how would you define a mathematical point?

[PG1995]

Hello

How would you define a mathematical point? Here is my 'crude' opinion. A point is simply a supposition from mathematician's point of view that a certain number is there where the point or dot is; although it (the number) is not there. But it serves the purpose. What do you say? Please let me hear. Thanks.

 

[misterT]

"The number is not there, but it serves the purpose".. I can agree with that.

In geometry there are these rules:

point - point = vector
point + vector = point
vector + vector = vector

You are not allowed to add points, unless you scale them with scalars that add up to 1:
0.7*point1 + 2*point2 - 1.7*point3 = point
above calculation is ok because 0.7 + 2 - 1.7 = 1

Can you see why the sum of scalars needs to be 1?

I did not answer your question, but I wanted to give you some more to think about. With points, the important thing is the distance between two points and other relations.. the actual position is not important (it can always be fixed arbitrarily).

 

[PG1995]

"The number is not there, but it serves the purpose".. I can agree with that.

Thank you for agreeing!

In geometry there are these rules:

point - point = vector
point + vector = point
vector + vector = vector

I have never heard of these rules and they don't make any sense to me! Suppose we have a vector displacement, then what kind of 'point' would you add to it?!

What kind of 'points' would you subtract to get a displacement vector?

You are not allowed to add points, unless you scale them with scalars that add up to 1:
0.7*point1 + 2*point2 - 1.7*point3 = point
above calculation is ok because 0.7 + 2 - 1.7 = 1

In the equation above, point equal what? If I ask you to substitute value for the 'point' what value would you use?

Please help me with the above queries. Please don't forget I'm not good at math. Thank you.

Regards
PG

 

[misterT]

Suppose we have a vector displacement, then what kind of 'point' would you add to it?!

You can add a displacement vector to a point. You get a point.

What kind of 'points' would you subtract to get a displacement vector?

I'm talking about coordinate points. For example in two dimensions we can have points P and Q:

P = (1,4)
Q = (5,3)

Subtracting these points is allowed, we get a vector v:
v = P-Q
v = [-4 1]

Adding a vector to a point, we get a point:
R= P + v
R = (-3 5)

we can add points only if we scale them so that the scaling factors add up to 1. (https://mathworld.wolfram.com/BarycentricCoordinates.html)
0.4P + 0.6Q = D
D = (3.4, 2.8)

This is valid because we can rewrite the equation:
D = (1 - 0.6)P + 0.6Q = P + 0.6*(Q-P)
Which is a point plus a scaled vector.

Usually points and vectors are treated like they are all vectors, But they are mathematically very different.
https://www.electro-tech-online.com/custompdfs/2011/09/Points-and-Vectors.pdf

 

[atferrari]

Basics

A point is the place that complies with 2 equations / 3 equations (depending) if defining it in 2D / 3D in a system with references defined beforehand (usually the zeros of each of 2 / 3 parameters).

 

[MrAl]

Hi,

Three dimensional space is an infinite set of two dimensional spaces.
Two dimensional space is an infinite set of one dimensional spaces.
One dimensional space is an infinite set of zero dimensional spaces.
A point is a zero dimensional space.

 

[Ratchit]

PG,

How would you define a mathematical point?

A position defined by a coordinate system.

Ratch