In a previous post, I argued that numbers and units are born together.
As we deal with quantities of increasing sophistication, we often have a choice between using more sophisticated units or more sophisticated numbers.
If we wish to deal with uncustomarily large lengths, we either start to use suitably large length units, or we learn to deal with uncustomarily large numbers. If we start to deal with unfamiliarly small weights, we can employ suitably small units, or we can start incorporating some kind fractions into our number system. If we start to discuss bidirectional quantities, we can use an opposed pair of units---e.g. debit and credit---or we can learn to use negative numbers. If we need to use quantities in the plane, we can use directional units like north, south, east, and west, or we can learn to use right and left (aka $\pm\sqrt{-1}$), and the other numbers of the plane (aka complex numbers).
Doesn't make sense yet? In subsequent posts, I hope to spill all the details.
For now, remember that increasingly sophisticated kinds of numbers can be understood by playing with increasingly sophisticated kinds of units. Also, notice that technical usage of the quantities convention supports real numbers, or at least ragged reals. It does not really support complex numbers. We shall require a broader view of quantities than is provided by customary usage of the quantities convention.
Wednesday, January 20, 2010
What is the proper place of units?
It is common to associate an algebraic name such as $x$ with, say, a length. There are two common conventions for how to do this:
In the numbers convention, common among mathematicians and often used informally by others, especially during calculations, one says e.g. :
amath \qquad\qquad\qquad x= endamath the length of the rod in feet.
and for given rod a mathematician might write e.g.
amath \qquad\qquad\qquad x=6 endamath.
The numbers convention keeps units on the sidelines. Units are treated as a way of turning quantities into numbers. Algebraic relations are thought of as relations between numbers.
In the quantities convention, common among physicists, engineers, and other technical folk, especially in formal settings, one says e.g. :
amath \qquad\qquad\qquad x= endamath the length of the rod.
For the same rod as before, one writes, e.g. :
amath\qquad\qquad\qquad x=6 endamath ft.
With the physical convention, the same quantity can be written in different units,
amath\qquad\qquad\qquad x = 6 \text( ft = 2 \text( yd =1 \text( fathom =182.88 \text( cm. endamath
The quantities convention lets units march on the playing field in company with numbers. Algebraic relations are thought of as relations between quantities.
The numbers convention is something of a straitjacket. For historical reasons, it persists in college textbooks for algebra, calculus, linear algebra, and multivariable calculus.
As it is usually practised, however, the quantities convention also suffers from unfortunate constraints. How this is so, and what to do about it, are matters for a later post.
In the numbers convention, common among mathematicians and often used informally by others, especially during calculations, one says e.g. :
amath \qquad\qquad\qquad x= endamath the length of the rod in feet.
and for given rod a mathematician might write e.g.
amath \qquad\qquad\qquad x=6 endamath.
The numbers convention keeps units on the sidelines. Units are treated as a way of turning quantities into numbers. Algebraic relations are thought of as relations between numbers.
In the quantities convention, common among physicists, engineers, and other technical folk, especially in formal settings, one says e.g. :
amath \qquad\qquad\qquad x= endamath the length of the rod.
For the same rod as before, one writes, e.g. :
amath\qquad\qquad\qquad x=6 endamath ft.
With the physical convention, the same quantity can be written in different units,
amath\qquad\qquad\qquad x = 6 \text( ft = 2 \text( yd =1 \text( fathom =182.88 \text( cm. endamath
The quantities convention lets units march on the playing field in company with numbers. Algebraic relations are thought of as relations between quantities.
The numbers convention is something of a straitjacket. For historical reasons, it persists in college textbooks for algebra, calculus, linear algebra, and multivariable calculus.
As it is usually practised, however, the quantities convention also suffers from unfortunate constraints. How this is so, and what to do about it, are matters for a later post.
To count is to count units
Numbers and units are born together. To count something, you have to know:
You also have to be able to distinguish the things you are counting from each other, or how will you be able to avoid counting the same thing twice, or know when you have finished counting?
In short, things to count have to be alike but also distinct: oranges, marbles, bus rides, dollars, gallons, votes, beats, hours. In other words, it must have exactly the properties required of a unit.
And this is why, whenever we count, we are are always counting some kind of unit, whether explicit or implicit, whether concrete or abstract.
- what you are counting, and
- when you have counted it.
You also have to be able to distinguish the things you are counting from each other, or how will you be able to avoid counting the same thing twice, or know when you have finished counting?
In short, things to count have to be alike but also distinct: oranges, marbles, bus rides, dollars, gallons, votes, beats, hours. In other words, it must have exactly the properties required of a unit.
And this is why, whenever we count, we are are always counting some kind of unit, whether explicit or implicit, whether concrete or abstract.
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