Prominent Variables

Variables in math are very prominent. It is a part and parcel in math, especially in the topic of algebra. Let us try to figure out what is variable and in the process let us find the variables definition. It takes place of an unknown quantity.

Suppose we say that I am 10 years older than my brother, then at any point of time my age is (x + 10). Since at any point of time my brother’s age is also not known, I assumed that as x. Therefore it can be described as letter that represents the unknown quantity and any value can be assigned to that depending on the situation and need.

It is a general practice to use the small case letters of latter halves of English letters to denote the unknown quantities. Of course, none of the terms ‘small case’, ‘latter half’ or ‘English letters’ is a strict requirement. Though it is a general convention, there are exceptions in many cases. The unknown measures of angles of triangles are denoted in capital letters, to be consistent with the symbols of the respective vertices. Again in the very same case of geometry, the first halves of English alphabets are used for unknown quantities.

For example, the prominent statement of Pythagorean Theorem is,              a2 + b2 = c2. Also, there are many Greek letters are used for the unknowns, mostly in trigonometry. For example, the measure of angles are mostly expressed as ? (in case of units in radians) and as a (in case of units in degrees).

In general two or more of these are used in expressions or functions. As a simple example let us take the case of a linear function y = f(x), where f(x) = mx + b. This is a case of input and output function where variable(s) x denotes the input quantity and the ‘y denotes the corresponding output quantity.
Now by change of variables and solving for the same on the left we can find the behavior of the inverse of the function f(x). Let us now see in what way the variables and expressions are related. Expressions are mostly parts of functions. The degrees (the powers) of these decide the type of functions.

The function behaviors can be predicted from the degree and the sign of the leading coefficient of the function. For example, a quadratic function which of degree 2 and with a single unknown will always have rising characteristics on both ends for a positive leading coefficient.