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Algebra 10 - The Cartesian Coordinate System
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- 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.