Measurement


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Measurement 

 

    Measurement is assigning a real number to some physical quantity such as length, width, height, volume, mass, brightness, etc.   A physical apparatus suitable for the measurement to be made is used to make the measurement.

    Measurement involves error, so it is appropriate to discuss accuracy and precision at some point.

    Examples:

          A yard stick measures length in yards.

          A meter stick measures length in meters.

          A ruler measures length in feet.

          A thermometer measures temperature in degrees F or C.

    Using these tools we can measure length and width to find area, and if we measure height, we can then find volume.

    With yardsticks and rulers we deal heavily with fractions and conversions between these fractions.  With meter sticks we deal with decimal points, moving them left or right as we convert, no fractions involved.

    Below is part of a ruler, exaggerated in size for clarity:

                

      The top scale is in inches (in), that means the major marks, labeled with numbers,  represent 1 inch in length.  We use the double quote " to mean inches, so each major mark represents 1".

       Counting all the marks between 0" and 1' we find 16 altogether that are equally spaced between 0" and 1'. So, each mark represents 1/16".  Continuing, taking  2 of these marks at a time we see there are 8 such pairs between 0" and 1".  As you can see this mark represents 2 of the 1/16" marks and there are 8 of them, so each of these  8 marks represents 1/8".   Likewise, the 3rd larger mark groups 2 of these 1/8" marks and 4 of the 1/16" marks and there are 4 of these groups, so this mark must represent 1/4".  Finally, the next larger mark represents 2 of these 1/4" marks, 4 of the 1/8" marks and 8 of the 1/16" marks, and 2 of these marks represents 1", so one mark would represent 1/2".

     Moving the other way,  each mark in order from larger to smaller represents 1/2 of the previous mark.  Starting with 1", then next largest mark is 1/2 of 1" or 1/2".   Now moving to the next mark, that mark is 1/2 of the 1/2" mark, so it represents 1/4".   Continuing, the next mark is 1/2 of 1/4" so it represents 1/8".  And finally the smallest mark is 1/2 of 1/8" which is 1/16".

     We say that this scale is calibrated to the nearest 1/16".  This means that it's precision is 1/16". But, since we cannot measure to 1/32" or smaller, the best we can say is a measurement made with this ruler is accurate to

                                  1/16" 1/32". 

     The error is expected to be 1/2 above or below the smallest unit of measurement with any instrument.  One cannot speak of an exact measurement, only of an approximation with an error bound.

 

     Now, let's investigate the metric side of that ruler.

                              

     Each major mark represents 1cm.   There are 10 smallest marks in equally spaced each of these marks. So each of these marks would represent 1/10 of 1cm.  1/10 of 1cm is 1mm.  Notice there are marks that group 5 of these smallest marks, these next larger marks would represent 1/2 of 1cm which is 0.5cm.

     In the English system we use powers of 2 to measure lengths; In the metric system we use powers of 10.  The metric system is easier since powers of ten require only movement of the decimal left or right.  The English system requires a multiplication or division by some power of 2 then a conversion to a decimal.

    Example, say we measured a length of 1 and 5/16" and wanted to express this in feet.  We know there are 12" in a foot. So we need to divide 1 5/16" by 12".  Let's see, 1 5/16" is 21/16"; now divide this by 12 and we get (21/(16*12)) = 21 / 192 ft = (I need my calculator here...)  = 0.0625 ft.

    Hmm, looking at the ruler above, 1 5/16" looks to be approximately 32mm.  1mm is 1/1000 of a meter, that is 1mm = 0.001m.  If we have 32 of these mm then that must be 0.032m.  And we're done.  Let's look at this a bit closer.

   1m = 10dm     (dm = decimeter, dec --> 10)

   1m = 100cm    (cm = centimeter, cent --> 100)

   1m = 1000mm  (mm = millimeter, mil --> 1000)

   We rarely speak of decimeters.  So let's focus on cm and mm.

      So  we measured 32 mm.   We want this in meters.

                     32.0mm      a mm is smaller than 1m by 3 decimal places

     look, 1m = 1000 mm, divide by 1000, and we get 1/1000 m = 1 mm.

     Dividing any number by 1000 involves shifting the decimal to the left by 3,  or multiplying by 3 factors of 10  (1000 = 10 * 10 * 10,   3 tens, 3 decimals)

      So.           3.20   (1 shift)  0.320  (2 shifts)  0.0320 (3 shifts)

                     0.0320 m

     Let's try another.  What is this measurement in cm?

                       32.0 mm = 3.20 cm 

       Ok,  since 1m = 100cm and 1m = 1000mm a meter is a meter so

              100cm = 1000mm,  dividing by 100 we get 1cm = 10mm (which we saw in the ruler above)   So, 1mm = 1/10 cm, and we have 32 mm, dividing by 10 moves the decimal to the left by 1.    32.0 mm = 3.20 cm.

 

     One last example.   Say we have 1.32 m.   How may cm is this?

                   1.32 m = 132 cm

     1m = 100cm.   we multiply by 100 that is, shift the decimal to the right by 2.  

              1.32m = 13.2 dm  (1 shift) = 132 cm (2 shifts)

Converting between the English system and the metric system requires a calculator.   For length we have

               2.54 cm = 1"           (this should be committed to memory)

all other length  conversions can then be done.

     For example:

             12in  * 2.54 cm/in = 30.48cm     (0.3048 m, 3048 mm)

             1 yard = 3 ft, so 1 yard = 3 * 30.48cm = 91.44cm  (0.9144m)

             (so a yard stick is slightly smaller than a meter stick. )

             A 100 yard football field would be 91.44m.

             1 mile is 5280ft/mi * (0.3048 m / ft) = 1609.3m

             We can  1km = 1000m, so we have 1609.3/1000 = 1.6093km

             1 mile = 1.6093 km.

             If you look at the speedometer  60 MPH is just under 100 KPH.

 

    Other metric scales follow this moving decimal pattern where the units are all powers of ten and all use the same prefixes.

 

 

    Ok, one more example.

              1) How many football fields one after the next are required to measure out one mile?

     A football field is 330 yards long.   A mile is 1760 yards. By dividing 1760 by 330  we get

         5.86666.... fields,  i.e., almost 6 fields

 

             2) How many football fields make up one quarter of a mile?

          About 6 / 4 = 1  fields

 

               

 

  


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