Techniques with Names, Laws, Properties, and Theorems with Names

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

The Vertical Line Test Theorem

If the vertical line at any position

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.