Symbols, variables, algebra, functions, equal signs, and Operators

STUDENT: Variables are represented by a symbol. Variables can be either known or unknown. When variables are known, they are given some value. I understand this. When I look at phasor and Laplace transform tables, I am confused. What is a function?

PROFESSOR: Variables are not "given some value." Variables are assigned (== := or '=') a function. Algebra problems are solved by substituting numbers in for known values. In this case a numeric constant function such as $100$ was substituted. But the language of "function" was never used .. on purpose.

A function is a mapping between two sets of values. By mapping I mean a correspondence. Take, for example, the square function for real numbers.

The first set consists of the numbers we are allowed to take the square of. We are careful to define this set for the function under discussion. This set is known as the domain of the function.

The second set consists of all of the numbers that can be a square of a real number. This set is called the co-domain of the function. We know from the properties of the square function that this set consists of the positive real numbers.

Both sets contain an infinite number of values.

In addition to the definition of the domain and the co-domain, functions have a rule that associates every value in the domain with a value from the co-domain. This is what I mean by a mapping. If I know some value from the domain, say 4.0, then I know its corresponding value in the co-domain, here 16.0.

The rule can be given in a number of ways

By a table that lists values of the domain in one column and the associated value from the co-domain in the other. This only works for finite sets, however.

By an algorithm, that is a description of a procedure that transforms a value from the domain to its corresponding value in the co-domain. For example, multiply the number by itself.

By an algebraic expression. Here, if

x

is a value in the domain, and

y

is a value in the co-domain, then the rule can be expressed as an equation:

y==x*x

So a function can also be described as a correspondence like $120*cos(377t+4\pi /3)$ . Here two different correspondences are assigned to $v_{S}$ :

V_{S}==100

... a constant function

V_{S}==120*cos(377t+4\pi /3)

... a function that varies with time

STUDENT: Isn't time $t$ above a variable? Isn't $V_{S}$ above now an equation? Isn't time just an unknown that appears through the math and is also in the solution?

PROFESSOR: NO. Time is a parameter of the function. A function, as a mapping, exists by itself. A variable name is a symbol. Symbols have functions assigned to them. Equations are not normally solved for time in terms of resistance and inductance. Time is normally the independent variable of the function mapping.

STUDENT: Why aren't problems solved for a specific time value?

PROFESSOR: Problems can have three types of answers:

numeric ... which you are most comfortable with .. pick a numeric value of time and plug it in

symbolic ... like numeric, but without decimal places, with the symbol

pi

and ratios of symbols or integers

functions ... the answer is a function of time ... a specific time value could be plugged in, but an answer that is a function can generate intuition, predict the future and be communicated through a graph better than a single number.

STUDENT: Ok. What is the difference between C = 1 and C = 1 + D and C = cos(377D + 1.14) ?

PROFESSOR: It depends on the meaning of the equal sign above. There are two possible meanings:

assignment of a function

a relationship among functions or an equation

This is why all computer programs start playing with the equal sign. Programming languages like C or perl use == to evaluate an expression as true or false and use = to assign or define functions. Programs like MatLab or Mathematica are the opposite.

Equal by itself means one thing: it says that two things have the same value. Double equal or colon equal means something else: it says that the value produced by the expression on the right-hand side of the equals is assigned to the symbol on the left-hand side. There are two fundamental activities of math:

functions that could be evaluated into a number, a thing

equations that relate values

STUDENT: You didn't answer my question.

PROFESSOR: C = 1 could be a assigning the constant function 1 to the symbol C, that is, the value of C is always 1. Or C = 1 could be an algebraic equation that is either true or false depending upon the value of C right now--if the value of C is 2 right now, the equation is false. The other expressions involve the symbol D. This does not change the fact that each could be interpreted two different ways. I have a question for you. What does your equal sign mean?

STUDENT: I don't know. How could I get this far without even understanding this was an issue?

PROFESSOR: You have been studying elementary algebra most of your life. There are many algebra's. Grammar, music and even art can be thought of as an algebra. Functions were not studied with algebra.

