# Things vs Coordinates-of-Things

• Stephen Tashi
In summary, the distinction between a mathematical object and its coordinates is not typically introduced in elementary mathematics, causing confusion when it comes to understanding angles. However, teaching angles in a way that incorporates both equivalence relations and coordinate systems could provide a logical and consistent approach. This approach would allow for different numbers, such as 0 and 360, to represent the same angle without contradicting their individual values.

#### Stephen Tashi

In advanced mathematics, one must eventually learn the distinction between a mathematical object and some coordinates of that object. For example, eventually students are supposed to understand that a "vector" is not an n-tuple of numbers.

I wonder why this distinction is not introduced in elementary mathematics when students are taught about angles. Perhaps it wouldn't be simple!

Teaching students that numbers like 0 degrees and 360 degrees are coordinates of angles instead of being angles would relieve the teacher of having to double-talk about them being "different, but really the same" angle. Yet there are situations when zero degrees and 360 degrees denote different things. For example, a moving object making a "turn of 360 degrees" is different than its making a "turn of zero degrees".

Perhaps teaching angles in a way that made sense would involve teaching both equivalance relations and coordinate systems.

It's not clear what 'coordinates of angles' means.

SteamKing said:
It's not clear what 'coordinates of angles' means.

Coordinate systems are permitted to be redundant. In some coordinate systems, the same thing can be represented by different coordinates. The numbers 0 and 360 are obviously different numbers. If you want to talk about them representing "the same angle" in a logically consistent manner then you have to do it without contradicting the fact that 0 and 360 are different numbers. Considering values in degrees to be a method of assigning coordinates to an angle would be one way of doing this. There might be others.

## 1. What is the difference between "things" and "coordinates-of-things"?

"Things" refer to physical objects or entities, while "coordinates-of-things" refer to the specific locations or positions of those objects in space.

## 2. How are "things" and "coordinates-of-things" related?

"Coordinates-of-things" are used to describe the location or position of "things" in space. They are closely related as one cannot exist without the other.

## 3. Can "things" and "coordinates-of-things" be used interchangeably?

No, they cannot be used interchangeably. While "things" refer to physical objects, "coordinates-of-things" refer to the specific locations or positions of those objects.

## 4. What are some examples of "things" and their corresponding "coordinates-of-things"?

An example of a "thing" could be a tree, and its "coordinates-of-things" would be its exact latitude and longitude coordinates on a map. Another example could be a car and its "coordinates-of-things" would be its position on a road relative to other objects.

## 5. Why is it important to understand the difference between "things" and "coordinates-of-things"?

Understanding the difference between "things" and "coordinates-of-things" is crucial for accurately describing and locating objects in space. It is also important for various scientific fields such as geography, astronomy, and physics where precise location information is necessary for research and analysis.