# Ambiguously Defined Mathematical Termsat the High School Level

Last revision: Sept. 30, 2018

This page attempts to show some of the ambiguities in defining some of the mathematical terms that might be encountered at the high school level. A number of these issues may seem quite trivial, and some are the result of consulting out-of-date texts. But many of these issues have led to disputed answers in mathematics competitions. Contributions and suggestions are welcome.

ADJACENT ANGLES. Kathy Borst writes (2011) that an item that formerly appeared on the California state test began, “Quadrilateral ABCD is a parallelogram. If adjacent angles are congruent, which statement must be true?” Normally, of course, adjacent angles share a vertex and the term here should be consecutive angles. I do not know whether this is a defective question or whether some authors define adjacent angles differently from the usual way.

AMPLITUDE OF A CURVE. Mathematics Dictionary by Glenn James and Robert C. James (1949) defines amplitude of a curve as "the greatest numerical value of the ordinates of a periodic curve." However, the third edition of this dictionary (1968) has the definition "half the difference between the greatest and the least values of the ordinates of a periodic curve." The more recent definition seems generally to be in use today.

ARCSECANT AND ARCCOSECANT, RANGE OF. Some textbooks give the range of the Arcsecant function to be [0, π] except for π/2. However, by defining Arcsec x to be Arccos (1/x), π/2 can be included in the range.

Other textbooks give the range of the Arcsecant function to be [0, π/2) U [π, 3π/2).

Some textbooks give the range of the Arccosecant function to be [-π/2, 0) U (0, π/2). However, by defining Arccsc x to be Arcsin (1/x), 0 can be included in the range.

Other textbooks give the range of the Arccosecant function to be (0, π/2] U (π, 3π/2].

BILLION, TRILLION, etc. There are two systems for naming large numbers. For example, in the American system a billion is 109, whereas in a system formerly used in Britain, a billion is 1012 and a milliard is 109.

CHARACTERISTIC AND MANTISSA. These terms have less significance now that logarithmic tables are no longer used. Older textbooks would write the logarithm of 0.56 as .7482 - 1 and identify .7482 as the mantissa and -1 as the characteristic. Since nowadays a calculator gives the logarithm of 0.56 as -0.2518 and modern dictionaries define the characteristic and mantissa as the integer and decimal parts of a logarithm respectively, it would seem that a requirement to identify the characteristic or mantissa of the logarithm of a number between 0 and 1 is ambiguous.

CONVERGENT and DIVERGENT SERIES. According to the third edition of A Treatise on Algebra (1892), an oscillating series is called "divergent by Cauchy, Bertrand, Laurent, and others." In 1893 in A Treatise on the Theory of Functions, James Harkness and Frank Morley write: "Strictly speaking, an oscillating series is distinct from a divergent series, but it is usual to speak of non-convergent series as divergent." In 1898 in Introduction to the theory of analytic functions by Harkness and Morley a footnote says, "Most English text-books regard oscillating series as not divergent." The 1968 third edition of Mathematics Dictionary by James and James defines a divergent series as: "A sequence which does not converge. It might either be properly divergent, or oscillate ..." This seems to be a generally accepted definition today.

COUNTABLE has a mathematical definition that is very different from its meaning in other contexts. For example, Integrated Mathematics, 2nd ed., Course I defines an infinite set as one whose elements "cannot be counted." But in mathematics a countable set can be countably infinite.

CRITICAL POINT. Calculus and Analytic Geometry by Sherman K. Stein and Anthony Barcellos (1992) defines a critical number as a number for which the derivative is zero, and the critical point as the corresponding point, and has the following remark: "Some texts define a critical number as a number where the derivative is 0 or else is not defined. Since we emphasize differentiable functions, a critical number is a number where the derivative is 0." Some textbooks include endpoints as critical points. For example, Calculus (1991) by Gilbert Strang has: "The maximum always occurs at a stationary point (where df/dx=0) or a rough point (no derivative) or an endpoint of the domain. These are the three types of critical points. All maxima and minima occur at critical points!"

CURVATURE OF A FUNCTION. According to Roman Arce, "Sometimes it’s defined as equivalent of its second derivative and other times as the inverse of the radius of a circle that fits the curve (the osculating circle), the relation is 1/R = f '' (x)/(1+f ' (x))3/2 so the two definitions are proportional but not equal and some times it’s used without specifying if they mean 1/R or f '' (x)."