Functions are things. They are more fundamental than an algebra. Functions can be defined in terms of other functions. The equal sign doesn't mean you can do the same things to either side of what looks like an algebra equation. The left side is usually the new function and the right side is a collection of other functions that are being combined.

I believe you have tried to make algebra and functions into one thing in your mind. Now you are being asked to separate the two. Trig was your introduction to functions. Functions are beautiful tools you need to become comfortable with. Some programming languages Scheme, Haskell, and FP try to do everything with the function concept.

STUDENT: Ok. I give up. Algebra and functions are different. What was the first function I studied?

PROFESSOR: Probably the first function you studied was a trig or circular function. Remember a function is a thing, a mapping, a description of reality. It is not algebra. We can do algebra with functions, just like we do algebra with numbers. We can have solutions that are functions just like we can have solutions that are numbers. We can have solutions that are symbols which in some cases could be either functions or numbers.

STUDENT: Ok. Say $y=sin(x)$ , or using your language the symbol "y" is assigned the function sin(x). What is $csc={\frac {1}{sin}}$ ? Why does $csc$ exist with a special name?

PROFESSOR: $csc$ is a function not a variable. $csc$ is a new function that is being defined in terms of a previously defined function. The "operation" is called a reciprocal or multiplicative inverse.

$csc$ , as a concept, predates calculators and computers. It existed as a calculation aid and was part of learning to do the physical number crunching by hand. It is not used that much any more, although there are still some very poor countries teaching their very best students how to memorize the trig tables.

STUDENT: Ok. A reciprocal. Sort of like ${\frac {1}{x}}$ . So what is an $arcsin$ or $sin^{-1}$ ?

PROFESSOR: $sin^{-1}$ is bad notation because if $sin^{2}$ is describing a function multiplied by itself, then logically, $sin^{-1}$ would be notation that is almost identical to a $csc$ . Unfortunately $arcsin$ is describing another type of operation on a function. This operation "arc" is called an inverse function or a compositional inverse. An inverse function "undos" the current function. It maps the co-domain of the original function to the domain of the original function. There is a lot of confusion between the multiplicative inverse:

g(x)={\frac {1}{f(x)}}

and the inverse function:

f^{'}(f(x))=x

STUDENT: Ok. Multiplicative inverse and inverse function are different things. What about the square root? Is it an "inverse function" of squaring a number?

PROFESSOR: No. Square root is an algebraic opposite operation to squared. It is not an inverse function because there answer may be 2 or -2. There is not a unique mapping. But it is possible to create an inverse function to the square function by defining the new co-domain as the set of positive real numbers.

Here, the English language confuses us. The word "inverse" is used in two different contexts. In an algebraic sense, the word means the opposite operations, such as addition and subtraction, or multiplication and division. When discussing functions, it means an operation that reverses the effect of the original function.

STUDENT: You keep introducing new words. What is an "operator" that performs "operations" on functions?

PROFESSOR: In elementary algebra, + and - are called operators. The definition of operator in the function context is something performed on a function that creates a new function. We have discussed the difference between a multiplicative inverse operation and the inverse function operation above. The context of algebra and functions has to be clear before "inverse" and "operation" have precise meaning. In calculus, you learned the similarities between the limit operation and the derivative and inverse derivative (integral) operations.

One goal of this class is to understand an "operation" called a convolution which combines two functions using a type of integral.

STUDENT: Why are we talking about this? How does this relate to phasors and Laplace Transforms?

PROFESSOR: It is hard to understand what the phasor or Laplace tables represent without understanding the difference between algebra and functions. Up until now we have been using algebraic operations (with some differences) to combine functions into new functions.

The word "transformation" is used to indicate a more complex mapping. Typically the objective is to find a short cut so that the math can be done by hand more easily. Transforms usually involve completely changing the parameter symbols, transforming functions and transforming operations.

Typically transformations involve tables rather than a linear equation. Phasor transforms are not really a transform in this sense. Time is just encapsulated in a away that it can be dropped during calculations and inserted at the very end. And all the phasor mappings are linear. Laplace and fourier transforms are mappings of both parameter symbols and operators.