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 ; once again I think I'm not getting the idea of accumulating sub-quantities(sum of sub-quantities) to gain the required quantity; and what I mean by that is actually my question down.


Well, I will visualize an example which will convey my problem with accumulating(sum) ; lets assume there's a bus which arrived at 7:00 at the station and his last left over that station is 10:00 and then the bus stops working ; given that between 7:00 and 10:00 the bus has passed over that station four time at 7:00, 8:00, 9:00, 10:00 .
the question is, find the total time that the bus is worked? it's simple to say from 10:00 till 7:00 which 10-7=3 ; but I'm confusing is how can I get the same answer if I added: (8:00-7:00) + (9:00-8:00) + (10:00-9:00) =3 ; my problem is why the accumulating the time between the pieces then I will get the total time between 7:00---10:00 ; exactly what I'm confusing at and what I'm thinking to solve the problem is like this:
frmo 7:00 till 8:00 the bus was working then this amount I must add it to the sum .., afterwards from 8:00+(not 8:00) till 9:00 I need to add this sum .. here is my problem I need to say in the second summation "8:00+" and not "8:00" .. so how why we are adding from 8:00 till 9:00 and not (8:00+) till 9:00 ?
actually why it doesn't matter to say 8:00 or 8:00+ is the same thing ?! isn't the amount(time) at concrete point "8:00" is smaller than the amount of time at concrete point "8:00+"?

To sum up; what I think and find it hard that I'm convinced that at boundaries - "between" - will have also a quantity so I need to say boundary+ or boundary- in accumulation .


thanks for helpers and sorry for that confusion.
 
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JeffM

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Your way of doing math was explored by Zeno two and a half millennia ago and leads to the conclusion that motion is impossible, which is obviously untrue of the physical world that contains things like clocks and buses. Therefore, it is not particularly useful to do math your way.

There are two ways to do math that avoid Zeno's paradoxes. The one that you would probably find more intellectually appealing is called non-standard analysis. Find a book on it in your native language because your English is nowhere close to being good enough to read such a book in English.

For information on Zeno's paradoxes, see

https://en.m.wikipedia.org/wiki/Zeno's_paradoxes
 
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Ryan$

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Excuse me? am I stupid that much to think like that?
 

JeffM

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Excuse me? am I stupid that much to think like that?
I did not say you were stupid. Zeno's paradoxes were not solved for thousands of years. Did you bother to read the article I cited?

There is no simple answer to Zeno's paradoxes. You can say that it is not useful to think like that because it appears to lead to a mathematics that is obviously not true of the universe that we live in. Or you can say that you can think like that if you do so in a sophisticated way. The branch of mathematics that deals with this kind of problem is called "analysis" in English. There are standard and non-standard versions of analysis. Analysis is generally taught after calculus.

I doubt that analysis can be taught on a site like this, but your university undoubtedly has a beginning course in analysis. I suggest that you sign up for it.

What I did say is that your English is not good. That does not mean that you are stupid. My Russian, Mandarin, Arabic, Swahili, and Hindi are not merely not good, but non-existent.
 
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Ryan$

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I did not say you were stupid. Zeno's paradoxes were not solved for thousands of years. Did you bother to read the article I cited?

There is no simple answer to Zeno's paradoxes. You can say that it is not useful to think like that because it appears to lead to a mathematics that is obviously not true of the universe that we live in. Or you can say that you can think like that if you do so in a sophisticated way. The branch of mathematics that deals with this kind of problem is called "analysis" in English. There are standard and non-standard versions of analysis. Analysis is generally taught after calculus.

I doubt that analysis can be taught on a site like this, but your university undoubtedly has a beginning course in analysis. I suggest that you sign up for it.

What I did say is that your English is not good. That does not mean that you are stupid. My Russian, Mandarin, Arabic, Swahili, and Hindi are not merely not good, but non-existent.

sorry for the miss understanding, if so .... please help me on how should I think/look at it? and by the way how can I visualize a number in my mind?! thanks
 

lev888

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Time is often compared to water. Let's accumulate water instead. What's the difference between 7 and 10 cups of water? 10-7=3 cups. Would it make a difference if we subtracted 1 cup 3 times? No. There is no water in the "boundary" between 8 and 9 cups or 9 and 10. Same with time.
 

JeffM

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sorry for the miss understanding, if so .... please help me on how should I think/look at it? and by the way how can I visualize a number in my mind?! thanks
I shall give a very informal and brief explanation of how I think about it.

In the physical world, all measurements are approximations based on measuring devices. What we call 7:00 o'clock and what we call 10:00 o'clock depend on a physical clock, and the measuring devices are not perfect. So the problem you are worrying about cannot come up because we can never say exactly how much time has past due to uncertainties in the accuracy of the clock. When physicists are being careful they will say that the elapsed time is

\(\displaystyle x \pm y.\)

Mathematicians mostly think about an imaginary world where the messy aspects of reality don't exist. In non-standard analysis, we imagine that we can divide time and space up into bits so tiny that the difference between them is zero even though the bits themselves are not zero. So in this imaginary world the difference between the moment before 8:00 o'clock and the moment after eight o'clock is no time at all, or zero bits of time. Whether you ignore that difference or add it twice makes no difference because the difference is zero.

What is amazing is that the mathematics that pertains to this imaginary world fits the real world to the utmost degree that we can measure the real world.

If you want a better philosophical answer, read Leibniz. If you want a more formal mathematical answer, take a course in analysis.
 

Ryan$

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I shall give a very informal and brief explanation of how I think about it.

In the physical world, all measurements are approximations based on measuring devices. What we call 7:00 o'clock and what we call 10:00 o'clock depend on a physical clock, and the measuring devices are not perfect. So the problem you are worrying about cannot come up because we can never say exactly how much time has past due to uncertainties in the accuracy of the clock. When physicists are being careful they will say that the elapsed time is

\(\displaystyle x \pm y.\)

Mathematicians mostly think about an imaginary world where the messy aspects of reality don't exist. In non-standard analysis, we imagine that we can divide time and space up into bits so tiny that the difference between them is zero even though the bits themselves are not zero. So in this imaginary world the difference between the moment before 8:00 o'clock and the moment after eight o'clock is no time at all, or zero bits of time. Whether you ignore that difference or add it twice makes no difference because the difference is zero.

What is amazing is that the mathematics that pertains to this imaginary world fits the real world to the utmost degree that we can measure the real world.

If you want a better philosophical answer, read Leibniz. If you want a more formal mathematical answer, take a course in analysis.
to sum up your words, you mean that the imaginary world is perfectly ideally concrete and we should visualize thing like this.
 

Subhotosh Khan

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to sum up your words, you mean that the imaginary world is perfectly ideally concrete and we should visualize thing like this.
We can visualize a "flat world" (instead of nearly spherical) and for most of our terrestrial problems that will not make a significant difference. However, that assumption (instead of the more realistic assumption of perfect sphere) will make building bridges, roads, etc. a lot easier with insignificant error.

Choose your poison!!!
 

Denis

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Ryan, what is your primary language?

Are you a student attending math classes?
 
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