point in math

Ryan$

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Jan 25, 2019
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133
Hi guys!
I'm new here and a new student in the university, I'm not accepting the idea how actually the "points" are not affecting the calculated quantity ; I mean:
lets assume that I have point a until point b and I want to calculate the distance, so it's b-a and if I wipe off point "b" so it will not included in the distance, but it's still the distance b-a , in other words I'm claiming that the distance between a and b is all the consecutive points between a and b (included a and b) ! ..so why if I wipe off the point b , the distance between a and b still b-a although I wiped off the point b ? what I claim that the distance must be b-a-(epsilon) and epsilon is because I wiped one point.

Another thing, why at integration, wiping points between the bounds that my integration is, will not affect the integral?


Maybe I miss understanding the definition of point and how it's going on math.


thanks!!
 

Otis

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Apr 22, 2015
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Let's start with this: points have no dimensions. That is, they don't exist as real objects; they are a concept of the mind.

Again, a point has no width; it is infinitely "small". You can't shave off point b (away from a point next to it) because there is no point next to b. If you think you've found a point adjacent to point b, then you're wrong. There are still an infinite number of points in between any two points you envision to be "next to" each other.

We describe this attribute (i.e., infinite points comprise any segment of the Real number line) by saying, "The Real number line is dense."

Infinity is also a concept of the mind (infinity is definitely not a Real number).

What is the smallest positive Real number? There isn't one. No matter how close you think you are, to the right side of zero, there are still an infinite quantity of Real numbers between where you are and zero.

At some point (a pun), the numbers become too small to comprehend and might serve no useful purpose other than exercising the mind.

If you "shave off" such an infinitely small quantity from your interval (ending in point b), nobody will ever notice on their own, heh.

Another example: if you "remove" all the points comprising the circumference of a circle, then the circle's area doesn't change one bit.

My reply is not very rigorous, but I hope it helps. Cheers :cool:
 
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tkhunny

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Apr 12, 2005
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9,777
Hi guys!
I'm new here and a new student in the university, I'm not accepting the idea how actually the "points" are not affecting the calculated quantity ; I mean:
lets assume that I have point a until point b and I want to calculate the distance, so it's b-a and if I wipe off point "b" so it will not included in the distance, but it's still the distance b-a , in other words I'm claiming that the distance between a and b is all the consecutive points between a and b (included a and b) ! ..so why if I wipe off the point b , the distance between a and b still b-a although I wiped off the point b ? what I claim that the distance must be b-a-(epsilon) and epsilon is because I wiped one point.

Another thing, why at integration, wiping points between the bounds that my integration is, will not affect the integral?


Maybe I miss understanding the definition of point and how it's going on math.


thanks!!
"definition of a point"? Where did you get one of those. It tends to be axiomatic not a definition.

"wiping points between the bounds...will not affect the integral"? Are you sure? You're basic Riemann Integral may simply fail to exist if you wipe too many. At least keep your wiping countable.
 
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Harry_the_cat

Full Member
Joined
Mar 16, 2016
Messages
939
Hi guys!
I'm new here and a new student in the university, I'm not accepting the idea how actually the "points" are not affecting the calculated quantity ; I mean:
lets assume that I have point a until point b and I want to calculate the distance, so it's b-a and if I wipe off point "b" so it will not included in the distance, but it's still the distance b-a , in other words I'm claiming that the distance between a and b is all the consecutive points between a and b (included a and b) ! ..so why if I wipe off the point b , the distance between a and b still b-a although I wiped off the point b ? what I claim that the distance must be b-a-(epsilon) and epsilon is because I wiped one point.

Another thing, why at integration, wiping points between the bounds that my integration is, will not affect the integral?


Maybe I miss understanding the definition of point and how it's going on math.


thanks!!
A question for you: Do you accept that 0.9999..... = 1?
 

Ryan$

Junior Member
Joined
Jan 25, 2019
Messages
133
A question for you: Do you accept that 0.9999..... = 1?

frankly NO , that's why I'm getting confused .. why would you accept that 0.999999 =1? so in other words we are not solving the problem "exactly" we are approximating it !
 

Ryan$

Junior Member
Joined
Jan 25, 2019
Messages
133
Let's start with this: points have no dimensions. That is, they don't exist as real objects; they are a concept of the mind.

Again, a point has no width; it is infinitely "small". You can't shave off point b (away from a point next to it) because there is no point next to b. If you think you've found a point adjacent to point b, then you're wrong. There are still an infinite number of points in between any two points you envision to be "next to" each other.

We describe this attribute (i.e., infinite points comprise any segment of the Real number line) by saying, "The Real number line is dense."

Infinity is also a concept of the mind (infinity is definitely not a Real number).

What is the smallest positive Real number? There isn't one. No matter how close you think you are, to the right side of zero, there are still an infinite quantity of Real numbers between where you are and zero.

At some point (a pun), the numbers become too small to comprehend and might serve no useful purpose other than exercising the mind.

If you "shave off" such an infinitely small quantity from your interval (ending in point b), no body will ever notice on their own, heh.

Another example: if you "remove" all the points comprising the circumference of a circle, then the circle's area doesn't change one bit.

