abstractmath.org 2.0
help with abstract math
Produced by Charles Wells Revised 20170223
Introduction to this website website TOC website index blog
Back to top of Understanding Math chapter
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:
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.

24

11000

18

$2^3\cdot3$

111

1101111

6F

$3\cdot37$

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 \[h(t):=25{{(t5)}^{2}}\] 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.
This work is licensed under a Creative Commons AttributionShareAlike 2.5 License.