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REPRESENTATIONS AND MODELS

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

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 the different kinds of repre­senta­tions, except that mental repre­senta­tions are discussed mostly in the chapter Images and metaphors.

Integers

An integer has many mathematical representations: positional notation (also called "base notation"), including decimal notation, binary notation, hexadecimal notation, and also in prime factorization notation:

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 and (perhaps) more useful. 

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.

Functions

Functions have mathematical representations by formulas, 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, but the picture like the one on the left is a physical repre­senta­tion.

Representations of functions are discussed in great detail in Functions: 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.

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

Mathematical and informal repre­senta­tions

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

Examples

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

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 modelTetrahedrons 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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