Entries marked with an asterisk are not commonly used terms in the general literature and correspond to concepts whose definitions are more specific to their use in this book.
big algebra*: Algebraic operations applied to expressions or equations that contain parameters, meaning that they are really multiple algebraic expressions or equations represented by a single general one.
break-even point: The point in a business where the revenue brought in equals the overall amount spent.
classroom word problems*: Manufactured problems not likely to be encountered outside of a mathematics education or recreational math context, or possible real-world problems with intentionally sculpted scenarios that may not be as realistic as they would be in actual practice.
cloud of possible values*: All of the possible numerical values that a variable expression can take on from its allowable input values; also known as the range of an expression. A synonym is “symphony of possible values.”
conceptual fuel*: Concepts, experiences, and ideas that can be put to effective use in a directly related area or in unrelated areas; something that provides conceptual enlightenment or sustenance.
conditional equation: An equation whose truth depends on the values chosen.
counting numbers: The numbers 1, 2, 3, 4, 5,…; in other words, the whole numbers without 0 or the integers greater than 0. They are also known as the natural numbers.
Descartes protocol: The algebraic protocol that represents regular variables or unknowns by letters late in the alphabet (such as x, y, z) and parameters by letters early in the alphabet (such as a, b, c). This protocol is still extant—though inconsistently applied—and was discussed by René Descartes in his 1637 book, La Géométrie (The Geometry).
Diophantine equation: Any equation, usually in several unknowns, that is studied in a problem whose solutions are required to be integers, or sometimes more general rational numbers (Nelson 2008).
first drama*: (a) Situations related to a numerical variation or numerical symphony or its capture as an algebraic expression, and the ensemble of all possible values generated by such a variation or expression. (b) Situations involving a function and its domain (all values that can be input) and range (all possible output values), or the relationship between the two.
identity equation: An equation that is always true in its domain of applicability regardless of the values chosen; often called an identity.
indeterminate equation: An equation for which the unknown or variable quantities can have many different values—sometimes infinite in number—that satisfy or solve the equation.
integers: The positive and negative whole numbers:…, –4, –3, –2, –1, 0, 1, 2, 3, 4,…
isotope: One of several varieties of the same element. Elements are identified by the number of protons in the atomic nucleus: If two atoms have a different number of protons, then they represent different elements. The number of neutrons in the nucleus of an element, however, can vary, which leads to different varieties of the same element. For example, the most commonly occurring type of nitrogen atom has seven protons and seven neutrons (nitrogen-14), yet another type of nitrogen atom has seven protons and eight neutrons (nitrogen-15). When two types of atoms of an element have different numbers of neutrons (such as nitrogen-14 and nitrogen-15), they are isotopes of that element. Some isotopes of an element may be stable while others can be radioactive.
multiplication of variables and values:
cross: The symbol × used to indicate multiplication: for example, 2 × 5 = 10.
dot: The symbol · also used to indicate multiplication. We may represent “2 times 3” or “8 times x” respectively as 2 · 3 and 8 · x. The dot symbol is often used when the traditional cross symbol × could be confusing due to its similarity to the letter x, which is often used in algebra.
juxtaposition: Algebra affords us the convenience of using juxtaposition as multiplication. For instance, x · y can be represented more succinctly as xy. This is not advisable using numerals alone because replacing 2 × 3 by the juxtaposition 23 gives two entirely different values (6 in the former and 23 in the latter).
asterisk: The symbol * is often used to represent multiplication in computer programming languages and software, for example, Excel or some computer algebra systems.
number of days and age problem*: Problem introduced in Chapter 1 as “Magical Three-Digit Numbers” and analyzed in detail in Chapter 2:
Pick the number of days you like to eat out in a week (choose from 1, 2, 3, 4, 5, 6, 7). Multiply this number by 4. Then add 17. Multiply that result by 25. Next add the number of calendar years it is past 2013. Now if you haven’t had a birthday this year, then add 1587, but if you have had a birthday this year, then add 1588. Finally, subtract the year that you were born from this. Reading the resulting three-digit number from left to right, the first digit is the number of times you like to eat out in a week and the last two digits are your age.
numerical symphony*: See symphony (numerical).
operative symbolism: A symbolism that is more than just a convenient shorthand. It does more than store information, but can allow manipulation and transformation according to well-defined rules to reproducibly yield new and different symbolic forms that remain useful—in that they may reveal new information and interpretations. For example, the addition of three hundred fifty-two plus one hundred forty-six is performed by first rewriting it in numerals as 352 + 146. Once in numeral form, systematic rules can be applied to yield the new value represented in symbols as 498. Rules such as 2 + 6 = 8, 5 + 4 = 9, and 3 + 1 = 4 still hold when we add together other numerals such as 523 + 461, allowing for the systematic computation of yet another new value represented in symbols as 984. The numerals used in this way form an operative symbolism.
