Simultaneity, Time, and Equations
Must the values of variables in an equation be
simultaneous? That is, if the variables are functions of time,
must all the variables be evaluated at the same instant?
If the answer is no, it is valid to perform arithmetic operations
(addition, subtraction, multiplication, division) on the symbols
representing those non-simultaneous values?
There is no true mathematical law. You must first understand the reality
that the equation represents. Here are two simple examples.
The number of marbles in the box today is twenty more than the number of
marbles in the box last week. One equation that works for this is Y=20+X.
In this case, Y is today's marble count and X is the count from last week.
They do not refer to values at the same instant.
As water is allowed to flow into a spherical balloon, the volume of water
can be determined by measuring the radius of the balloon: V=(4/3)(pi)R^3.
In this case, V is the volume of water in the balloon and R is the radius of
the balloon. They do refer to values at the same instant.
An equation is a short-hand way of writing a relationship between
quantities. An algebraic equation is much easier to write and easier to
work with than full sentences. If you do not understand the reality it
represents, you cannot know what you can do with the equation. An example
is negative numbers. If a variable represents the length of a rod, the
variable cannot be negative. If it represents Fahrenheit temperature, it
can be negative.
Learn what the equation represents. Learn how the quantities will be
measured or calculated. If you understand these qualities of the equation,
you will then be able to decide whether the quantities are simultaneous.
Dr. Ken Mellendorf
Illinois Central College
There are two cases here:
1. Any set of equations involving a variable, e.g. "time". There is
no reason why the value given to time be the same for all equations
2. Parametric equations are different in this regard. Suppose you
have a function F(X,Y,Z,...) and each independent variable X, Y, Z
is given in terms of a parameter, p. That is X = X(p), Y= Y(p),
Z = Z(p), then it is understood that usually you mean the same input
value of 'p' for each variable. The parameter 'p' may be "time" but
that is not required.
If I understand your question correctly, the answer is no.
Most formulas describe a pattern of change, such as how fast a
dropped object is moving, how much profit a business can make
selling its product, etc.
On the other hand, the typical use for such formulas is looking for
one answer, or one range in which the answer is most suitable. (such
as how to make the most profit by selling a product) In these
cases, there are often other factors involved as well, so there is a
frequent search for where two different formulas give the same
answer. It is not that the formulas do not have multiple
possibilities, it is just that you are frequently only looking for one of them.
Click here to return to the Mathematics Archives
Update: June 2012