In this lesson, our instructor Vincent Selhorst-Jones gives an introduction on coordinate systems. He explains the inherent order in real numbers, less than and greater than, ordered pairs, the one dimension number line, two dimension plane, and three dimension space. He also explains quadrants and higher dimensions.
The real numbers (ℝ) have an inherent order to them. Large negatives are lowest, then small negatives, then 0, then small positives, and finally large positives.
We can show this order with the symbols < ('less than') and > ('greater than'). Examples: −7 < 2, 100 > 47.
If we want to indicate that the relationship between two numbers might be equal, we can use ≤ ('less than or equal') and ≥ ('greater than or equal'). Example: x ≤ 5 means that x can be any number up to and
including 5, while y < 5 means that y must be strictly less than 5.
If we have a mathematical relationship based on one of the above, we call it an inequality because the two sides are not equal.
We can graphically represent this idea of order with the number line. We build out from 0 (the origin) to −∞ on the left and ∞ on the right.
If we want to talk about two numbers at the same time, we can create an ordered pair. We can represent these ordered pairs of numbers with the plane: two number lines crossed perpendicularly.
In the plane, we call the point of intersection the origin: (0,0). By convention, the first number in an ordered pair always goes by the horizontal, and the second by the vertical. While it changes, we often call the horizontal axis the x-axis,
and the vertical axis the y-axis.
Sometimes we'll talk about which quadrant-quarters of the plane-a point is located in. We start with where both coordinates are positive: the top-right, then work counter-clockwise, counting off the four quadrants.
We can continue with the idea or ordered pairs by creating ordered triplets. These can be represented visually with another perpendicular number line to create a third dimension. We call this a (three-dimensional) space. This course won't
explore much in three dimensions, but it's interesting to think about.
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.