parameter: An object that has aspects of both a constant and a variable. It acts as a constant within a given scenario but can change value from one scenario to another scenario, then taking on the characteristics of variable. It is often referred to in this text as a scenario variable and more generally is sometimes referred to as a coefficient, given, or known.
quantitative cocktail*: A numerical measure whose value depends on a systematic mixture in varying strengths of other, more basic measures; a numerical measure that can be broken up into a discrete set (spectrum) of more basic measures of varying contribution strengths (or intensities).
real-world problems/applications: Quantitative problems that might naturally occur outside of a mathematics class or a recreational math book.
rhetorical algebra: A type of algebra where the primary algorithms are expressed in a non-operative symbolism, usually in the words or word abbreviations of a language such as Arabic, Chinese, Sanskrit, Latin, Italian, German, or English.
rhyme (common): Two or more words or phrases that end in the same sounds.
rhyme (metaphorical)*: Two or more different situations that share a framework of similarity, regularity, or repetition. For example, (a) the expressions 16h, 16h2, 16h3, and 16h4 can be viewed as metaphorically rhyming in the sense that they are different but represent the similar idea of the variable h being raised to various powers; and (b) the witty saying “History never repeats itself, but it rhymes” implies that situations in history are never exactly the same but share important and deep similarities. This is similar to Alfred North Whitehead’s definition of rhythm as the conveyance of difference within a framework of repetition (The Aims of Education, p. 17).
second drama*: (a) Situations related to finding specific input value(s) of a numerical variation (or its capture as an algebraic expression) that make the variation (expression) take on particular output value(s). (b) In algebra, situations involving finding a particular (domain) value of a function that makes the function take on a specific (range) value. The algebraic treatment of such situations usually involves solving equation(s).
symbolic algebra: A type of algebra where the primary algorithms are expressed in an operative symbolism (usually involving letters from a language alphabet such as Latin or Greek but can involve words or abbreviations in any written language).
symbolic maneuver*: Any introduction, combination, movement, and/or manipulation of symbols (including tables and diagrams) to gain an advantage in knowledge, insight, organization, identification, clarity, efficiency, etc.
symphony (music): A lengthy form of musical composition for orchestra, normally consisting of several large sections or movements often tied together around a central theme or emotion.
symphony (numerical)*: An ensemble of varying numbers, expressions, or objects that are tied together around a well-defined or easy-to-recognize procedure, rule, or theme.
term: A product of numbers and variables: for example, 6x, 6x3y5, and –90x4yz7. The expression 8xy + 40w10y5 contains two terms (8xy and 40w10y5). In some cases, such as in the three-term expression 3x + 4x + 25x, the terms can be combined to become a simpler expression (the single-term 32x in this case).
unknown: A yet-to-be-determined value (or set of values) that satisfy a condition (or set of conditions). The algebraic representation of such conditions is usually in the form of equations or inequalities. Variables in an algebraic expression are often called unknowns whenever the algebraic expression becomes part of an equation. It is often used interchangeably with “variable.” For example, in the algebraic expression 5x + 6, the x is viewed as a variable that can take on infinitely many values. However, when placed into the equation 5x + 6 = 66, it is not uncommon for the x to be viewed as an unknown whose value is to be determined (which is 12 in this case).
variable: A quantity that can take on different values. In algebra, such quantities are often incorporated as part of a larger expression. Variables are useful in modeling changing quantities in nature or human situations. See also unknown.
variable (regular): A quantity (when viewed together with parameter quantities in an algebraic expression) that can take on different values within a given scenario (while the parameter is fixed). See also parameter.
variable (scenario): See parameter.
variable expression: An algebraic expression that contains at least one variable quantity, meaning that the expression itself can vary in value.
Viète/Harriot protocol: The protocol that represents regular variables or unknowns by vowels (such as A, E, or I) and parameters by consonants (such as B, G, or D). Francois Viète introduced this protocol using capital letters in his 1591 book, In artem analyticam isagoge (Introduction to the Analytic Art). The English mathematician Thomas Harriot preferred the protocol in lowercase letters and used it to tremendous effect. This protocol is no longer in common use.
whole numbers: The numbers 0, 1, 2, 3,…; in other words, the counting numbers and 0 or the nonnegative integers.