Continuity/Discrete, Point vs Unit, Infinity question

aprovehh

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Feb 15, 2019
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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

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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

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

Anybody else have any other thoughts.
 

Harry_the_cat

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aprovehh

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

HallsofIvy

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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

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

Jomo

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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

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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

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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!
 
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