point in math

Ryan$

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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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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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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

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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$

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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$

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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), 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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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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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 [I] 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 (eg: location) 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

😎
 

Harry_the_cat

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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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Hi guys, I have to confusions which please illustrate for me them in a simple manner.
first is there more than smaller to the point? I mean if I imagine a point .. then I can also imagine a point inside the point that I imagined which it's smaller than what I imagined .. so?

second question, is continuous function means that at everypoint there's a possibility that the values of function over it will be changed? thanks alot

and who said that at every point I have value? maybe at semi-point I have value? who corresponded that a value corresponded to a point and not to semi-point?! thanks alot
 
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Dr.Peterson

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No, if you correctly imagine a point, then you can't imagine anything smaller! A point is to be thought of as a mere location, with no size at all. Yes, it's hard to imagine that; but if you imagine a size, then you are imagining wrongly, just as you would be wrong to imagine a vacuum from which you could suck more air than you already did to form it.

As for functions, a function isn't something that changes; it is a particular relationship between an input and an output. For any give input, the output is fixed. So it's not at all clear what you are thinking. A continuous function (leaving out some technical details) is one whose graph can be drawn in one continuous line, without ever lifting your pencil. It has to do with how values of x close to one another affect values of y -- if x doesn't change much, y shouldn't change much. (You need calculus to really express this correctly.)
 

Ryan$

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Wow convinced me about point in away that I cant say anything :) 😊
Another thing then why we assume that value is corresponded to a point and not for example a value corresponded to semi-point? I mean maybe value occupy two points at one time? Who claims that value number is corresponded to one point at a time?
 

Dr.Peterson

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Math is invented so that it represents the concepts we are trying to represent, as simply as possible. Assuming that each number corresponds to one location on a number line agrees with what we expect from generalizations of the real world, and makes calculations easy, so we go with that (and have for a very long time).

Now, perhaps you could invent a new kind of geometry in which there was such a thing as a "semi-point" and there were two numbers in the same place on a number line; but you probably couldn't get anyone to try using it, because it would be too cumbersome with no benefits. That's not to say it would necessarily be "wrong".

Mathematicians do invent new mathematical objects or systems, just to see what will happen! And sometimes the results turn out to be useful, even if they weren't trying to make something that corresponds to the real world. But if what they invent is not either useful or interesting, their paper will just gather dust, even if it's perfectly valid, because there will be no motivation for anyone to pursue it further.
 

Ryan$

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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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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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… 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.

😎
 

Ryan$

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Hi guys, I know that I already opened a thread about that subject but still struggling and by you I really boost myself.

when I imagine point, then I imagine a black box which if I split one of its points then its place will be white so it has dimension .. and that's wrong analogy .. can anyone help me how should I imagine point?! thanks alot
 

pka

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when I imagine point, then I imagine a black box which if I split one of its points then its place will be white so it has dimension .. and that's wrong analogy .. can anyone help me how should I imagine point?!
In mathematics it is impossible to imagine a a point. Point is an undefined term. A point is something that just is.
In a famous example of a finite geometry bee hives are points.
 

Ryan$

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Hi guys, I really want to verify about something it might be silly but I face it every time, and I want to verify if I just alone facing it or actually it's likely to others.
for example once I face something like subtraction such as : 4-5 then in my mind I imagine it like I have something continuous like this ----------------------------------------------- which its length is 5 and if I want to subtract 4 then I just remove 4 units from that line, what's confusing me that the mutual area or "point" is found between the removed area (the empty) and the reminder area .. exactly what I mean ----------------------------------- ------ , the right line is the subtracted area, and once I removed the left line (which it's like 4 because we are doing 5-4) then the point between the line removed and the left line is mutual between two lines(left and right) so doesn't it matter and change the quantity of 5-4 ?! I mean I claim that it would be matter because there's one point left from the ( "removed line" = 4 ) on the reminder amount of 5-4 ... and it's found because it's mutual between two lines so if I removed it from the left line , it would be still on the right line, doesn't that matter?!
 
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