DEGENERATE CONIC SECTIONS. The degenerate conic sections are normally considered to be a point, a line, and a pair of lines. True or false: "x2/a2 + y2/b2 = 0 is an ellipse." It is a point, but it is also a degenerate ellipse. Mathematics Dictionary by James and James, both in the 1949 and 1968 editions, calls this a point ellipse.

Various web pages say that a degenerate ellipse is a circle or a parabola. Steve Wilson, Professor of Mathematics at Johnson County Community College, writes, "Personally, I could consider circles and parabolas as special or limiting cases, but not as degenerates."

DISCRIMINANT. The Wikipedia article has “the discriminant of a polynomial”; an Algebra II textbook has “the discriminant of the quadratic equation.”

DOMAIN OF A FUNCTION. This term seems not to be precisely defined in algebra textbooks. Mathematics Dictionary by James and James (1949) says that it is “the set of values which the independent variable may take on, or the range of the independent variable.Merriam-Webster’s 11th Collegiate Dictionary has: “the set of elements to which a mathematical or logical variable is limited; specifically, the set on which a function is defined.” Heath Algebra 2 by Larson, Kanold, and Stiff (1998) defines the domain, or input, of a function as "the x-values of a function y = f(x)."

GEOMETRIC MEAN. According to the Wikipedia article on the subject, retrieved in 2011, “The geometric mean only applies to positive numbers.” However, at least one textbook and Wolfram Alpha give 2i as the geometric mean of 2 and –2.

IMAGINARY NUMBER. There are two modern meanings of the term imaginary number. In Merriam-Webster’s Collegiate Dictionary, 10th ed., an imaginary number is a number of the form a + bi where b is not equal to 0. In Calculus and Analytic Geometry (1992) by Stein and Barcellos, "a complex number that lies on the y axis is called imaginary."

Algebra 2 (2015) published by Houghton Mifflin Harcourt defines an imaginary number as a number written in the form bi where b is not 0.

Steve Wilson, referring to the 1968 Third edition of Mathematics Dictionary by James and James, writes:

JJ3’s discussion of imaginary numbers is interesting. Under complex numbers, they say "Any number, real or imaginary, of the form a + bi, where a and b are real numbers and i2 = -1. Called imaginary numbers when b is not equal to 0, and pure imaginary when a = 0 and b is not equal to 0 (although complex numbers are not imaginary in the usual sense)." If that parenthetical comment had not been made, their position would have been clear. Most current college algebra texts appear to agree with JJ3, although some will avoid the term "imaginary number" altogether, and Gustafson & Frisk’s "College Algebra", 8th ed., 2004, agrees with Stein & Barcellos.

IMAGINARY PART. According to Merriam-Webster’s Collegiate Dictionary, 11th ed., the imaginary part of 2 + 3i is 3i. However, according to the definition found on several web sites, the imaginary part is 3. Algebra 2 (2015) published by Houghton Mifflin Harcourt has: “Note that ‘imaginary part’ refers to the real multiplier of i; it does not refer to the imaginary number bi.” This textbook defines an imaginary number as a number written in the form bi where b is not 0. Roman Arce points out that the function Im[] of Mathematica 3.0 gives 3.

INCREASING/DECREASING FUNCTION. In Stein and Barcellos (1992), "a function f is said to be increasing if whenever x1 < x2, one has f(x1) < f (x2)." The definition in Ellis and Gulick (1986) requires only that f(x1) is less than or equal to f (x2). The latter text used strictly increasing to describe the other situation.

INFLECTION POINT. Some texts require that a tangent line exist at an inflection point, but other texts do not.

INVERSE TRIGONOMETRIC FUNCTIONS, RANGE OF. Precalculus With Limits: A Graphing Approach, 3rd. edition, by Larson, Hostetler, and Edwards has: "In Example 4, if you had set the calculator to degree mode, the display would have been in degrees rather than in radians. This convention is peculiar to calculators. By definition, the values of inverse trigonometric functions are always in radians." Other textbooks, however, allow degrees.

ISOSCELES TRIANGLE. Older definitions specified that an isosceles triangle has exactly two congruent sides, as, for example, a 1570 translation of Euclid:

Isosceles, is a triangle, which hath onely two sides equall.
Many texts define an isosceles triangle as a triangle having two sides equal; such a definition could perhaps be interpreted either way. Heath Geometry: An Integrated Approach (1998) has: “An isosceles triangle has at least two congruent sides.”

LIMIT, EXISTENCE OF. Some texts say carefully that, although we write lim f(x) = oo, the limit does not exist. Ellis and Gulick (1986) says that "f has an infinite limit at a" although it also has "Caution: If f has an infinite limit at a, then f does not have a limit at a in the sense of Definition 2.1."

