GRE Prep Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized
for You
Track
Your Progress
Practice
Pays
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
Mathematical Conventions for the Quantitative Reasoning
[#permalink]
01 May 2018, 08:12
01 May 2018, 08:12
2
Source: OG 2nd Edition
Mathematical Conventions for the Quantitative Reasoning Measure of the GRE revised General Test
Numbers and Quantities
1. 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.
2. 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.
3. 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
4. Some other conventions involving numbers: one billion means 1,000,000,000, or 10^9(not as in some countries); one dozen means 12; the Greek letter π represents the ratio of the circumference of a circle to its diameter and is approximately 3.14.
5. 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, −36.5 rounded to the nearest integer is −37.
6. Repeating decimals are sometimes written with a bar over the digits that repeat, as in 25/12 = 2.0(83) ̅ an 1/7 = 0.(142857) ̅.
7. If r, s, and t are integers and then rs = t, 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, −7 is a factor of 35, 8 is a factor of -40 and the integer 4 has six factors: -4,-2,-1,1, 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 is a −2π is a multiple of π.
8. The least common multiple of two nonzero 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 positive integer that is a divisor of both a and b.
9. If an integer n is divided by a nonzero integer d resulting in a quotient q with remainder r, then n = qd + r,where 0 ≤ r < ⎪d⎪. Furthermore, r = 0 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 -17 is divided by 7, the quotient is -3 and the remainder is 4.
10. 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.
11. Odd and even integers are not necessarily positive; for example, -7 is odd, and −18 and 0 are even.
12. The integer 0 is neither positive nor negative.
Mathematical Expressions, Symbols, and Variables
1. 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 l , 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.
2. 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; and 1 ≤ x < π; T is the tens digits of a two-digit positive integer, so T is an integer from 1 to 9.
3. Standard mathematical symbols at the high school level are used. These include the arithmetic operations +,-,×,÷ though multiplication is usually denoted by juxtaposition, often with parentheses, e.g., 2y and (3)(4.5); and division is usually denoted with a horizontal fraction bar, e.g.\(\frac{w}{3}\)Sometimes mixed numbers or mixed fractions, are used, like \(4 \frac{3}{8}\) and\(-10 \frac{1}{2}\) These two numbers are equal to \(\frac{35}{8}\) and \(\frac{-21}{2}\),respectively. Exponents are also used, e.g.\(2^1^0\) = 1024, \(10^-^2\) = 1/100, and \(x^0\) = 1 for all nonzero numbers x.
4. Mathematical expressions are to be interpreted with respect to order of operations, which establishes which operations are performed before others in an expression. The order is as follows: parentheses; exponentiation; negation; multiplication and division (from left to right); addition and subtraction (from left to right). For example, the value of the expression 1 + 2 × 4 is 9, because the expression is evaluated by first multiplying 2 and 4 and then adding 1 to the result. Also, −32 means “the negative of ‘3 squared’ ” because exponentiation takes precedence over negation. Therefore, −\(3^2\) = -9, but (−3)^2 = 9 because parentheses takes precedence over exponentiation.
5. Here are examples of other standard symbols with their meanings:
x ≤ y; x is less than or equal to y
x ≠ y; x and y are not equal
x≈y; x and y are approximately equal
⎪x⎪ ; the absolute value of x
√x ; the nonnegative square root of x, where x ≥ 0
−√x ; the nonpositive square root of x, where x ≥ 0
n! ;the product of all positive integers less than or equal to n, where n is any positive integer and, as a special definition, 0! = 1.
L ǁ m ;lines L and m are parallel
L ⊥ m ;lines L and m are perpendicular
6. Because all numbers are assumed to be real, some expressions are not defined. For example, for every number x, the expression \(\frac{x}{0}\) is not defined; if x < 0, then √x is not defined; and \(0^0\) is not defined.
7. Sometimes special symbols or notation are introduced in a question. Here are two examples:
The operation ◊is defined for all integers r and s by r ◊ s = \(\frac{rs}{(1+r^2)}\)
The operation ~ is defined for all nonzero numbers x by ~ x = \(\frac{- 1}{x}\)
8. Sometimes juxtaposition of letters does not denote multiplication, as in “consider a three-digit integer denoted by XYZ, where X, Y, and Z are digits.” The meaning is taken from the context.
9.Standard function notation is used in the test. For example, “the function g is defined for all x ≥ 0 by g(x)=2x+√x.” If the domain of a function f is not given explicitly, it is assumed to be the set of all real numbers x for which f (x) is a real number. If f and g are two functions, then the composition of g with f is denoted by g(f (x)).
Geometry
1.In questions involving geometry, the conventions of plane (or Euclidean) geometry are followed, including the assumption that the sum of the measures of the interior angles of a triangle is 180 degrees.
2.Lines are assumed to be “straight” lines that extend in both directions without end.
3.Angle measures are in degrees and are assumed to be positive and less than or equal to 360 degrees.
4.When a square, circle, polygon, or other closed geometric figure is described in words but not shown, the figure is assumed to enclose a convex region. It is also assumed that such a closed geometric figure is not just a single point or a line segment.
