Reading a Ruler

Date: 09/18/97 at 10:48:42
From: Kyle Reynolds
Subject: Ruler

I need to read a ruler, and I can't. Can you help me?

Date: 09/24/97 at 13:35:23
From: Doctor Rob
Subject: Re: Ruler

I hope so.

There are two main kinds of rulers in general use, and other, more 
obscure kinds.  We will ignore the obscure ones.

First there is the ruler marked in inches, and each inch is subdivided 
into 16 parts.  The lines on it look something like this sketch of a 
part of a one-foot ruler:

 8                               9
 |                               |
 |               |               |
 |       |       |       |       |
 |   |   |   |   |   |   |   |   |
 | | | | | | | | | | | | | | | | |  
-----------------------------------
   D C D B D C D A D C D B D C D

The lines have different lengths to help you figure out what lengths 
they represent. 

   The shortest lines (D) represent an odd number of sixteenths 
   of an inch. 

   The next shortest lines (C) represent an odd number of eighths 
   of an inch. 

   The next shortest lines (B) represent an odd number of quarters
   of an inch.  

   The next shortest lines (A) represent an odd number of halves
   of an inch.  

   The longest lines (8 or 9) represent whole inches, and are
   numbered.

   The lines labeled 8 and 9 above mark points on the edge of the 
   ruler that are eight and nine inches from the left-hand end of 
   the ruler.  

All distances are measured from that same left-hand end of the ruler, 
which could be, but probably isn't, marked "0".

The line halfway between them labeled A above marks a point on the 
edge of the ruler, which is 8 1/2 inches from the end. That makes 
sense because 8 1/2 is halfway between 8 and 9.

The next shorter lines labeled B above are halfway between 8 and 
8 1/2, and halfway between 8 1/2 and 9. The former one marks 8 1/4 
inches, and the latter marks 8 3/4 inches. These make sense because 
8 1/4 is halfway between 8 and 8 1/2, and 8 3/4 is halfway between 
8 1/2 and 9. In the words of arithmetic,

8 + ([8+1/2]-8)*(1/2) = [8+1/4],
     ^^^^^^^^^
     distance from 8 to [8+1/2]

and [8+1/2] + (9-[8+1/2])*(1/2) = [8+3/4].
              ^^^^^^^^^^^
              distance from [8+1/2] to 9

Likewise, the next shorter lines labeled C above are halfway between 
8 and [8+1/4], between [8+1/4] and [8+1/2], between [8+1/2] and 
[8+3/4], and between [8+3/4] and 9. They must therefore mark the 
distances [8+1/8], [8+3/8], [8+5/8], and [8+7/8], respectively.

Finally, the shortest lines labeled D above are halfway between 
adjacent pairs of longer lines, and mark [8+1/16], [8+3/16], ..., 
[8+15/16].

When I measure a distance, I put the "0" end of the ruler at one end, 
and then pick the mark on the ruler which is closest to the other end 
of the distance. The nearest inch line to the left gives me the number 
of whole inches. I then figure out whether this line is a 1/16 line 
(shortest), a 1/8 line, a 1/4 line, a 1/2 line, or an inch line. That 
tells me what the denominator of the fraction of an inch will be.  
From the inch line I count the lines the same length as my chosen one 
using odd number:  "1, 3, 5, 7, ..." until I find my line. That tells 
me what the numerator of the fraction of an inch will be. I then 
combine the number of whole inches with the fraction to get the 
distance.

The other common kind of ruler measures centimeters instead of inches.
Each centimeter is divided into 10 parts (each called a millimeter).
The lines on it look something like this sketch of a part of such a 
ruler:

 13        14
 |         |
 |    |    |
 |||||||||||
-------------
 CAAAABAAAAC

   The longest lines labeled C represent whole centimeters.  

   The next longest line labeled B represents a half centimeter.  

   The shortest lines labeled A represent tenths of centimeters, 
   or millimeters.  Since 1/2 = 5/10, the B line also represents 
   5 millimeters.

To measure a length, put the left end of the ruler, which could be 
labeled "0" but probably isn't, at one end, and pick the closest mark 
on the ruler to the other end. Find the closest centimeter mark to the 
left of your mark. That will tell you the number of whole centimeters 
(13 in the above example). The denominator of the fraction of a 
centimeter is fixed at 10.

The numerator is found by counting from the whole centimeter mark you 
found above, and the medium-length lines B help you count by showing 
you where 5 tenths or half a centimeter is. If you are close to the 
"13" mark, you count up as you move to the right starting with "0" for 
the "13" mark itself. If you are close to the "B" mark, you can count 
up as you move to the right or down as you move to the left, starting 
with "5" for the B mark itself.  If you are close to the "14" mark, 
you can count down as you move to the left, starting with "10" for the 
"14" mark itself.  This will tell you the numerator of the fraction of 
a centimeter.  

When you have the fraction, you may find that it is not in lowest 
terms: 4/10, for example, is not and can be reduced to 2/5, and 
5/10 can be reduced to 1/2. When you have reduced the fraction, put it 
together with the whole number of centimeters (13 in this case), and 
you will have your answer.

I hope this explanation has helped. It sounds more complicated when I
write out all the little steps than it feels in practice. With some
experience, I think you will find that it all works very naturally and
easily.

-Doctor Rob,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/