Correct way to calculate distance (taking negatives into account)

[navneet9431]

See this theorem of my textbook along with its proof.

You will see that my book simply calculates the distance LM by subtracting the distance OL from OM without taking into consideration that the distance is always positive.I can prove that this method of calculating the distance is inappropriate.

See this image.

Here we have to calculate the distance between points L and M.So if we subtract the distance OL from OM then we would get a negative value and that negative value can't be the distance(because the distance is always positive).
In my textbook, the author has done the same error. Though, I think that his final result will always be correct because he has squared OM-OL, which makes the distance positive. But, I think it is not correct to use some incorrect method even if we are getting the correct result.(Do you agree with me?)

so, what according to you should be correct method to calculate the distance between L and M(so that the distance is always positive)?

I will be thankful for help!

Note: I am a high school student and English is my second language!

 

[tkhunny]

It should not read "OL-LM". It should read "|OL-LM|". If you know you are squaring, the effect is the same.

"Incorrect" can be in the eye of the beholder.

 

[Dr.Peterson]

The distance ML is |OM-OL|, because O, M, and L are collinear. The absolute value is necessary. But if you replace parentheses with absolute value bars as needed, the proof becomes valid.

Now, it is barely possible that the author is thinking of "signed distances" (which in effect is what coordinates are); but that concept is not really appropriate here. I think the author is just being careless, making assumptions based on the picture that are not justified from the statement of the theorem. You are right to question it.

When a textbook has such errors, you can practice careful reading, and ask others as you have done here; but then you can go back and read for what is valid in the book. Focus on the main ideas in the book; and be glad you know better!

 

[JeffM]

The distance ML is |OM-OL|, because O, M, and L are collinear. The absolute value is necessary. But if you replace parentheses with absolute value bars as needed, the proof becomes valid.

Now, it is barely possible that the author is thinking of "signed distances" (which in effect is what coordinates are); but that concept is not really appropriate here.

As the author has established his co-ordinate system,

\(\displaystyle (OM - OL) \equiv |OM - OL|,\) correct?

And because we can set the origin anywhere, I am not sure I see the logical error although it may be a pedagogical one.

What am I missing?

 

[Dr.Peterson]

As the author has established his co-ordinate system,

\(\displaystyle (OM - OL) \equiv |OM - OL|,\) correct?

And because we can set the origin anywhere, I am not sure I see the logical error although it may be a pedagogical one.

What am I missing?

We can't set the origin anywhere, because the theorem specifies the coordinates of the points (relative to a fixed origin). And because the theorem does not restrict the signs or magnitudes of the coordinates, we can't assume that x2>x1>0 and y2>y1>0, as is supposed in the drawing. As a result, |OM-OL| need not equal (OM-OL).

Actually, my statement was still wrong, assuming that OM and OL are intended to be (positive) lengths; if they are in the order M - O - L, then LM = OM + OL!

I observe that this author is not precise in use of notation (perhaps for ease of typesetting); some authors carefully distinguish between LM as a line, a segment, and a length, by putting different symbols above it; whereas here AL in one place means a line, and in another the length of a segment. But when it says "OL = x1", it is equating a length to a number that, as far as we know, could be negative. So either he is just wrong, or is making an unstated assumption about the coordinates, or is using signed magnitudes without saying so.

Navneet, does this book ever talk about signed lengths? And is there any notation used to distinguish lines from lengths? Some books I have recently seen would use |LM| to mean the length of segment LM, and I very much like that notation.

 

[JeffM]

We can't set the origin anywhere, because the theorem specifies the coordinates of the points (relative to a fixed origin). And because the theorem does not restrict the signs or magnitudes of the coordinates, we can't assume that x2>x1>0 and y2>y1>0, as is supposed in the drawing. As a result, |OM-OL| need not equal (OM-OL).

Actually, my statement was still wrong, assuming that OM and OL are intended to be (positive) lengths; if they are in the order M - O - L, then LM = OM + OL!

I observe that this author is not precise in use of notation (perhaps for ease of typesetting); some authors carefully distinguish between LM as a line, a segment, and a length, by putting different symbols above it; whereas here AL in one place means a line, and in another the length of a segment. But when it says "OL = x1", it is equating a length to a number that, as far as we know, could be negative. So either he is just wrong, or is making an unstated assumption about the coordinates, or is using signed magnitudes without saying so.

Navneet, does this book ever talk about signed lengths? And is there any notation used to distinguish lines from lengths? Some books I have recently seen would use |LM| to mean the length of segment LM, and I very much like that notation.

Ahh right. His theorem is about arbitrary \(\displaystyle (x_1,\ y_1) \text { and } (x_2,\ y_2)\), which already imply an origin.

 

[navneet9431]

Navneet, does this book ever talk about signed lengths? And is there any notation used to distinguish lines from lengths? Some books I have recently seen would use |LM| to mean the length of segment LM, and I very much like that notation.

No, my textbook does not use any notation to distinguish lines from lengths.

I really did not get the meaning of signed lengths! Can you please give an example of signed length?(As far as I know, lengths do not have any sign)

Thanks!

