gre_math_conventions

® www.ets.org Mathematical Conventions for the Quantitative Reasoning Measure of the GRE® revised General Test www.ets.org Overview The mathematical symbols and terminology used in the Quantitative Reasoning measure of the test are conventional at the high school level, and most of these appear in the Math Review. Whenever nonstandard or special notation or terminology is used in a test question, it is explicitly introduced in the question. However, there are some assumptions about numbers and geometric figures that are particular to the test. These assumptions appear in the test at the beginning of the Quantitative Reasoning sections, and they are elaborated below. Also, some notation and terminology, while standard at the high school level in many countries, may be different from those used in other countries or from those used at higher or lower levels of mathematics. Such notation and terminology are clarified below. Because it is impossible to ascertain which notation and terminology should be clarified for an individual test taker, more material than necessary may be included. Finally, there are some guidelines for how certain information given in test questions should be interpreted and used in the context of answering the questions—information such as certain words, phrases, quantities, mathematical expressions, and displays of data. These guidelines appear at the end. Copyright © 2009 by Educational Testing Service. All rights reserved. ETS, the ETS logo, LISTENING. LEARNING. LEADING. and GRE are registered trademarks of Educational Testing Service (ETS). Numbers and quantities • All numbers used in the test questions are real numbers. In particular, integers and both rational and irrational numbers are to be considered, but imaginary numbers are not. This is the main assumption regarding numbers. Also, all quantities are real numbers, although quantities may involve units of measurement. • Numbers are expressed in base 10 unless otherwise noted, using the 10 digits 0 through 9 and a period to the right of the ones digit, or units digit, for the decimal point. Also, in numbers that are 1,000 or greater, commas are used to separate groups of three digits to the left of the decimal point. • When a positive integer is described by the number of its digits, e.g., a two-digit integer, the digits that are counted include the ones digit and all the digits further to the left, where the left-most digit is not 0. For example, 5,000 is a four-digit integer, whereas 031 is not considered to be a three-digit integer. • Some other conventions involving numbers: one billion means 1,000,000,000, or (not as in some countries); one dozen means 12; the Greek letter 910 1210 , p represents the ratio of the circumference of a circle to its diameter and is approximately 3.14. • When a positive number is to be rounded to a certain decimal place and the number is halfway between the two nearest possibilities, the number should be rounded to the greater possibility. For example, 23.5 rounded to the nearest integer is 24, and 123.985 rounded to the nearest 0.01 is 123.99. When the number to be rounded is negative, the number should be rounded to the lesser possibility. For example, rounded to the nearest integer is 36.5- 37.- • Repeating decimals are sometimes written with a bar over the digits that repeat, as in 25 2.08312 = and 1 0.142857.7 = • If r, s, and t are integers and then r and s are factors, or divisors, of t; also, t is a multiple of r (and of s) and t is divisible by r (and by s). The factors of an integer include positive and negative integers. For example, is a factor of 35, 8 is a factor of and the integer 4 has six factors: 1, 2, and 4. The terms factor, divisor, and divisible are used only when r, s, and t are integers. However, the term multiple can be used with any real numbers s and t provided r is an integer. For example, 1.2 is a multiple of 0.4, and ,rs t= 7- 40,- 4,- 2,- 1,- 2p- is a multiple of p . • The least common multiple of two integers a and b is the least positive integer that is a multiple of both a and b. The greatest common divisor (or greatest common factor) of a and b is the greatest integer that is a divisor of both a and b. • When an integer n is divided by a nonzero integer d resulting in a quotient q with remainder r, then where ,n qd r= + 0 . Furthermore, if and only if n is a multiple of d. For example, when 20 is divided by 7, the quotient is 2 and the remainder is 6; when 21 is divided by 7, the quotient is 3 and the remainder is 0; and when is divided by 7, the quotient is and the remainder is 4. r d£ < 0=r 17- 3- • A prime number is an integer greater than 1 that has only two positive divisors: 1 and itself. The first five prime numbers are 2, 3, 5, 7, and 11. A composite number is an integer greater than 1 that is not a prime number. The first five composite numbers are 4, 6, 8, 9, and 10. • Odd and even integers are not necessarily positive; for example, is odd, and and 0 are even. 7- 18- • The integer 0 is neither positive nor negative. - 1 - Mathematical expressions, symbols, and variables • As is common in algebra, italic letters like x are used to denote numbers, constants, and variables. Letters are also used to label various objects, such as line point P, function f, set S, list T, event E, random variable X, Brand X, City Y, and Company Z. The meaning of a letter is determined by the context. ,A • When numbers, constants, or variables are given, their possible values are all real numbers unless otherwise restricted. It is common to restrict the possible values in various ways. Here are some examples: n is a nonzero integer; 1 ;p£ o The measures of angles BAD and BDA are equal. o The measure of angle DBC is less than x degrees. o The area of triangle ABD is greater than the area of triangle DBC. o Angle SRT is a right angle. o Line m is parallel to line AC. - 4 - Coordinate systems • Coordinate systems, such as xy-planes and number lines, are drawn to scale. Therefore, you can read, estimate, or compare quantities in such figures by sight or by measurement, including geometric figures that appear in coordinate systems. • The positive direction of a number line is to the right. • As in geometry, distances in a coordinate system are nonnegative. • The rectangular coordinate plane, or rectangular coordinate system, commonly