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Basic Math Facts – Numbers, Variables, Equations, and Sets

In order to master mathematics, at least at the basic levels, you need to know the language. Without being able to recognize basic facts and concepts, you can never truly understand this subject. Armed with a solid knowledge of the building blocks such as numbers, variables, equations, and sets, the student has a good shot at doing well in this subject and moving progressively along the ranks.

Constants and Variables

Math hinges on numbers. A constant is a quantity that does not change. Thus any number such as 3, -10, or 1/2 are all constants. A variable, on the other hand, is an expression that can change, or vary. Variables are represented by letters such as x or y. Algebra is a branch of mathematics that uses variables and symbols to represent models that describe physical processes or other phenomena of the real world. For example, if a person makes $8 per hour and works 40 hours per week, the pay can be calculated by 40*8 = $320. If we wanted to calculate the pay for each of a group of employees who each make the same hourly wage, but whose hours varied, we would derive the expression 8h, where h represents the hours of each. Thus if Stan worked 30 hours, Stan’s pay would be 30*8 or $240.

Variables serve different purposes. In the equation x + 9 = 10, x serves as a placeholder for the number 1 since 1 makes the equation true. In y = x + 3, y changes as x changes. Because y is determined from values of x, we call y the dependent variable and x the independent variable. If we choose x from the values 1, 3, and 5 and substitute them into the equation, we obtain for y, 4, 6, and 8, respectively. The set of values from which we choose x is called the domain, or replacement set; and the set of values obtained for y is called the range.

Equations and Inequations

An equation is a mathematical expression that two quantities or expressions have the same value. For example, 3 = 3 or 3×4 = 12, are basic examples of equations. In the expression, y = 2x, we are saying that y is always twice the value of x. That is, when we plug in values for x, we obtain the equivalent for y. An inequation is an expression that two quantities or expressions do not have the same value; an inequation is represented by the “=” sign symbol with a line through it (here we will represent this by “/=,” which is the symbol used in some computer programming languages). For example, y /= x means that y and x represent different values; 3 /= 4 is obviously an inequation.

Equations are ubiquitous in mathematics, and range from the simplest linear equations to the most complex integral equations. Linear equations deal with lines, quadratic equations deal with parabolas, and polynomial equations deal with more complex shapes. In more advanced courses, students study transcendental equations, which involve such functions as the sine and cosine, and differential equations, which involve derivatives. Many problems require that solutions be found to various equations, or to determine whether there are no solutions.

Sets and Subsets

A set is simply a group or collection of distinct objects. For example, the set A = John, Sue, Steve or B = 1, 3, 6. The members that make up a set are called elements. The order in which the elements appear in a set does not matter and thus the sets 1, 3, 5 and 3, 1, 5 are the same. Subsets are sets that are portions of other sets. For example, the set John, Sue is a subset of A stated previously. The union of two sets is the combination of all the distinct elements in the two sets and is represented by the symbol “U.” For example, let C = 4, 9, 10, 12, 20 and D = 1, 2, 4. Then CUD is the set 1, 2, 4, 9, 10, 12, 20. Notice that both sets contain the element 4, and therefore it was not repeated in the union. The intersection of two sets, represented by an inverted U (here we will use “/”) is the set containing the common elements of the sets in question. Thus C/D is the set 4.

With these basic math facts in hand, any student will be better prepared to tackle the rigors of mathematics. Once a solid foundation in the core concepts is laid, and the development of a robust vocabulary of the terms and definitions is obtained, any individual will stand better equipped to achieve success in the bugbear of all bugbear subjects—mathematics.

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