Reading a Ruler II

Date: 11/08/2000 at 19:30:05
From: John Lambert
Subject: How do I read a ruler?

What do the other little lines on the ruler stand for?

Date: 11/10/2000 at 17:09:33
From: Doctor Ian
Subject: Re: How do I read a ruler?

Hi John,

It depends on whether you're using a metric ruler or the fractional 
feet-and-inches kind.

METRIC RULER
A metric ruler divides a meter into 100 centimeters,

     0   1   2          99  100
     |___|___|___ ... ___|___|

and divides each centimeter into millimeters,

     0         1         2
     |         |         |
     |||||||||||||||||||||
               ^        ^
             10 mm    19 mm
              1 cm   1.9 cm
            0.1 m   0.19 m

which means that the distance between any of the smallest lines is 
1/1000 of a meter.

FRACTIONAL RULER
The largest lines on a fractional ruler divide a foot (or a yard) 
into inches:

     0   1   2          11  12
     |   |   |           |   |
     |___|___|___ ... ___|___|

Smaller lines divide each inch into two equal pieces, each 1/2 inch 
long:

     0         1         2
     |         |         |
     |    |    |    |    |
     |____|____|____|____|

          ^____^
          1/2 in

Still smaller lines divide each half-inch into two equal pieces, each 
1/4 inch long:

     0           1           2
     |           |           |
     |     |     |     |     |
     |__|__|__|__|__|__|__|__|

           ^__^
          1/4 in

And so on,

     0               1
     |               |     a = 1/8
     |       |       |     b = 2/8 = 1/4
     |   |   |   |   |     c = 3/8
     |_|_|_|_|_|_|_|_|     d = 4/8 = 1/2
       a b c d e f g h     e = 5/8
                           f = 6/8 = 3/4
                           g = 7/8
                           h = 8/8 = 1

These divisions can get smaller and smaller (although I've never seen 
a ruler that was divided into more than 1/64 of an inch).

So let's say I want to measure something with a ruler. I set it down 
next to the ruler, 

  0               1               2               3
  |               |               |               |
  |       |       |       |       |       |       |
  |   |   |   |   |   |   |   |   |   |   |   |   |  
  |_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|  
  ############################################
  ############################################
  ############################################
  ############################################  
                                  ^_______________^  2 to 3 in
                                    
                                          ^_______^  2-1/2 to 3 in
                  
                                          ^___^      2-1/2 to 2-3/4 in

                                            ^_^      2-5/8 to 2-3/4 in
                        
and I find the two closest inch-marks that are on either side of it. 
In the picture above, these are the 2- and 3-inch marks. So we know 
that the thing is between 2 and 3 inches long.

Next, we look at the 2-1/2 inch mark. The thing extends past it, so we 
know that the thing is between 2-1/2 and 3 inches long.

Next, we look at the mark halfway between those, the 2-3/4 inch mark. 
The thing doesn't extend quite that far, so we know that the thing is 
between 2-1/2 and 2-3/4 inches long.

And so on, until we've identified the two closest marks that are on 
either side of the end of the thing we're measuring. At this point, if 
it's very close to either mark, we can just call that the measurement. 
Or if it's right in between, we can take the average. In the example 
above, that would be:

              5/8 + 3/4
       2 and ----------- inches
                  2            

              5/8 + 6/8
     = 2 and ----------- inches
                  2     
  
                11/8
     = 2 and ----------- inches
                  2         
               
     = 2 and 11/16 inches    

Note that you can measure something to the nearest 16th of an inch, 
even though the ruler is only marked to the nearest 8th of an inch.

This is the main idea. Write back if you'd like to talk about this 
some more, or if you have any other questions. 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/