Symbols
Symbols are part of the language of mathematics. The use of symbols gives mathematics the power of generality: The ability to focus not only on a specific object, but also on a collection of objects. For instance, variables— symbols with the ability to vary in value used in place of numbers—make it possible to write general equations. Consider the equation 3 + 4 = 7. This equation is specific. Suppose we are interested in all pairs of numbers whose sum is equal to 7. Using variables, x and y, to denote the two numbers, we would write the general equation x + y = 7. The variables, x and y, stand for many different pairs of numbers; x can be 5 and y can be 2 or x can be 9 and y can be -2. The only condition is that the sum of x and y is 7.
Symbols also include special notation for mathematical operations, like + for addition and 
for taking the square root. Some common symbols and their uses are explained in the accompanying table.
Basic Mathematical Symbols
The Set. The foundation of mathematics rests on the concept of a set, which is a collection of objects that share a well-defined characteristic. The elements of a set are written in braces {}. An element x that belongs to a set A is written as x ∈ A. An element y that does not belong to A is written as y ∉ A. {0} is a set that contains zero as its only element. This set has one element. {} is the empty set and does not contain any element. The empty set is also called a null set and sometimes written with the symbol Ø.
Operations. Numbers are the building blocks of mathematics. They can be combined by four operations: addition and its inverse operation,
subtraction, and multiplication and its inverse operation, division. The additive inverse of a number b is -b and the sum of a number and its additive inverse is always 0, that is, b + (-b) = 0. The additive inverse of 4 is -4. Similarly, the multiplicative inverse of a nonzero number b is b-1, or 
, and the product of a number and its multiplicative inverse is always 1, that is, 
. The multiplicative inverse of 7 is .1. The multiplicative inverse of 7 is
.
Size. Numbers can be compared according to their magnitude, or size. Given any two numbers, say x and y, there are three possibilities: x is equal to y (x = y), or x is less than y (x < y), or x is greater than y (x > y). Two numbers, x and y, can also be compared by taking their ratio, x:y, or x/y. A number percent, say 5 percent denotes the ratio of 5 to 100 or 5/100. In some problems, it is not important to know if a given number, say x, is positive or negative. The only important thing is its magnitude and it is represented by its absolute value, |x|. For instance, |5| = 5 and |-5| = 5.
Exponents. If a number is multiplied by itself two or more times, it can be written more compactly using an exponent that appears as a superscript to the right of the number. For example, if 4 is multiplied by itself three times, then it can be written as 4 ×4 ×4, or 43, where 3 is the exponent. An exponent is also called a power. In general, xn is read as "x raised to the power n." Raising a number to powers that are fractional, like ½ or ⅓, is called a radical, or root of the number.
denotes the nth root of the number b; where n is called the index and b is called the radicand. For n = 2, the index is omitted and 
is the square root of b. For n = 3, 
is called the cube root of b.
Functions. A fundamental idea in mathematics is that of a function that describes a relationship between two quantities. In symbols, it is written as y = f(x) and read as "y is the function of x," which means that the value of y depends on x. A function is sometimes called a map, or mapping. Logarithm, y loga x, is a particular type of function. By definition, a logarithm is an exponent. If 10x = y, then log10 y = x.
Algebra. In algebra, numbers are represented as both variables and constants. A variable is a number whose value can change. Variables are mostly denoted by lowercase, terminating letters of the alphabet like x, y, or z. A constant is a fixed number, denoted mostly by first few lower case letters a, b, and c, or k. Algebraic expressions called polynomials contain one or more variable and/or constant terms. For example, 3x, 
and x2+3x+2 are all polynomials: 3x is a monomial (one term), 
is a binomial (two terms), and x2+3x+2 is a trinomial (three terms). A constant number that multiplies a variable in an equation, or polynomial, is called a coefficient. For example, 2 is a coefficient of 2x in the equation 2x+3 =0. In ax2, a is the coefficient of x2.
Equations. Solving different types of equations appears in many situations in mathematics. The simplest equation is linear, in which the highest power of the variable, x, is 1, for example, 3x+2 =1. If the highest power of the variable is 2, then the equation is a quadratic equation. For example, x2 = 4 and 2x2+3x+1 =0 are quadratic equations.
Calculus. Calculus is a branch of mathematics that partly deals with rate of changes. For a variable x, Δx denotes a small change in x. Further, 
measures the change in variable y with respect to change in x. When Δx becomes increasingly smaller and approaches 0, the rate of change 
equals the instantaneous rate of change and is denoted by 
and is called the derivative of y with respect to x. Also, calculus deals with finding the area of a region under the curve, or graph, of a function f(x). If the region is bounded by lines x=a and x=b, then the area is denoted by the integral of the function f. In symbols, it is written as 
Algebra; Calculus; Functions and Equations; Number System, Real.
Bibliography
Amdahl, Kenn, and Jim Loats. Algebra Unplugged. Broomfield, CO: Clearwater Publishing Co., 1995.
Dugopolski, Mark. Elementary Algebra, 2nd ed. Boston: Addison-Wesley Publishing Company, 1996.
Dunham, William. The Mathematical Universe. New York: John Wiley & Sons Inc., 1994.
Miller, Charles D., Vern E. Heeren, and E. John Hornsby, Jr. Mathematical Ideas, 6th ed. New York: Harper Collins Publishers, 1990.
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