MULTIPLE. Do multiples of 4 include negative numbers and zero? Apparently, yes, although several textbooks consulted do not define the word multiple. Merriam-Webster’s 11th Collegiate Dictionary defines multiple as “the product of a quantity by an integer,” and most dictionaries have similar definitions. That precise definition is in marked contrast with the definition in the third edition of that dictionary: “The product of one number multiplied by another.” Several dictionaries say that a multiple is a number that is contained in another “an exact number of times.” Funk & Wagnalls Practical Standard Dictionary (1935) has “A resultant of multiplying a quantity by a whole number,” but that dictionary defines whole number in an unusual way (see below). The 1949 edition of Mathematics Dictionary by James and James has:

MULTIPLE, n. In arithmetic, a number which is the product of a given number and another factor; 12 is a multiple of 2, 3, 4, 6, and trivially of 1 and 12. In general, a product, no matter whether it be arithmetic or algebraic, is said to be a multiple of any of its factors.

NATURAL NUMBER. Modern dictionaries, except for the Random House dictionaries, exclude 0 in the definition of this term. The 1771 Encyclopaedia Britannica excludes 0. In 1889 Elements of Algebra by G. A. Wentworth includes 0. In 1919 in Introduction to Mathematical Philosophy Bertrand Russell includes 0.

NO SLOPE. This is a term which should probably be avoided, since it could be interpreted to mean "slope of zero" or "undefined slope."

PASCAL’S TRIANGLE. In Heath Algebra 2: An Integrated Approach (1998), the row of Pascal’s Triangle consisting only of 1 is Row 0. However, on the following page, it has: "The binomial coefficients in Row 4 of Pascal’s Triangle are 1, 3, 3, 1." Presumably, this is an oversight, but some texts call the row consisting only of 1 Row 1.

Paul Sisson’s College Algebra, 2003, says, "Pascal’s triangle is a useful way of generating binomial coefficients, with the nth row containing the coefficients of a binomial raised to the (n - 1)st power."

In December 2003, Steve Wilson wrote, "Having just examined 15 college algebra textbooks, they seem to come in the ratio 2 agreeing with Sisson to 1 recognizing a zeroth row to 1 finessing the issue (e.g., "the row for n = 6")."

PRIME NUMBER. Is 1 a prime number? Most textbooks today call it neither prime nor composite, but older texts generally considered it to be prime. In 1859, Lebesgue stated explicitly that 1 is prime in Exercices d'analyse numérique. It is prime in Primary Elements of Algebra for Common Schools and Academies (1866) by Joseph Ray and Standard Arithmetic (1892) by William J. Milne. A list of primes to 10,006,721 published in 1914 by D. N. Lehmer includes 1. [David Cantrell suggests this is more a historical issue and is inappropriate for this page. He writes, "No high-school students (unless they're doing historical research) are going to encounter a legitimate definition of prime which would include 1." This entry may be removed.]

RATIONAL EXPRESSION. Algebra 2 (2015) published by Houghton Mifflin Harcourt requires that the bottom function in a rational expression have degree at least one. This seems to be an unusual definition. Thus, for this textbook a polynomial is not a rational expression.

REFLEX ANGLE. Older geometry texts sometimes defined a reflex angle as an angle with measure greater than 180 degrees, whereas modern geometry textbooks seem to consider an angle in such a way that no angle measures greater than 180 degrees.

RELATIVE EXTREMUM. Textbook authors differ on whether an endpoint can be a local minimum or maximum. According to a post in the AP Calculus mailing list, this issue is avoided in AP exam questions because of the inconsistency.

RHOMBUS. Mathematics Dictionary in 1949 and in the 1968 third edition has: "Some authors require that a rhombus not be a square, but the preference seems to be to call the square a special case of the rhombus." [Can any readers of this page cite any modern writer who makes this requirement?]

ROUNDING. There is not a universally accepted procedure for rounding, for example, 2.315 to the nearest hundredth. However, one commonly accepted rule, called the computer’s rule or the odd-even rule, is to make the digit before the 5 even, thus to add 1 if it is odd and leave it alone if it is already even.

SECANT. Prentice Hall Geometry defines secant as "Line, ray, or segment that contains a chord of a circle." Is a chord also a secant? If a chord contains itself, it would be a secant. This text seems not to define contain.

