Why does adding a the beginning and end of a number and dividing them by 2 tell us the middle of those numbers?

Question

This is probably a relatively simple question and one that betrays my lack of knowledge of basic mathematics but it seems to be something I can't quite grasp even though I suspect the answer is quite intuitive.

Suppose we have a number line from 1 to 10. Were I to count the distance from 1 to 10, I would get 9 and it can be expressed as 10-1 = 9. Were I to want to identify in what position on this number line I were to find the middle part of this segment, I would simply need to take 9, or the distance between 1 and 10, and divide by 2 to get the half point of this line segment. Thus the middle of the number line between 1 and 10 occurs at the 4.5th position. I could in theory and in practice count 4 and a half steps starting at 1 and arrive at the 4th and a half step. This step would find itself between 5 and 6, or 5.5.

One of the ways I'm conceiving this is by thinking of the point starting at 1 and ending at 10 as a line that can be measured in say inches. If I took a ruler and measured from 1 to 10, I would get 9 inches in length and it makes sense that the middle of that line would simply be the number of inches measured divided by 2. This could be applied to any set of numbers such as from 11 to 22. I could use a ruler to measure from 11 to 22 and find that I would get 11 inches and then divide that 11 by 2 to get 5.5 inches, or the number of inches needed to move to get to the middle of 11 and 22. So far so good.

What I don't understand is why we are able to identify the middle number--not the position as was true in the previous case--between two numbers simply by adding the beginning number and the ending number and dividing them by 2. It doesn't seem to compute in the same way when I think of it in terms of a number line that can be measured. It doesn't seem intuitive to me. For example, suppose we have a line starting at 1 and going to 10. I know that were I want to identify the middle number between these two numbers, I would need to add 1 and 10 together and divide it by 2 to get 5.5, the number and not the position that is directly between 1 and 10. Yet something feels off about this. I can conceive of it as the following. 1 is 1 inch plus 10 inches that yields 11 inches and were I to divide that by 2, I expect it to also yield the position and not the exact number that is between these two values. Yet this is not the case. If I were to add 10 and 1 and divide that sum by 2, I would get 5.5 the number in the middle of them.

Part of my misunderstanding I suspect comes from something very elementary. For example, were I to draw a line starting at 1 and ending at 10, to count the distance of that line, I would count the interval between 1 and 2 as the first part of that distance, the interval between 2 and 3 as the second part of that distance until I finally reached 10. The number of intervals is not 10 but rather 9. Explaining it in different terms, I can say that I start counting at 2 and not at 1 when measuring the distance between 1 and 10 on this number line. However, this doesn't seem to be happening when counting the position that I derived when dividing 9/2 and getting 4.5. To get the numbers in the 4.5th position, I would need to start counting at 1 and stop when reaching the 4.5th position on this number line. It is possible I am complicating the matter slightly but as I go down this rabbit hole, I seem to be realizing how many things I'm taking for granted when counting on a number line such as knowing that the number of intervals between two numbers is one less than the number of numbers. This is maybe part of the reason that adding the starting number and the ending number and dividing by 2 doesn't seem to make sense to me as a way to identify the middle number between these two numbers. I found a link on Quora that appears to deal with some parts of my question but I am still unable to grasp why my intuition is wrong and adding the two extremes of a number a line, it's lowest and highest values and dividing them by 2 yields the value in between them.

Answer 1

Your first method for finding the exact middle of numbers $a$ and $b$ (say $a <b$) was to find:

1. the distance between them, $b-a$
2. Half of that, so $\frac{b-a}{2}$
3. Add that to the "beginning" number $a$, getting $a + \frac{b-a}{2}$.

Your second method is about the average, which is $\frac{a+b}{2}$, and why this gives the same result as the exact middle. But these two things are indeed the same, since the first is $$ a + \frac{b-a}{2} = \frac{2a}{2} + \frac{b-a}{2} = \frac{2a+b-a}{2} = \frac{a+b}{2}. $$

Answer 2

Get in the habit of always calling the starting point $0$ because the starting point is $0$ distance from itself. So if you start at $1$ and end at $10$ that in terms of distance subtract $1$ from all: you start at $0$ and end at $9$. Middle distance is $4.5$ (from $0$). Now to go back to the original idea where you start at $1$ add $1$ do everything. The midpoint is at $4.5 + 1=5.5$.

What I don't understand is why we are able to identify the middle number--not the position as was true in the previous case--between two numbers simply by adding the beginning number and the ending number and dividing them by 2

Algebra:

The starting position is $S$ and the ending position is $E$.

readjust to make this in terms of $0$.

The starting position is as $S-S$ distance and the ending position is at $E-S$.

So the midpoint has a distince of $\frac {E-S}2$.

Now to shift back the the original position of $S$ so add $S$.

The starting position is $0 + S = S$. And the ending position is $E-S + S = E$....

And the middle position is $\frac {E-S}2 + S$.

The question is why does $\frac {E-S}2 + S = \frac {E+S}2$?

Well:

$\frac {E-S}2 + S =$

$\frac {E-S}2 + S\cdot \frac 22 =$

$\frac {E-S}2 + \frac {2S}2 =$

$\frac {(E-S) + 2S} 2= $

$\frac {E + 2S - S}2 =$

$\frac {E+S}2$.

.....

ANother way of thinking of this is:

At the starting point you are at $S$ which is the nearest point on your road from $0$; it is a distance of $S$ from $0$. And the end point you are at $E$ which is the furthest point on your road from $0$; it is at a distance of $S$ from $0$. And in the very middle you are at the absolute most AVERAGE distance from $0$. The point $M$ is average of $S$ and $E$. Well, to find the average you add them $S+E$ and divide by how many points you are considering. $M = $average of $(S,E)=\frac {S+E}2$.

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A thing to get used to:

If $a < b$ are your points.

Then the distance between them is $b-a$.

Half that distance is $\frac {b-a}2$.

If you start and the begining and go forward half the distance you end up in the middle.

So $M = a + \frac {b-a}2$.

And if you start and the end and go backward half the distance you end up in the middle.

So $M = b - \frac {b-a}2$.

And if you start at $0$ and decide to go the average of the distances $a$ and $b$ you will end in the middle.

So $M = average(a,b) = \frac {a+b}2$.

So we should have $a + \frac {b-a}2 = b-\frac {b-a}2 =\frac {a+b}2$.

Which the do:

$\frac {a+b}2 = \frac a2 + \frac b2$.

And $a +\frac {b-a}2 = a+\frac b2 - \frac a2=(a -\frac a2) + \frac b2 =\frac a2 + \frac b2$.

And $b -\frac {b-a}2= b-(\frac b2 -\frac a2)=b-\frac b2 + \frac a2 =\frac b2 + \frac a2=\frac a2 + \frac b2$.

So all three of those way of thinking, that are all legitimate, all result in the same value $\frac a2 + \frac b2$.