Produced by Charles Wells Revised 2015-09-12 Introduction to this website website TOC website index blog Back to top of Understanding Math chapter
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.
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 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.
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.
A representation of a math object may help in one way, and another one may help in a different way.
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.
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.
Some representations have a mathematical definition and others have a more informal status.
All models are wrong... some models are useful. --George Box
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.
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.
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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