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24 August 2001
Functions, One of the Most Important Tools in ScienceGeorge E. Hrabovsky
President, MAST
Sad News
On 27 November 2000 my wife of three years (and constant companion for thirteen) lost her battle to breast cancer. Heather was the most important thing in my life. I will miss her, probably more than I can imagine. I would like to observe a moment of silent contemplation, if you would indulge me. Thank you. MAST, and I, will continue on despite her loss; Heather would have wanted it this way.
A Word About MAST
I have received a lot of email about the URL from my previous column not working correctly. Here is the official MAST URL:
This works (I just checked it to make sure). I have not updated the web page for a long time. I plan to update it by the end of February.
We have finished the first course, "Self-Study," and it is on the web-page. This course consists of four lessons that guide you to planning and executing a "Learning Project." The next course we put up will be on "Basic Mathematical Skills," and this will cover basic topics in arithmetic, algebra, and geometry that are necessary to do science.
What is a Function?
There seems to be a lot of confusion about what, exactly, a function is. Any rule that changes one thing into another thing (perhaps even the same thing) is a function. We can write such a rule in several different ways.
(1) function of thing 1 = thing 2,
or,
(2) f(thing 1) = thing 2,
or, if we get tired of writing the word "thing" we can abbreviate it by writing t1 for thing 1 and t2 for thing 2,
(3) f(t1) = t2,
or, if we want to avoid confusion about the things we can vary the symbols we use for the things. Traditionally we use x and y, but any different symbols will work so long as we remember what we are calling the different things. Using x and y,
(3) f(x) = y.
It is important to understand that all we have done up to now is state that y is determined by changing x in some specific way, we have not yet defined the rule that does the changing. It is the rule that is the function. If we say that the rule that changes x into y is (arbitrarily) a multiplication by 5, then we can write the function, f, as
(4) f = ___ x 5.
To avoid confusion between x the thing being changed, and x the symbol for multiplication we will simply understand that a space represents the multiplication. (4) then becomes,
(5) f = 5 ___,
where the underscore ___ serves as a place-holder for the thing to be changed.
The thing being changed is called the independent variable because it can take any allowable value. What determines whether or not a value is allowable is beyond the scope of this column (consult the references below for further details). The thing the independent variable becomes is called the dependent variable because it depends on the independent variable.
Why are Functions Important?
Functions allow us to understand the interdependency of different quantities. Ultimately this allows us to understand how things change with respect to other things. In a previous column, "The More Things Change, the More Some Things Stay the Same," I pointed out some of the rules about conservation laws. One of the principle definitions of a conserved quantity is where a dependent variable does not change with respect to an independent variable. By identifying such quantities we learn about the constants of the universe.
Since most things change with respect to something else, by understanding the functional relationships between quantities we can predict the values of quantities as yet unmeasured. This is at the heart of theoretical science.
Where Can I Learn More About Functions?
Below are what I consider to be very good presentations of the theory of functions at various levels:
Elementary
Morris Kline, 1977 (Reprinted by Dover Books in 1998), Calculus an Inuitive and Physical Approach. This is an excellent book containing a strong introduction to the concept of functions. I highly recommend this book for self-study in calculus in general and functions in particular.
Donald W. Hight, 1966 (Reprinted by Dover Books in 1977), A Concept of Limits. This is an truly wonderful little book on limits in general, and two-thirds of the book deal with limits of functions (with a good introduction to functions in general).
Richard Courant, Fritz John, 1989, Introduction to Calculus and Analysis, Volume 1, Springer. This is an excellent calculus book that has a wonderful first chapter of over one hundred pages dealing with the topics discussed in this chapter. It is a good book, perhaps even a great book, but it is not an easy book. There are no shortcuts here. It goes right to the heart of the subject with no holds barred. But if you intend to do theoretical work, these are all topics you should master. This book is a good place to start.
Intermediate
Douglas Smith, Maurice Eggen, and Richard St. Andre, 1990, A Transition to Higher Mathematics, Brooks/Cole Publishing Company. This is a book that develops the theoretical and logical side of functions, and it does an excellent job. Beginning with a strong introduction to logic and proof techniques, it continues to set theory, relations, and then functions. I strongly recommend this book to anyone truly interested in doing theoretical work. I recommend that you have a good grasp of calculus before starting this book, as a good many examples and practice problems are related to calculus.
Bruce E. Meserve, 1953 (Reprinted by Dover Books in 1981), Fundamental Concepts of Algebra. This is a very nice little book on elementary algebra, including the notion of functions, but using some of the tools of abstract mathematics. I recommend this book to anyone who is familiar with elementary calculus.
R. P. Burn, 1992, Numbers and Functions, Cambridge Unviersity Press. This is a truly remarkable book. It is more of a work-book than a standard text. It presents the subject through a series of guided exercises. In this way the reader develops the subject from scratch. For the subject of this column I recommend chapters 1-7 (covering the same material as the Hight book).
Advanced
Saunders MacLane, Garret Birkhoff, 1988, Algebra, Chelsea Publishing Company. This is a complete textbook on abstract algebra. It presents various types of functions and how they operate on various algebraic structures throughout. This is not a book for casual reading though, it can take a whole day to really understand a page of the text in some places, but it is highly worth-while. Abstract algebra is a current tool of cutting-edge scientific theory research.
Georgi. E. Shilov, 1973 (Reprinted by Dover Books in 1996), Elementary Real and Complex Analysis. This book is the first in a two-volume set on analysis. It is a thorough treatment of the properties of real and complex numbers and their functions. I recommend it, but it is at a fairly high level.