SIMPLIFIED FORM. Mathematics Dictionary (1949) calls this term "probably the most indefinite term used seriously in mathematics." One particular case is that some newer textbooks seem not to require rationalizing the denominators of some fractions.

SQUARE PRISM. This term probably should be avoided. Some Internet web pages say it is a prism with a square base; others say it is a cube.

The 1968 third edition of Mathematics Dictionary by James and James has: "A prism with a triangle as base is a triangular prism; one with a quadrilateral as base is a quadrangular prism; etc." This might imply that a square prism should be a prism with a square base.

STANDARD FORM OF A LINEAR EQUATION. Standard form is given as both Ax + By = C and Ax + By + C = 0. Some writers require that A be non-negative, but others do not.

It is usually required that A, B, and C be integers, but Prentice Hall Mathematics Algebra 2 (Florida, 2004) has:

The standard form of a linear equation is Ax + By = C, where A, B, and C are real numbers, and A and B are not both zero.

STANDARD FORM OF A QUADRATIC FUNCTION. According to Algebra and Trigonometry, 6th ed., by Larson and Hostetler, the standard form of a quadratic function is f(x) = a(xh)2 + k.

According to Florida Prentice Hall Mathematics Algebra 2 (2004), the standard form of a quadratic equation is y = ax2 + bx + c.

TRAPEZOID. The usual definition of a trapezoid requires that it have exactly one pair of parallel sides. However, the UCSMP textbook Geometry (1997) has the definition "a quadrilateral with at least one pair of parallel sides." (In this textbook, an isosceles trapezoid is defined as "a trapezoid with a pair of base angles equal in measure.") According to Chris White, the Cresent Dictionary of Mathematics by William Karush (1962) defines trapezoid as "A quadrilateral which has exactly one pair of parallel sides (sometimes, the parallelogram, with both pairs of sides parallel, is included)."

TRAPEZOID/TRAPEZIUM. In the United States, a trapezoid is generally defined as a quadrilateral with exactly one pair of parallel sides and a trapezium is a quadrilateral with no sides parallel. However, until the end of the eighteenth century, the definitions for these two terms were reversed, and outside the United States even today the definitions are often reversed.

TRIGONOMETRIC FUNCTIONS, DOMAIN OF. When using radians, are the arguments of the trigonometric functions numbers of radians or just real numbers? Textbook writers seem to differ.

The definition of radian in the third edition of Mathematics Dictionary by James and James is: "A central angle subtended in a circle by an arc whose length is equal to the radius of the circle. Thus the radian measure of an angle is the ratio of the arc it subtends to the radius of the circle in which it is the central angle." Thus technically this dictionary distinguishes between "radian" and "radian measure," and since radian measure is a ratio of lengths, it would be a real number.

VERTICES OF AN ELLIPSE. Most textbooks define the vertices of an ellipse to be the end points of the major axis. Some textbooks call the end points of the minor axis co-vertices. However, Advanced Mathematical Concepts (2004, Glencoe/McGraw Hill) uses the term vertices for the end points of both axes.

WHOLE NUMBER. The generally accepted definition for this term is an integer greater than or equal to 0, but Funk & Wagnalls Practical Standard Dictionary (1935) has “a unit or a number composed of units; an integral number or integer: opposed to fraction and mixed number.

ZERO TO THE ZERO POWER. Most textbook writers either leave 00 undefined, or state that it is undefined. Others say it should be defined as 1. Euler argued for 00 = 1; Cauchy considered it undefined. Most calculators give “error” for this expression, although some give “1.” Wolframalpha.com gives “indeterminate.”

## Some Ambiguously Worded Test Questions at the High School Level

(Under Construction)
Item Comment
1. What is the sum of the measures of the exterior angles of a triangle? The item should specify whether there is only one angle at each vertex, as this is usually the intent.
2. If you roll a pair of dice, what is the probability you will get a 6? The item should say "a sum of six," since if you roll a 6 and a 3 you did get a 6.
3. What is the fourth decimal digit of e4? Are the digits to the left of the decimal point counted? All of the digits are decimal digits, since the number is written in decimal notation (base 10).

This page was created by Jeff Miller, a teacher at Gulf High School in New Port Richey, Florida. I also maintain these pages: Earliest Uses of Various Mathematical Symbols and Earliest Known Uses of Some of the Words of Mathematics. Assistance for this page was provided by Susan Hammer, Gaither High School in Tampa; Steve Wilson, Professor of Mathematics, Johnson County Community College; Michael Hamm; David Cantrell; Roman Arce; Don Johnson; Judy Ann Brown; and Nan Jackson.