Meaning of "Commas and Periods" in Mathematical Functions?

Question

Can someone please help me understand the meaning of "commas and periods" that appear on the right side of mathematical functions?

For example, in another question (Making the Mandelbrot Fractal in Desmos Online Graphing Calculator), I learned how to make the "Mandelbrot Fractal" using an online graphing calculator: https://www.desmos.com/calculator/hvrhuvaue5

I noticed that in some of these mathematical equations, there are "periods and commas" that appear on the right hand side of these equations:

Normally, I would have thought that:

• "Commas" generally appear on the left hand side of mathematical equations to indicate which variables are included in the function. For example, a function F(x,y,z) = z*y - x : commas are used to indicate that the function "F" is made up of variables "x", "y" and "z". For instance, what does "x,2z" mean in the function f(z)?

• On the other hand, I have usually seen "periods" as another notation for indicating "multiplication" (apart from indicating "decimals", which is clearly not the case here) - but for some reason, I do not think that the "period" is referring to multiplication here. I was thinking that maybe the "period" might indicate a "nested function" , but I also do not think that this is the case here. For instance, what does "z.y^2" mean in the function D(z)?

Can someone please help me understand the meaning of "periods and commas" in mathematical functions?

Thank you!

Answer 1

These are two very different questions.

Dot notation

This is actually not standard mathematics notation, but special notation used in Desmos that other software/textbooks/mathematicians would not necessarily understand. In Desmos, if p is a point $(a,b)$, then p.x denotes its $x$-coordinate $a$. Similarly, p.y denotes its $y$-coordinate. This works whether or not the coordinates are named x and y, or whether or not they're abstract: (3,4).y works fine and is $4$. But this is just Desmos notation.

Sometimes, in mathematics we might write $p_x$ to denote the $x$-coordinate of the point $p$, but the subscript $x$ could mean something else/be ambiguous, so it's best to explain your notation.

Commas

Commas have many different meanings in mathematics. The relevant one in the definition of $f(z)$ in your image is to separate two coordinates of a point. $f(z)$ takes in a point $z=(z_x,z_y)$ (with coordinates that one would write in Desmos as z.x and z.y instead of subscripts), and outputs another point with first coordinate $z_x^2-z_y^2+x$ and second coordinate $2z_xz_y+y$ (here $+x$ and $+y$ are other parameters not necessarily related to $z$, which is a little confusing), so we write the coordinates of the output point together with a comma as $(z_x^2-z_y^2+x,2z_xz_y+y)$. A much simpler usage would be the point $(3,4)$ with $x$-coordinate $3$ and $y$-coordinate $4$. If you are not familiar with $x$ and $y$-coordinates of points in the plane, you may need to study the cartesian/coordinate plane.