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abstractmath.org help with abstract math Produced by Charles Wells. Home. Website Contents Website Index Posted 19 June 2017 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.