I doubt many people would be surprised if I said that squares are a special case of rectangle. But, they do have certain properties that you don’t see in a general rectangle, foremost being that they have their own name for when you multiple the two sides together to get the area of the object.

Normally, the x- and y-axis will be measured in the same units when they represent the same thing, such as with the previous example of the backyard. Both sides were measured in feet, which represent “unit squares” when area is calculated. Hence 100 feet times 40 feet equals 4,000 feet-feet units. Since “feet-feet” is kind of awkward to say, especially when you’re at parties, it’s more proper to use “square feet”, although “foot squared” might make a little more sense (given that we use “two-squared” to mean “2*2” or “2^2”).

(One rectangle moved around on the graph.)

If we look at the graph, we can see that the position of the backyard rectangle doesn’t change its area. For simplicity, it’s easier if one corner is located at (0,0). What’s interesting, right now, is if we notice that the rectangle could be said to have one side that’s 40 units tall, and 100 units wide.

But, we could also say that there are 100 smaller rectangles set side-by-side that are 40 units tall and 1 unit wide. If we step through each unit stack and add up their heights, we get 40 + 40 + 40… 100 times. This could be a simple empty-headed mindgame, except that this is how we could approximate the area of a less linear backyard, one that has been landscaped.

(Finding the area of a backyard in Beverly Hills.)

We use the formula: area = sum(f(x)*delta x)

to represent this summation. This kind of looks like the Riemann Zeta function…

(Identical squares located in each of the four quadrants.)

But, let’s get back to squares. It is possible to get a negative area. This can be thought of as removing dirt from a hole. In one sense, the removed dirt is positive, but in another, the hole is negative.

If you remember your multiplication properties:

positive times positive is positive

negative times negative is positive

negative times positive is negative

positive times negative is negative

Look at those squares. They’re the same square, just shifted into a different quadrant of the graph. What if there was something about the number -1 that had a magic ability to kick points around on the graph. Like, maybe that one corner at (0,0) was nailed in place and the square was rotating around the origin?

I was taught in school to write out points on a graph as (x0, y0). This is fine when you want to record the corner points of the backyard on a piece of paper, but it gets messy for anything more practical.

Rene Descartes gave us the cartesian coordinate system, which I’ve been using up to this point. In simple systems, the x-axis represents the variable in the formula that changes, while the y-axis is the output from the formula for that value of x.

y(x) = 2*x^2 + 3

(Graph where the vertical axis represents the output from f(x).

But, what if the part that changes has two movable parts? Such as changing “x” and “y” at the same time?

Multiplying x by a real number (any value between positive and negative infinity) acts like a scalar – it scales the size of x.

1 * 2 = (2, 3)

2 * 2 = (4, 3)

3 * 2 = (6, 3)

All we’re doing here is moving a point (x, 3) back and forth along the horizontal axis.

Conversely, multiplying y by a real number still acts as a scalar – scaling the size of y.

1 * 3 = (2, 3)

2 * 3 = (2, 6)

3 * 3 = (2, 9)

This just moves the point (2, y) up and down along the vertical axis.

How can we change axes? How can we jump from the (x, 3) line to the (2, y) line?

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