My reply is not very rigorous, but I hope it helps. Cheers :cool:
so I can say that "point" can be visualized in mind as something isn't found and at math point is about "nullity" nothing and doesn't affect my solutions?


thanks
 

Ryan$

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Jan 25, 2019
Messages
133
Actually what I imagine a point as something that have quantity(value) that's why I'm finding it hard to solve problems in math, and why I imagine that is, because I'm not convinced that point is something not found or actually "empty"
 

Otis

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Apr 22, 2015
Messages
1,154
so I can say that [a] "point" can be visualized in [the] mind as something [that] isn't found [in the real world] …
Yes. We use the idea of points to help us visualize numerical objects.


… and [in] math [a] point is about "nullity" [or] nothing and doesn't affect my solutions?
Well, the word nullity infers no usefulness. Points are very useful, in mathematics. We use them as models, to represent numerical values.

The Real number line is composed of an infinite collection of points. Each point on the number line represents a specific Real number. Likewise, for every Real number, there is a point on the line.

When we measure a specific distance on the Real number line, we subtract the smaller number from the larger number, yes? For example, what is the distance from -4 to positive 4?

We subtract the smaller number from the larger number:

4 - (-4) = 8

The number 4 is represented by the point which is exactly four units to the right of zero. The number -4 is represented by the point which is exactly four units to the left of zero. These two points are the 'endpoints' of the interval.

In the real world, if we use a measuring device (like a ruler), we can't measure exactly 8 units, but what we see is close enough. That is, if we were to use a very precise measuring device, the measurement might show 8.0000000000000000000000000000000000203956…

In the real world, the average person won't care about those non-zero digits (starting around the 40th decimal place). Whether or not a mathematician cares about them, the mathematician understands they do exist.

Maybe you are perplexed because you're not yet thinking about infinity. Did you think about my earlier question? What is the smallest positive number? :cool:
 
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Otis

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Actually imagine a point as something that [has] quantity (value) that's why I'm finding it hard to solve problems in math, …
A point represents a value. It is only a model (i.e., part of a system for organizing and visualizing relationships between numbers).


… I'm not convinced that point is something not found or actually "empty"
Fair enough. If you think a point is something that you can find in the real world, please give me an example.

A point represents a number, and that number has value. If you want to say that some point has value 4, that's okay, but saying it doesn't give the point mass (in any sense).

When you say that a point cannot be "empty", are you thinking about width? A point has no width. You cannot measure the width of a point. You cannot assign any value to a point, other than the specific number it represents.

Points are dimensionless.

If you think that a point has width (or some value different from the Real number it models), please give me a specific example.

Also, tell me in your own words what 'infinity' means to you. Cheers :cool:
 
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Harry_the_cat

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Mar 16, 2016
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frankly NO , that's why I'm getting confused .. why would you accept that 0.999999 =1? so in other words we are not solving the problem "exactly" we are approximating it !
NO! 0.999999 does NOT equal 1. I did not say that.
What I said was 0.9999... = 1 , the … indicating a recurring decimal.
 

Ryan$

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Jan 25, 2019
Messages
133
Hi guys, I totally know that I already opened this thread before, and I will not discuss more .. but just for my heart and my soul to be satisfied..... at the end, in briefly, a point isn't anything .. yeah? I mean if I want to calculate a quantity between A and B (lets assume quantity represented a distance ) so that quantity is irrelevant to A and B, I mean irrelevant to point A and B .. right? thanks alot
 

pka

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Jan 29, 2005
Messages
7,797
Hi guys, I totally know that I already opened this thread before, and I will not discuss more .. but just for my heart and my soul to be satisfied..... at the end, in briefly, a point isn't anything .. yeah? I mean if I want to calculate a quantity between A and B (lets assume quantity represented a distance ) so that quantity is irrelevant to A and B, I mean irrelevant to point A and B .. right?
Here is a favorite quote of mine. “As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.” Albert Einstein in Geometry & Experience, 1929. Einstein was warning against being to literal in referring to thing mathematical. Almost all axiom systems begin with a listing of undefined terms. Point is on many, many of those lists.
You asked about distance. What you may not realize is that distance is a measure usually call a metric. \(\displaystyle d(P,Q)\) is the distance between points
\(\displaystyle P\;\&\;Q\) BUT that has very strict rules: 1) \(\displaystyle d(P,Q)\ge 0\) 2) \(\displaystyle d(P,Q)=d(Q,P)\) & 3) \(\displaystyle d(P,Q)=0\) if and only if \(\displaystyle P=Q\).
Absolute value is a metric. \(\displaystyle |x-y|\) is the distance from \(\displaystyle x\text{ to }y\).
If \(\displaystyle |x-7|<4\) then \(\displaystyle x\) is within four units of seven or \(\displaystyle 3<x<11\) note that 7 is midpoint and \(\displaystyle 2(4)=8\) is the diameter of that interval.
But more to your point, \(\displaystyle |X|=|X-X|=|0|=0\) the distance of \(\displaystyle X\text{ to itself is }0\), the measure of a point is \(\displaystyle 0\).

So lighten up, take Einstein's advice to heart.
 

Otis

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Apr 22, 2015
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1,154
… will not discuss more …
Okay. But I will … :p

… a point isn't anything .. yeah? …
A point is not anything physical, in the real world.

However, a point is something. As people have said, a point is a concept, an idea, a useful model for things like numbers, locations, intervals, graphs, etc.

😎
 
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