within the grid.
It is not necessary for each operand of a Cartesian product to be a different set.
For instance, we can form the Cartesian product of set A with itself.
This can be written as the Cartesian product of A and A
or "A squared".
Just as before, we can locate any horizontal and vertical position within the grid
by specifying the colors of each element.
Or, if instead of set A containing colored squares, set A contained numbers
we could locate any position in the grid by specifying the numerical value
of the first and second elements of an ordered pair.
Set A does not have to be a finite set of numbers.
We can also create Cartesian products of infinite sets.
For instance, set A could be the set of integers Z.
Forming a Cartesian product of the set of integers with itself
creates an infinite set of ordered pairs "Z-squared"
whose elements are every possible combination of two integers.
Of course, we would need an infinitely large grid
to represent all the ordered pairs in Z-squared.
Instead of writing every ordered pair
each ordered pair can be represented by its position in the grid
each pair corresponding to a unique point.
We can locate the positions of these points using a pair of number lines.
For each ordered pair, the first element corresponds to a position on the horizontal number line
and the second element corresponds to a position on the vertical number line.
The points formed by the Cartesian product of the set of integers with itself
form an infinite grid of points spaced one unit apart.
Now, if instead of using the set of integers Z
we form the Cartesian product of the set of real numbers R with itself
we create a continuum of points which completely fill the plane.
Then every ordered pair of two real numbers
corresponds to a unique point in this 2-dimensional space.
This system for visualizing ordered pairs of real numbers as points
is called the "Cartesian coordinate system".
And the elements of an ordered pair which corresponds to a point
are called the "coordinates" of the point.
The ideas which led to this system were developed by Rene Descartes in his book "La Geometrie".
La Geometrie, published in 1637
united algebra and geometry into a single subject "analytic geometry"
which describes geometric shapes by algebraic equations.
Likewise, algebraic equations can be visualized as geometric shapes.
This is possible because, as we will soon see
algebraic equations define sets of points
which when viewed in the Cartesian coordinate system, appear as shapes.
The perpendicular number lines in the Cartesian coordinate system are referred to as "axes".
The horizontal and vertical axes are often called the x-axis and y-axis.
Sometimes these two axes are referred to as the "abscissa" and the "ordinate".
The point where the axes meet represents the number zero on each axis.
This is called the "origin" of the coordinate system.
The origin corresponds to the ordered pair (0,0).
The infinite plane containing the x and y axes is referred to as the "Cartesian plane"
or the "xy-plane".
The axes divide the xy-plane into four regions called "quadrants".
These are numbered from the first to fourth
starting with the upper right quadrant and continuing counter-clockwise.
The quadrants are usually denoted with Roman numerals.
Many mathematicians prefer to draw number lines and axes
with arrows pointing towards the positive direction only
indicating the direction of increasing value.
We have seen that the number line corresponds to the set of real numbers R.
So forming the Cartesian product of the set of real number with itself
is equivalent to forming the Cartesian product of the number line with itself
to form the Cartesian plane.
And just as the number line consists of a continuum of points
residing in 1-dimensional space
where each point corresponds to a unique real number
the Cartesian plane consists of a continuum of points residing in 2-dimensional space
where each point corresponds to a unique ordered pair of real numbers.
Using a 2-dimensional Cartesian coordinate system
we can graphically display sets of ordered pairs as groups of points in this space.
Later we will see how algebraic equations can describe infinite sets of points
which when viewed in this system appear as shapes in two dimensions.
In the next lecture we will see how a 3-dimensional Cartesian coordinate system can be constructed
which will allow us to visualize sets of ordered triples in three dimensions.