known as the xy- plane, is shown below. The x-axis and y-axis intersect at the origin O, and they partition the plane into four quadrants. Each point in the xy-plane has coordinates ( , )x y that give its location with respect to the axes; for example, the point is located 2 units to the right of the y-axis and 8 units below the x-axis. The units on the x-axis have the same length as the units on the y-axis, unless otherwise noted. (2, 8-P ) • Intermediate grid lines or tick marks in a coordinate system are evenly spaced unless otherwise noted. • The term x-intercept refers to the x-coordinate of the point at which a graph in the xy-plane intersects the x-axis; it does not refer to the point itself. The term y-intercept is used analogously. Sets, lists, and sequences • Sets of numbers or other elements appear in some questions. Some sets are infinite, such as the set of integers; other sets are finite and may have all of their elements listed within curly brackets, such as the set { }2, 4, 6, 8 . When the elements of a set are given, repetitions are not counted as additional elements and the order of the elements is not relevant. Elements are also called members. A set with one or more members is called nonempty; there is a set with no members, called the empty set and denoted by If A and B are sets, then the intersection of A and B, denoted by .∆ ,«A B is the set of elements that are in both A and B, and the union of A and B, denoted by ,»A B is the set of elements that are in either A or B or both. If all of the elements in A are also in B, then A is a subset of B. By convention, the empty set is a subset of every set. If A and B have no elements in common, they are called disjoint sets or mutually exclusive sets. - 5 - • Lists of numbers or other elements are also used in the test. When the elements of a list are given, repetitions are counted as additional elements and the order of the elements is relevant. For example, the list 3, 1, 2, 3, 3 contains five numbers, and the first, fourth, and last numbers in the list are each 3. • The terms data set and set of data are not sets in the mathematical sense given above. Rather they refer to a list of data because there may be repetitions in the data, and if there are repetitions, they would be relevant. • Sequences are lists that often have an infinite number of elements, or terms. The terms of a sequence are often represented by a fixed letter along with a subscript that indicates the order of a term in the sequence. For example, represents an infinite sequence in which the first term is the second term is and more generally, the nth term is for every positive integer n. Sometimes the nth term of a sequence is given by a formula, such as Sometimes the first few terms of a sequence are given explicitly, as in the following sequence of consecutive even negative integers: 1 2 3, , , . . . , , . . .na a a a 1,a 2 ,a na 2 1nnb = + . 2, 4, 6, 8, 10, . . . .- - - - - • Sets of consecutive integers are sometimes described by indicating the first and last integer, as in “the integers from 0 to 9, inclusive.” This phrase refers to 10 integers, with or without “inclusive” at the end. Thus, the phrase “during the years from 1985 to 2005” refers to 21 years. Data and statistics • Numerical data are sometimes given in lists and sometimes displayed in other ways, such as in tables, bar graphs, or circle graphs. Various statistics, or measures of data, appear in questions: measures of central tendency—mean, median, and mode; measures of position—quartiles and percentiles; and measures of dispersion—standard deviation, range, and interquartile range. • The term average is used in two ways, with and without the qualification “(arithmetic mean).” For a list of data, the average (arithmetic mean) of the data is the sum of the data divided by the number of data. The term average does not refer to either median or mode in the test. Without the qualification of “arithmetic mean,” average can refer to a rate or the ratio of one quantity to another, as in “average number of miles per hour” or “average weight per truckload.” • When mean is used in the context of data, it means arithmetic mean. • The median of an odd number of data is the middle number when the data are listed in increasing order; the median of an even number of data is the arithmetic mean of the two middle numbers when the data are listed in increasing order. • For a list of data, the mode of the data is the most frequently occurring number in the list. Thus, there may be more than one mode for a list of data. • For data listed in increasing order, the first quartile, second quartile, and third quartile of the data are three numbers that divide the data into four groups that are roughly equal in size. The first group of numbers is from the least number up to the first quartile. The second group is from the first quartile up to the second quartile, which is also the median of the data. The third group is from the second quartile up to the third quartile, and the fourth group is from the third quartile up to the greatest number. Note that the four groups themselves are sometimes referred to as quartiles—first quartile, second quartile, third quartile, and fourth quartile. The latter usage is clarified by the word “in” as in the phrase “the cow’s weight is in the third quartile of the herd.” • For data listed in increasing order, the percentiles of the data are 99 numbers that divide the data into 100 groups that are roughly equal in size. The 25th percentile equals the first quartile; the 50th percentile equals the second quartile, or median; and the 75th percentile equals the third quartile. • For a list of data, where the arithmetic mean is denoted by m, the standard deviation of the data refers to the nonnegative square root of the mean of the squared differences between m and each of - 6 - the data. This statistic is also known as the population standard deviation and is not to be confused with the sample standard deviation. • For a list of data, the range of the data is the greatest number in the list minus the least number. The interquartile range of the data is the third quartile minus the first quartile. Data distributions and probability distributions • Some questions display data in frequency distributions, where discrete data values are repeated with various frequencies, or where p