Working with Measurement Units

When we are measure anything we should always write down the units of measurement with the number. 12 in., 16.34 ft or 12 deg. The number is meaningless unless we know the units. The units are part of the number. There are some special rules for dealing with numbers with units attached that we need to follow.

Adding and Subtracting:

We can only add and subtract numbers if the units are the same.

12 ft. + 6 ft. = 18 ft.

but

12 ft + 6 cm = ???

We could add 12 and 6 to get 18, but 18 what? It is meaningless. We can’t do it unless we convert 12 ft to cm or 6 cm to ft.

Multipling and dividing

When we multiply or divide numbers with units we must do the same thing with the units that we do to the numbers.

12 ft x 6 ft = 12 x 6 ft x ft = 72 ft² multiplying two distances gives us an area

If we divide a unit by itself, the units “cancel” and we can remove them from the number.

12 ft / 6 ft = 12/6 ft/ft = 2 the top length is twice the bottom length. It doesn’t matter if we measured it in ft or inches, the answer would still be 2

12 ft x 6 cm = 72 ft cm This might seem meaningless, but as long as we keep

the units, it is not. With a little more math we can make it into a useful number

Converting units

If we say 2 + 2 = 4 then we know that both sides of the equation mean the same thing, and = 1.

So if we say 1 inch = 2.54 cm then

or or if we do the math 1 / 2.54 = 0.396 and

We can multiply any number by one and not change its meaning, so we can multiply any number with units by one of these unit fractions or conversion factors and not change the measurement it represents.

10 in = The in. in the top and bottom of the units cancel and we are left with 10 in = 25.4 cm

Note that if we approach converting units this way, we never have to remember, “do I multiply or divide by this conversion factor?” If you keep the units in the number and the conversion factor, the units will tell you if you are right. If I accidentally write down

I’ll quickly see that my units are “in x in/cm” and something is wrong.

Also note that you don’t have to remember different conversion factors for inches to cm and cm to inches. One conversion equality works either way

Finally, if you need to convert mm to yards you can write all the steps down at once, for a neater write up and reduced chance of errors.

On the right side mm on top cancels mm on the bottom, cm on top cancels cm on the bottom and so on, and I’m left with yd. 10,000/10/2.54/12/3 = 32.8 yds

Photo or map scales

When we are working with air photos or maps, we have one additional thing to keep track of with our numbers, which is whether they represent a measurement on the map or on the ground. To avoid confusion, write it down! Don’t just write 3.5 in., write 3.5 in photo. This “photo” and “ground” can be treated like another unit that has to appear in the right place to cancel it’s corresponding unit in the top or bottom of the fraction.

We say that the representative fraction is unit-less, but it really means

1 unit on photo= x units on the ground, and it is good to write that down when you are using a scale. So if the scale is 1/6000 it is 1 photo = 6000 gnd and if we measure 1.5 in on the photo and we want to turn that into a measurement.

Then we need to convert 9000 in to feet or yards (or meters)

Of course, some scales do have units. 1” = 1 mi. Then we need to write the units down when we use this scale in a calculation to make sure we have the numbers in the right places.

In this class, if I ask for the scale I usually mean the representative fraction. If you have to figure the scale out you will usually end up with some numbers like 1.52 in on photo = 4250 ft on the ground. We need to end up with a unit-less fraction with 1 on top.

scale =