# 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

##### Senior 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

##### Senior 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
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

##### Elite Member
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

#### Otis

##### Senior Member
… will not discuss more …
Okay. But I will … … 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$#### Ryan$

##### Full Member
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

##### Elite Member
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\$

##### Full Member
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?!