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Posted 7 May 2009
REPRESENTATIONS AND MODELS

A representation of a math object can be one of the several types:
¨
Mathematical representation.
These types are fuzzy and overlap. You can argue about whether some examples are of one type or another. The word “representation” is not used for all of them and many mathematicians would not use the word in as general a sense as I use it here. This section discusses representations, except that mental representations are discussed mostly in the chapter Images and metaphors.
Contents
What are representations good for?
Mathematical and informal
representations
An integer can be represented in decimal notation, binary notation, hexadecimal notation and in prime factorization notation:
Decimal 
Binary 
Hexadecimal 
Prime Fac. 
199 
11000111 
C7 
199 
200 
11001000 
C8 

201 
11001001 
C9 

202 
11001010 
CA 

203 
11001011 
CB 

204 
11001100 
CC 

205 
11001101 
CD 

206 
11001110 
CE 

207 
11001111 
CF 

208 
11010000 
D0 

2,747,072,786 
10100011101111010000000100010010 
A3BD0112 

The decimal representation of an integer may be more familiar to you than one of the other representations given above, but it is not the only genuine or legal one it is only more familiar and (perhaps) more useful.
When you see the expression “ ” you know what the integer is.
You may crave to know the decimal representation
because the integer does not somehow seem real to you until you know it,
but that is a human feeling not based on a mathematical property of integers.
The decimal representation “207” is a mathematical representation, but when you think about the number 207 you may in fact visualize the sequence “207” of digits, so the decimal representation, at least up to (perhaps) seven digits, can also correspond to a mental representation in the form of notation. More about this here.
¨ A function may be given by a formula.
¨ A continuous function on the real numbers has a graph. The graph of a function is a mathematical object, but the drawing you may make of the graph (like the picture to the left) is a physical representation.
¨ A finite function may be given by a table of
values.
¨ A linear transformation on a finite dimensional vector space can be represented by a matrix.
The chapter on images and metaphors for functions describes many ways to think about functions, including the first three above.
The picture to the right represents the rectangle with sides 2, 3, 2, 3 . You might draw a picture of it on a chalkboard that would look like this picture. These are physical representations of the rectangle. When you think about it you may visualize a very similar picture.
You may represent this rectangle in the real plane by giving coordinates of its corners, for example (0, 0), (0, 2), (3, 0), (3, 2). Of course, the corners (0, 1), (0, 3), (3, 1), (3, 3) gives another representation in the plane of the same rectangle.
The family of rectangles of different sizes may given by parameters, two real numbers representing the lengths of two adjacent sides. (If you give the length of two adjacent sides, the other two sides are determined by the fact that it is a rectangle.) The rectangle with sides 2, 3, 2, 3 then has parameters (2, 3). This is a representation of the rectangle that, for example, allows you to calculate the area easily.
Rectangles and their parametrization and representation will eventually be discussed in the chapters on parameters and on isomorphism and identity.
A representation of a math object helps in many ways.
The decimal notation ‘2,747,072,786’ and the prime factor representation identify the same positive integer. Both identify it completely; there is no doubt about which integer it is. Representations that identify the object are commonly used as symbolic names of the object. See structural notation.
Other representations do not completely identify the object.
¨ A sketch of the graph of a function defined on the reals does not determine the function completely because it can’t be perfectly accurate and it can’t show all the values.
¨ A picture of a rectangle is also not prefectly accurate, so does not completely identify it.
¨ The decimal notation ‘2,747,072,786’ gives you a good idea of the size of that integer. You can tell at a glance that it is between two and three billion. It is more difficult to use that representation to determine the prime factors.
¨ The prime factor representation of the same number makes it immediately obvious what its prime factors are but does not make it easy to tell how big it is.
¨ If you know about how integers are represented in computers, you can tell at a glance from the hexadecimal notation A3BD0112 that it is too big to be represented as a “long” integer (on 32 bit machines). That is because it uses eight hex digits and the leftmost digit is bigger than 7.
¨ You can calculate easily because the numbers are represented in
decimal notation, for which there is an easy algorithm for addition that you
learned in elementary school. Using the
prime factor representation
or the Roman Numeral Representation
MCCCVIII
+ CCCLXXV = MDCLXXXIII
it is much harder to add
them up because there is no efficient algorithm for computing sums using those
representations.
¨ The prime factor representation makes it easy to calculate the prime factorization representation of the product of the two numbers: .
Some representations have a mathematical definition and others have a more informal status.
¨ The representation of a linear transformation on a finite dimensional vector space as a matrix has a strict mathematical
definition.
¨ The representation of a number in decimal notation can be defined as a mathematical object, but in practice it is treated more informally. Is the string of symbols ‘42’ a mathematical object or a typographical object? You can think of it either way and most math texts discussing such a number won’t be precise about its status. Sometimes, especially in computing science or logic, it is necessary to consider it as a mathematical object.
¨ The representation of a function by its graph (as here) is
clearly informal, but the phrase “graph of a function” has a technical mathematical definition (a certain set of ordered pairs) as well.
¨ The representation of a symmetry of a square described here is a mathematical representation of the symmetry.
In one of its uses in mathematical discourse,
a model, or mathematical model,
of a phenomenon is a mathematical object that represents the phenomenon in some
sense. The phenomenon being modeled may be physical or another mathematical
object.
¨ A moving physical object has a location at each instant. This may be modeled by a function. (Example). You can then determine the velocity of the object at different times by taking the derivative of the function.
¨ A word problem in algebra or calculus texts is an invitation to find a mathematical model of the problem. You must set it up as a mathematical expression using appropriate operations and then solve for the appropriate variable.
¨ Mathematical logic has a concept of model of a theory. Both the theory and the models are math objects.
¨ Computing science defines various mathematical models of the concept of algorithm, for example Turing machines. See also my discussion of the word “algorithm”.
As the examples just discussed illustrate, a
model and
the thing it models may be called by the same name.
Thus one refers to the velocity of an object (a physical property) and one also
says
“The
derivative of the velocity is the acceleration.”
Of course, the derivative is a function and
the acceleration is a physical property, but that is the way we talk. A mathematical model is a special kind of
metaphor
and to
refer to the mathematical model as if it were the thing modeled is a normal way of using metaphors. Indeed, in rhetoric (but not here) the word
“metaphor” is typically restricted to referring to this way of using them.
A mathematical object may also have a physical modell, which is a particular kind of external
representation. Tetrahedrons and Möbius strips are
mathematical objects that you can build models of out of paper. The drawing above of the graph of
a function is also a physical model.