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Chapter 12 - Chapter 12 Symbolic Mathematics When...

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Chapter 12 Symbolic Mathematics W hen mathematicians work, they use symbols, such as x and y , to represent numbers and the functional notation f(x) to represent a function of x . We have used the same symbols in Matlab, and yet the meanings are subtly different. The Matlab symbols x and y refer to floating point numbers or integers of the types we considered in Chapter 8, whose ranges and accuracies are limited and to functions that operate on these numbers. A common feature of all the numbers that can be represented as floating point numbers or integers is that they are rational numbers, i.e. , numbers that are equal to ratios of integers. The x and y of mathematics, on the other hand, refer to real (or complex) numbers whose values are often defined only by being solutions to equations. These solutions are sometimes not rational, as for example the solution to the equation, 2 2 x = , which we can represent by 2 x = or 1 2 2 x = . In Matlab we can easily set x = 2^(1/2) or x = -2^(1/2) , but the meaning is not the same. The result of Matlab’s calculation of 2^(1/2) is approximately equal to 1 2 2 , but it is certainly not exact because 2 is an irrational number and cannot be represented to perfect accuracy with any floating point number. The statement 1 2 2 x = , on the other hand, means “ x is equal to the number that solves the equation, 2 2 x = ”. The equation is the central feature here, not the number. Furthermore, suppose a mathematician is interested in the function ( 29 2 f x x = and wants to know the rate at which ( 29 f x changes with x . The rate of change of ( 29 f x with respect to x is the derivative of ( 29 f x , expressed symbolically in standard calculus notation as ( 29 df x dx . We learn from calculus that the derivative is another function, in this case, ( 29 2 df x x dx = . In this case there is no number involved at all. We have, by taking a derivative, manipulated one function to produce another function. There are many such relationships between functions of importance in mathematics. For example, if we treat 2 x as a function of x , then its integral gives a family of functions: 2 2 xdx x c = + , 1
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each with a different value for c . As another example of the manipulation of functions, if ( 29 2 2 2 g x x x = - + , then it can be shown that ( 29 ( 29 ( 29 g x p x q x = , where ( 29 1 p x x = + and ( 29 2 q x x = - . In each of each of these cases, manipulations of equations and functions are the central problems; specific numbers play only a supporting role. Such manipulations cannot be done by the Matlab with which we have become familiar in the previous chapters. That Matlab deals with rational numbers and with functions and programs that manipulate rational numbers to produce other rational numbers. What we have seen is the numerical side of Matlab. Numerical mathematics is the greatest strength of Matlab, and it is the most well known part of Matlab. Indeed it is the only side of Matlab that most people know. However, Matlab has another side—symbolic mathematics. Symbolic mathematics is the manipulation of symbols according to mathematical rules. It is the sort of thing that we learn first in algebra and then in
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