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REPRESENTATIONS AND MODELS

A repre­senta­tion of a math object can be one of the several types:

• Mathematical repre­senta­tion
• Mental repre­senta­tion
• Physical repre­senta­tion
• Model

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.

Contents of this chapter

This chapter gives some examples of different kinds of repre­senta­tions of math objects. Three other chapters discuss some types of representations in more detail:

• Mental repre­senta­tions are discussed mostly in the chapters Images and metaphors and Images and metaphors for functions.
• The chapter on Representations of functions discusses many kinds of (mostly mathematical) representations of functions.

Integers

An integer has many mathematical representations: decimal notation, binary notation, hexadecimal notation, and also in prime factorization notation. The table below provides a few examples of these representations.

Representations of integers are also discussed in the section on Natural Numbers.

Decimal	Binary	Hex	Prime Fac.
199	11000111	C7	199
200	11001000	C8	${{2}^{3}}\cdot {{5}^{2}}$
201	11001001	C9	$3\cdot 37$
202	11001010	CA	$2\cdot 101$
203	11001011	CB	$7\cdot 29$
204	11001100	CC	$2\cdot 3\cdot 17$
205	11001101	CD	$5\cdot 41$
206	11001110	CE	$2\cdot 103$
207	11001111	CF	${{3}^{2}}\cdot 23$
208	11010000	D0	${{2}^{4}}\cdot 13$
$2$,747,07$2$,786	10100011101111010000000100010010	A3BD0112	$2\cdot {{53}^{2}}\cdot {{71}^{2}}\cdot 97$

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.

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 can also correspond to a mental repre­senta­tion in the form of notation.  More about this in the article Images and Metaphors.

Functions

Functions have mathematical representations by formulas, graphs, algorithms (such as computer programs), tables (for finite functions), matrices (for linear maps) and many other ways.

• The graph of a function is a mathematical object which is a mathematical repre­senta­tion of a function.
• The picture on the left is a physical repre­senta­tion (of part of) the graph of the arch function defined by \[h(t):=25-{{(t-5)}^{2}}\] This function is also discussed in the articles Images and metaphors) and Images and metaphors for functions.
• If you visualize this graph you have a mental representation of the function.
• You can also visualize the process of moving along the graph from left to right, going up the left side, then at $x=5$ turning to go back downward. This gives a kinetic understanding of the function, discussed in detail in Images and metaphors.

Rectangles

The picture below 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.  This is a physical repre­senta­tion of the rectangle.
• When you think about it you may visualize a very similar picture. That visualization is a mental repre­senta­tion.
• You may give a mathematical representation of this rectangle in the real plane by giving coordinates of its corners, for example \[\{(0, 0), (0, 2), (3, 0), (3, 2)\}\]
• 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 $2$, $3$, $2$, $3$ mentioned above then has parameters $(2, 3)$.  This is also a mathe­mati­cal repre­senta­tion of the rectangle that, for example, allows you to calculate the area easily.
• The set \[\{(x,y)\,|\,x\gt0\text{ and }y\gt0\}\] is a parameter space for rectangles.

What are repre­senta­tions good for?

A repre­senta­tion of a math object may help in one way, and another one may help in a different way.

The repre­senta­tion may identify the object.

The decimal notation ‘$2$,747,07$2$,786’  and the prime factor repre­senta­tion $2\cdot {{53}^{2}}\cdot {{71}^{2}}\cdot 97$ 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.

Some 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 perfectly accurate, so does not completely identify it.

The repre­senta­tion may enable you to deduce some properties of the object.

• The decimal notation "$2$,747,07$2$,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 $2\cdot {{53}^{2}}\cdot {{71}^{2}}\cdot 97$ 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 $7$.

The repre­senta­tion may enable you to calculate something about a math object. 

• You can calculate \[1308+375=1683\] 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\[{{2}^{2}}\cdot 3\cdot 109+3\cdot {{5}^{3}}={{3}^{2}}\cdot 11\cdot 17\] 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: \[({{2}^{2}}\cdot 3\cdot 109)\cdot (3\cdot {{5}^{3}})={{2}^{2}}\cdot {{3}^{2}}\cdot {{5}^{3}}\cdot 109.\]

Mathematical and informal repre­senta­tions

Some repre­senta­tions have a mathematical definition and others have a more informal status. 

Examples

• 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 is clearly informal, but the phrase “graph of a function” has a technical mathematical definition (a certain set of ordered pairs) as well.
• The repre­senta­tions of a symmetry of a square described in Images and Metaphors for Functions is a mathematical repre­senta­tion of the symmetry.

Models

All models are wrong... some models are useful. --George Box

Model as mathematical object

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.

Examples

• A moving physical object has a location at each instant.  This may be modeled by a function.  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”.

Remark

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. My use of the word may be called conceptual metaphor in contrast to metaphors in rhetoric.

Physical models

A mathematical object may also have a physical model.  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.

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