 

[Dr.Peterson]

I really did not get the meaning of signed lengths! Can you please give an example of signed length?(As far as I know, lengths do not have any sign)

It happens that you asked about signed lengths last March on Ask Dr. Math! Here is my answer from March 4:

Lengths are ordinarily considered positive only; in the case of the
sides of a triangle, that is definitely true. But ...

We do sometimes talk about DIRECTED distances, or "signed lengths",
which are considered positive or negative according to which direction
they go relative to some standard. These are used in some theorems
that explicitly call for directed distances. Here is an example:

  https://en.wikipedia.org/wiki/Ceva%27s_theorem

Directed distances are also used in talking about trigonometric
functions on a coordinate system, where one might draw a triangle as

        -3
     X------O---------Q
     |     /
   -4|   /
     |  / 5
     |/
     P

(thinking of the angle at the origin O in standard position, starting
along the positive x-axis OQ and rotating to OP). Here we label the
sides OX and XP with the coordinates of point P, (-3, -4), thinking of
them as directed distances in the x and y directions. But this is
mostly just a shorthand for the coordinates themselves, relating them
to the right-triangle definitions of the trig functions.

Apart from special cases like these, and in the absence of specific
indications, you can assume "length" always means a positive number.

Does that sound familiar?

To find more about this, in addition to the page I referred to (which uses the term "signed length"), search for the phrase "directed distance".

 

[navneet9431]

Actually, my statement was still wrong, assuming that OM and OL are intended to be (positive) lengths; if they are in the order M - O - L, then LM = OM + OL!

Where did you assume that OM and OL are intended to be positive lengths?
Please mark here that sentence in which you assumed this very thing!
Thanks!

 

[mmm4444bot]

I really did not get the meaning of signed lengths! Can you please give an example of signed length?(As far as I know, lengths do not have any sign)

Not sure what happened to my post from an hour ago … it was here for awhile.

Basically, I gave a couple examples of signed lengths. I use them in trigonometry, when labeling sides of a representative triangle, to help me keep track in which quadrant the terminal ray of some (triangle's) angle lies, after introducing an xy-coordinate system.

I also mentioned vectors (a special type of number you might study later on), where each value has a direction associated with it.

But, my other example comes from real-life, from an old job. The ship had a robotic arm with a winch, to raise and lower a cage from below and above the water's surface. The software was coded to recognize positive distances as above the water and negative distances as submerged in water. So, if the primary control tower ordered "go negative 15", that meant they wanted the cage 15 feet below the surface; we entered the distance as -15; likewise, "go positive 8" meant 8 feet above the surface, and we entered the length as +8.

Often, signed lengths are used as a convenience, to consolidate notations or combine different steps into a single process. They are very important, in working with vectors, like the directed distances (arrows) shown on a wind map below. :cool:

 

[Dr.Peterson]

Where did you assume that OM and OL are intended to be positive lengths?
Please mark here that sentence in which you assumed this very thing!
Thanks!

My statement that I said was wrong was, "The distance ML is |OM-OL|, because O, M, and L are collinear." Here is what I said about it:

Actually, my statement was still wrong, assuming that OM and OL are intended to be (positive) lengths; if they are in the order M - O - L, then LM = OM + OL!

What I am saying there is that, if they are taking lengths as always positive, as we understand to be the case, then my statement that ML = |OM-OL|, is not always true.

In particular, if, say, M is a negative number on the number line (say, -5), O is the origin, and L is a positive number (say, +3), then OM = 5, OL = 3, and LM = 8. Thus LM = OM + OL. But |OM - OL| = |5 - 3| = 2, not 8.

The fact is that the distance ML is |OM-OL| when O, M, and L are collinear, AND O is between L and M.

In saying this, I am assuming that lengths are taken to be positive, as usual.

If, to the contrary, they took distances as signed (which they can't really do consistently), then in my example we could say that OM = -5, OL = 3, and LM = -8; then |OM - OL| = |-5 - 3| = |-8| = 8 = |LM|. So you might say that the claim was true.

 

[navneet9431]

My statement that I said was wrong was, "The distance ML is |OM-OL|, because O, M, and L are collinear." Here is what I said about it:



What I am saying there is that, if they are taking lengths as always positive, as we understand to be the case, then my statement that ML = |OM-OL|, is not always true.

In particular, if, say, M is a negative number on the number line (say, -5), O is the origin, and L is a positive number (say, +3), then OM = 5, OL = 3, and LM = 8. Thus LM = OM + OL. But |OM - OL| = |5 - 3| = 2, not 8.

The fact is that the distance ML is |OM-OL| when O, M, and L are collinear, AND O is between L and M.

In saying this, I am assuming that lengths are taken to be positive, as usual.

If, to the contrary, they took distances as signed (which they can't really do consistently), then in my example we could say that OM = -5, OL = 3, and LM = -8; then |OM - OL| = |-5 - 3| = |-8| = 8 = |LM|. So you might say that the claim was true.

Thank you so much!
Everything is clear now.☺

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