Whenever anything occurs in mathematics with a name attached to it, then it should be studied carefully and thoroughly. Why? Because such mathematical items have earned their names for their value and usefulness in understanding and doing mathematics. By having names, then it becomes convenient to refer to them and remember them whenever they needed or used.
An example of a named technique is the technique of
“completing the square.” You should be familiar with this technique from your
high school Algebra II course. Are you? Another example of a named technique is
the technique of “synthetic division.” You have learn
this techniques in College Algebra. Completing the square is one technique that
is important for your understanding of conic sections in technical calculus.
Laws are also important. Just as you should know the
laws of the community and, of course, obey them, you should also know the
mathematical laws and learn them well so as not to violate them. The penalty
for breaking a speeding law is a fine. The penalty for breaking a law of
mathematics is incorrect work and when this occurs on an exam, the cost is the
loss of valuable points and a lower grade. This is too high a price to pay.
Properties, such as, the properties of the real
numbers, the properties of inequalities, the properties of radicals, and the
properties of logarithms, tell us about the characteristics of a mathematical
operation, relation, or function. Knowing these characteristics is essential to
using and applying these operations properly. For example, a property that many
students remember from their previous study of algebra, but only vaguely, is
the following property:
, provided and
This property, about
applying square roots to a product of numbers, works only for nonnegative
numbers a and b. Secondly, it works only for products.
It is useful when we wish to combine and simplify radicals, e.g., by applying
the above property of radicals, we have and we can simplify as follows
Although students remember
the property of radicals for square roots and products, they also believe,
incorrectly, that it applies to situations involving radicals and the sum of
two numbers. A simple numerical example shows that the property is not valid for sums.
Taking a = 1 and b = 1 yields
Whenever you wonder about whether you have remembered a property from your past mathematics courses correctly or incorrectly take several simple numerical examples and try them out, analogous to what we did above. If your numerical examples verify the property, then it is a good bet that it is remembered correctly.
Recall earlier in these notes we stressed the
importance of knowing items that were highlighted in a box. Well here is
something to remember for all time about radicals.
This is true! provided and This is not true! |
That previous discussion illustrates the importance of observing the details related to a mathematical definition, property, law, technique, or theorem. As you study mathematics it is valuable to get in the habit of asking questions about what you are reading or studying. Under what conditions is the property true? Under what conditions is the property false? Can I think of a simple numerical example to help me keep straight when a property holds and when it does not? Usually your instructor will provide you with clarifying examples which will enable you to keep your understanding on the right track. At the risk of sounding preachy, students will be successful in Technical Calculus if they are observant of the details of mathematics.
Theorems are mathematical statements usually written
in the form “if . . . , then . . . .” For example, a typical theorem from
College Algebra is The Vertical line Test Theorem
Along
the x-axis intersects the graph
in more than one point, then the graph
is not the graph of a
function.
All
theorems are statements that can be proved from previously known mathematical
facts. Theorems are statements which tend to summarize important facts about
mathematical objects of study. For example, the Vertical Line Test Theorem
above gives a useful graphical condition to look for to determine when a graph
represents a function. When theorems have names, as it is with any named
mathematical item, it is a sign that it is frequently called upon as a tool to
solve problems or to help with understanding mathematics.