# Quantities, Number Units and Counting

We have a simple and effective way in gist to represent a wide range of physical quantities such as ‘82 kg’, ‘3 meters’ and ‘20 minutes’.  Each quantity has a number and a unit, such as ‘meter’ or ‘second’.  In addition to these simple units, we have unit multiplication and division to represent more complex units, e.g. for speed and acceleration. A standard speed unit is meters per second [m/s] and a standard acceleration unit is meters per second per second [(m/s)/s] or simply  [m/s^2].

Physicists as well as business people like to avoid the inconvenience of working with very large or very small numbers like 1000 meters, or .00000001 meters (a trillionth of a meter).  If you counted to see if the number of zeros was correct, you understand the problem.  So we create units like kilometer and picometer and give them conversion factors.   This works for any kind of unit (time, electric current, mass).  Note that the standard units have a conversion of 1 (which in normal parlance, means there is no conversion necessary). See figure 1 for some examples. Figure 1: Example Quantities

We also have found a need for counting units like dozen or gross. For example, a wine merchant stocks and sells cases of 12 bottles of wine, so counting in dozens is more convenient than counting single bottles of wine.  What is interesting is that we can use the exact same structure for representing ‘4 dozen’ or  ‘7 gross’ as we do for representing things like ‘82 kg’ and ‘20 minutes’.   Take ‘4 dozen’, the number is 4, and the unit is ‘dozen’ and the conversion is 12.

In gist there is also a way to represent percentages, which we have always treated as a ratio. After all, when speaking of a percentage, there is always an explicit or implicit ratio somewhere.  For example:

1. “Shipment A has only 65% as much oil as shipment B” corresponds to the ratio:
(No. of barrels in shipment A) / (No of barrels in shipment B) = .65
2. “There are 20% more grams of chocolate in the new package size” corresponds to the ratio:
(NewQuantity – OldQuantity) / (OldQuantity) = .20

The units for the first example are barrels/barrels which cancel out leaving a pure number. Similarly, the units for the second example are grams/grams which again cancel out. In fact, every ratio unit that corresponds to a percentage will cancel out and leave a pure number. This means that although it may be useful to do so, we don’t need to represent gist:Percentage using a ratio unit.

Another thing that we never realized before is that, being a pure number,  a percentage can be represented in the same way we represent dozen or gross. The only difference is the conversion (12 vs. .01).  We can use this same structure to represent:

• parts per million (ppm), used by toxicologists say to measure amounts of mercury in tuna
• basis points (used by the Fed for describing interest rates)
Investopedia defines a basis point as “a unit that is equal to 1/100 of 1%”

See figure 2 for the representational structures. Figure 2: One structure for number units and ordinary units

Notice how ‘ 4 cm’ is very similar to ‘4 percent’:

• to convert 4 cm to its standard unit, we multiply 4 by the conversion factor of .01 resulting in .04 meters
• to convert 4 percent to its standard unit, we multiply 4 by the conversion factor of .01 resulting in .04 ??.

This means we can use the same computational mechanism to perform units conversion for pure numbers like 4 dozen and 4% as we do for ordinary physical quantities like 4 cm or 82 kg.

One question remains. Whereas we can readily see that the conversion factor for kilometer is based on the standard unit of meter, and the conversion factor for hour is based on the standard unit of second, what are the conversion factors of 12, .01 and .00001 (for dozen, percent and basis point) based on? What does it mean to have a standard unit for these pure numbers with a conversion of 1?

Let’s look to see how gist represents dozen and kilometer to see if that gives us any insight.

1. gist:kilometer is an instance of gist:DistanceUnit &
‘3 meters’ is an instance of gist:Extent &
the base unit is gist:meterAnalogously:
2. gist:dozen is an instance of gist:CountingUnit,
‘4 dozen’ is an instance of gist:Count &
the base unit is gist:each

Curiously, while ‘meter’ actually means something to us, and we know what it means to say ‘3 meters’, it strange to think what ‘3 eaches’ could possibly mean.  I invite you to stare at the following table for a while and see some analogies. Figure 3: Standard Unit for Pure Number Quanties

Then notice that:

1. 4 dozen = 48 eaches
2. 4 dozen = 48 (just a simple number)
3. Therefore, 48 must equal 48 eaches (because both are equal to 4 dozen).

But what is it, such that if you have 48 of them gives you the number 48?  The answer is the number one:  48 x 1 = 1.  So the meaning of gist:each is the number one acting as a unit. This is a mathematical abstraction. The ??’s in figure 2 stand for ‘each’ which is the standard number unit. So when you say ‘3 eaches’ it is just 3 of the number one which is just the pure number 3.  As an aside, we can also say that ‘each’ is the identity element for unit multiplication and division. This is analogous to the number 1 being the identity element for multiplication and division of numbers.

• You can multiply or divide any number by 1 and you get that number back.
• You can multiply or divide any unit by each (which means one) and get that unit back.

Note that while conceptually they mean the same thing, syntactically gist:each is very different from the number one as a number whose datatype is say integer, or float.

Notice that for these pure numbers in convenient sized units, we are usually counting things: how many dozens, how many basis or percentage points, or how many parts per million.  We refer to ‘each’ thing as ‘one’ thing being counted.  So that links gist:each to the number one.  Thus, despite the awkwardness of speaking of ‘3 eaches’ the names ‘Count’, ‘CountingUnit’ and ‘each’ are quite reasonable.

Finally, insofar as all instances of CountingUnits are based on the number one, and all instances of Count represent pure numbers, we can think of every CountingUnits as a degenerate unit, and we can think of gist:Count as a degenerate quantity.  A ‘real’ quantity is not just a number, it has a number and has a non-numeric unit.

So in conclusion:

1. We have extended the notion of gist:Count and gist:CountingUnit to apply to pure numbers that are less than one as well as those that are greater than one.
2. We can represent pure numbers expressed in dozens, percentages, basis points and ppm just like we express the more usual quantities: ‘82 kg’, ‘3 meters’ and ‘20 minutes’.
3. We can use the same computational mechanism to do units conversions on pure numbers as we can for ordinary physical quantities.
4. We can represent gist:Percentage using a new unit called gist:percent with a conversion of .01 instead of using a ratio unit, making a more uniform representation.
5. It will often be helpful to represent a gist:Percentage using a ratio, but it is no longer required.
6. gist:Count could meaningfully and accurately be called gist:PureNumber since every instance of gist:Count (e.g. ‘4 dozen’, ‘65%’) is a pure number (e.g. 48, .65)
7. gist:CountingUnit could meaningfully and accurately be called gist:PureNumberUnit because every instance of gist:CountingUnit is used to express pure numbers.
8. gist:each corresponds to the number one
9. We can think of Counts (pure numbers) and CountingUnits (number units) as degenerate cases of ordinary quantities and units like ’82 kg’ and ‘kg’

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