# point in math

##### Full Member
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 !

##### Full Member
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

##### Elite Member
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

##### Elite Member
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

##### Senior Member
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.

##### Full Member
Excuse me? am I stupid that much to think like that?

#### JeffM

##### Elite Member
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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##### Full Member
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

##### Super Moderator
Staff member
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.