# Math objects with properties

I'm trying to write a simple formula based on some work I've had to do with programming. In programming, I have the ability to write a class with certain properties. For example: A coordinate with x & y values.

I can then access these values like... point.x & point.y

What is the correct way to describe these objects and use their values in mathematical notation?

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We usually write them as $(x,y)$. – wj32 Oct 31 '12 at 11:24
What kind of properties are you talking about? Ints, strings, vectors, lists...? – rbm Oct 31 '12 at 11:36
Ints and floats mainly. I'm just summarizing the solution to the problem given to me. – Dan Oct 31 '12 at 11:40
If we have a point $P$, we usually first write $P=(x, y)$ somewhere in the text, and after that, we can use $x$ and $y$ just like a programmer would write P.x and P.y. If you have several points (say $P$ and $Q$), we differ between them by adding indices, like $P = (x_P, y_P)$ and $Q = (x_Q, y_Q)$. – Arthur Oct 31 '12 at 11:45
Fantastic, Arthur. Just what I needed. – Dan Oct 31 '12 at 11:49

The dot-notation is not common in Maths. Instead, you usually use indices, where the order of element and property might be reversed. Thus, $element.property$ is usually referred to as $property_{element}$, where any name is usually single-lettered, and elements are also often identified by some number.

For example let $M=\{point1, point2\}$ be some set of two points, both with properties x and y. In the programming lanuage you would refer to them by point1.x, point1.y, point2.x, point2.y.

In Maths, you would usually identify the two points with numbers, thus, $M = \{1, 2\}$, and write $x_1, y_1, x_2, y_2$. But things can also be totally different, depending on how it is most comfortable for your purpose.

It is also ok to write $p_x$, $p_y$ for any $p \in M$, where it might be that $p = point1$ or $p = point2$.

Another alternative: it is always possible to represent properties by a function on the set of elements (just as properties can also be accessed with a function, e.g., point1.getX()). Thus, we could define the function $x : M \rightarrow \mathbb{R}$ to map to the x-coordinate of any given point from the set $M$ to its real value, and refer to it by writing $x(p)$ for any $p \in M$.

However, the universal answer to this question is that there is no universal answer :)

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Haha, thanks for that. I guess as long as my documentation also explains what is happening correctly, then I can use any of these options. Thanks for your help :) – Dan Oct 31 '12 at 11:55

For properties of mathematical values one uses functional notation, though notational variations exists. For instance for a complex number $z$ one could note its real and imaginary parts $\Re z$ (or $\operatorname{Re}z$) and $\Im z$ (or $\operatorname{Im}z$), its absolute value is written $|z|$ and its argument $\arg z$. However one often uses an indirect way to access these properties, so instead of saying "let $a=\Re z$ and $b=\Im z$" one would say "let $z=a+bi$", and instead of "let $r=|z|$ and $\phi=\arg z$" one would say "let $z=re^{\phi i}$" (with some luck the author would add: $a,b,r,\phi\in\Bbb R$ and $r\geq0$, to make this less ambiguous). In some cases the latter is virtually the only style used, for instance one could write the coefficient of $X^3$ in a polynomial $P$ as $\operatorname{coef}(X^3,P)$, but this is not often done; rather one would say "let $P=c_nX^n+\cdots+c_1X+c_0$" and then talk about the coefficient $c_3$.

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