Continuity/Discrete, Point vs Unit, Infinity question

aprovehh

New member
I was thinking about what is the location of the median of a set of 10 numbers which led to my thought of difference between a point and unit.

Obviously writing out 10 numbers like so:

1 2 3 4 5 6 7 8 9 10

Clearly shows that the median is between 5 and 6 since that would mean you have the same number of items above and below.

But if we think of 10 as a length of 10 on a number line for instance and try to find the "spot" where half the data is above and below, I would think it would be 5. i.e. 10/2 (halving the length).

Does this have something to do with how we define each item from a discrete sense (writing out 10 integers like 1-10) vs the continuous representation of drawing a line?

Since with the number line, each unit (or discrete object if we need to think of that) is actually a span that stretches between two integers, do we need to find a middle span and not just a middle point?

I guess I'm trying to reconcile what the differences are between these two models and when they lead to different or non-intuitive results.

Can anybody please comment / let me know what they think as I think this kind of has to do with how I represent / think of what the nature of a number really is.

Related to this question is, can somebody help me understand the formula (n+1)/2 as the position of the median from a geometric (length based) representation? It's not clicking with me.

Harry_the_cat

Senior Member
I was thinking about what is the location of the median of a set of 10 numbers which led to my thought of difference between a point and unit.

Obviously writing out 10 numbers like so:

1 2 3 4 5 6 7 8 9 10

Clearly shows that the median is between 5 and 6 since that would mean you have the same number of items above and below.

But if we think of 10 as a length of 10 on a number line for instance and try to find the "spot" where half the data is above and below, I would think it would be 5. i.e. 10/2 (halving the length).

Does this have something to do with how we define each item from a discrete sense (writing out 10 integers like 1-10) vs the continuous representation of drawing a line?

Since with the number line, each unit (or discrete object if we need to think of that) is actually a span that stretches between two integers, do we need to find a middle span and not just a middle point?

I guess I'm trying to reconcile what the differences are between these two models and when they lead to different or non-intuitive results.

Can anybody please comment / let me know what they think as I think this kind of has to do with how I represent / think of what the nature of a number really is.

Related to this question is, can somebody help me understand the formula (n+1)/2 as the position of the median from a geometric (length based) representation? It's not clicking with me.
A number line would go from 0 to 10, not 1 to 10.
If you write out the integers from 0 to 10 (ie 11 integers), the median will be 5.

aprovehh

New member
Ok well then how does 1 - 10 and 0 -10 represent the same thing?

Anybody else have any other thoughts.

Harry_the_cat

Senior Member
Ok well then how does 1 - 10 and 0 -10 represent the same thing?
They don't represent the same thing!

aprovehh

New member
Do we not represent 10 items by a length of 10 on the number line and by counting ten items?

HallsofIvy

Elite Member
When "counting 10 items" we start counting at 1, not 0. You should NOT "represent 10 items by a length of 10 on the number line"! You "represent 10 items" on a number line by marking the individual points 1 through 10. It is not an interval at all and has NO "length".

aprovehh

New member
so length is different than counting? Plenty of people represent 10 as the distance 10 units away from 0 on the number line.

Jomo

Elite Member
Yes, length is different from counting. Say I give you 5 different lengths. If I ask you how many lengths I gave you, you can count them 1-2-3-4-5 and the answer would be 5. Now how would you count the actual lengths??? You could add them together but count them, I don't know about that.

aprovehh

New member
So is this just an interpretation thing and the flexibility of using numbers in a couple ways? Like on the number line seeing 9 as the middle of 18 is true whether we are thinking about length or a count of items

HallsofIvy

Elite Member
There exist segments with irrational length (the most famous is the hypotenuse of a right triangle with both legs of length 1) so measuring a length can't be the same as counting!