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EXTERNAL REPRESENTATIONS
Table of contents
Representations ............................................................................................ Error! Bookmark not defined.
Example: Representations of integers ......................................................................................................... 1
What are representations good for? ............................................................................................................. 2
Models ........................................................................................................... Error! Bookmark not defined.
Model as mathematical object ..................................................................... Error! Bookmark not defined.
Physical models ........................................................................................ Error! Bookmark not defined.
Appendix ....................................................................................................................................................... 3
Mathematical and informal representations .................................................................................................. 3
Example: Representations of integers
An integer can be represented in decimal notation, binary notation, hexadecimal notation and in
prime factorization notation:
Decimal
199
200
Binary
11000111
11001000
Hexadecimal
C7
C8
Prime Fac.
199
201
202
203
204
205
206
207
11001001
11001010
11001011
11001100
11001101
11001110
11001111
C9
CA
CB
CC
CD
CE
CF
208
11010000
D0
24 ×13
2,747,072,786
10100011101111010000000100010010
A3BD0112
2 ×532 ·712 ×97
23 ×52
3 ×37
2 ×101
7 ×29
2 ×3 ×17
5 ×41
2 ×103
3 2 ×23
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 “ 3 2 ×23 ” 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.
Other examples
Functions
 A function may be given by a formula.
 You may display the graph of a continuous function on the reals. Example.
 A 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. Some of those ways deserve to be called representations of the functions.
Rectangles
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.
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 are also discussed in the chapters on parameters and on isomorphism and identity.
What are representations good for?
A representation of a math object helps in many ways.
The representation may identify the object.
The decimal notation ‘2,747,072,786’ and the prime factor representation 2 ×532 ·712 ×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. See structural notation.
Other representations do not completely identify the object. A sketch of the graph of a function does
not determine the function completely, and neither does a picture of a rectangle.
The representation may enable you to deduce some properties of the object.
 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 2 ×532 ·712 ×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
22 ×3 ×109 + 3 ×53 = 32 ×11×17
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: (22 ×3 ×109) ×(3 ×53 ) = 22 ×32 ×53 ×109 .
Appendix
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 (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.