5.The phrase area of a rectangle means the area of the region enclosed by the rectangle. The same terminology applies to circles, triangles, and other closed figures.
6.The distance between a point and a line is the length of the perpendicular line segment from the point to the line, which is the shortest distance between the point and the line. Similarly, the distance between two parallel lines is the distance between a point on one line and the other line.
7.In a geometric context, the phrase similar triangles (or other figures) means that the figures have the same shape. See the Geometry section of the Math Review for further explanation of the terms similar and congruent.
Mathematical Conventions for the Quantitative Reasoning Measure of the GRE revised General Test
Numbers and Quantities
1. 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.
2. 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.
3. 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
4. Some other conventions involving numbers: one billion means 1,000,000,000, or 10^9(not as in some countries); one dozen means 12; the Greek letter π represents the ratio of the circumference of a circle to its diameter and is approximately 3.14.
5. 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, −36.5 rounded to the nearest integer is −37.
6. Repeating decimals are sometimes written with a bar over the digits that repeat, as in 25/12 = 2.0(83) ̅ an 1/7 = 0.(142857) ̅.
7. If r, s, and t are integers and then rs = t, 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, −7 is a factor of 35, 8 is a factor of -40 and the integer 4 has six factors: -4,-2,-1,1, 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 is a −2π is a multiple of π.
8. The least common multiple of two nonzero 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 positive integer that is a divisor of both a and b.
9. If an integer n is divided by a nonzero integer d resulting in a quotient q with remainder r, then n = qd + r,where 0 ≤ r < ⎪d⎪. Furthermore, r = 0 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 -17 is divided by 7, the quotient is -3 and the remainder is 4.
10. 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.
11. Odd and even integers are not necessarily positive; for example, -7 is odd, and −18 and 0 are even.
12. The integer 0 is neither positive nor negative.
Mathematical Expressions, Symbols, and Variables
1. 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 l , 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.
2. 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; and 1 ≤ x < π; T is the tens digits of a two-digit positive integer, so T is an integer from 1 to 9.
3. Standard mathematical symbols at the high school level are used. These include the arithmetic operations +,-,×,÷ though multiplication is usually denoted by juxtaposition, often with parentheses, e.g., 2y and (3)(4.5); and division is usually denoted with a horizontal fraction bar, e.g.\(\frac{w}{3}\)Sometimes mixed numbers or mixed fractions, are used, like \(4 \frac{3}{8}\) and\(-10 \frac{1}{2}\) These two numbers are equal to \(\frac{35}{8}\) and \(\frac{-21}{2}\),respectively. Exponents are also used, e.g.\(2^1^0\) = 1024, \(10^-^2\) = 1/100, and \(x^0\) = 1 for all nonzero numbers x.
4. Mathematical expressions are to be interpreted with respect to order of operations, which establishes which operations are performed before others in an expression. The order is as follows: parentheses; exponentiation; negation; multiplication and division (from left to right); addition and subtraction (from left to right). For example, the value of the expression 1 + 2 × 4 is 9, because the expression is evaluated by first multiplying 2 and 4 and then adding 1 to the result. Also, −32 means “the negative of ‘3 squared’ ” because exponentiation takes precedence over negation. Therefore, −\(3^2\) = -9, but (−3)^2 = 9 because parentheses takes precedence over exponentiation.
5. Here are examples of other standard symbols with their meanings:
x ≤ y; x is less than or equal to y
x ≠ y; x and y are not equal
x≈y; x and y are approximately equal
⎪x⎪ ; the absolute value of x
√x ; the nonnegative square root of x, where x ≥ 0
−√x ; the nonpositive square root of x, where x ≥ 0
n! ;the product of all positive integers less than or equal to n, where n is any positive integer and, as a special definition, 0! = 1.
L ǁ m ;lines L and m are parallel
L ⊥ m ;lines L and m are perpendicular
6. Because all numbers are assumed to be real, some expressions are not defined. For example, for every number x, the expression \(\frac{x}{0}\) is not defined; if x < 0, then √x is not defined; and \(0^0\) is not defined.
7. Sometimes special symbols or notation are introduced in a question. Here are two examples:
The operation ◊is defined for all integers r and s by r ◊ s = \(\frac{rs}{(1+r^2)}\)
The operation ~ is defined for all nonzero numbers x by ~ x = \(\frac{- 1}{x}\)
8. Sometimes juxtaposition of letters does not denote multiplication, as in “consider a three-digit integer denoted by XYZ, where X, Y, and Z are digits.” The meaning is taken from the context.
9.Standard function notation is used in the test. For example, “the function g is defined for all x ≥ 0 by g(x)=2x+√x.” If the domain of a function f is not given explicitly, it is assumed to be the set of all real numbers x for which f (x) is a real number. If f and g are two functions, then the composition of g with f is denoted by g(f (x)).
Geometry
1.In questions involving geometry, the conventions of plane (or Euclidean) geometry are followed, including the assumption that the sum of the measures of the interior angles of a triangle is 180 degrees.
2.Lines are assumed to be “straight” lines that extend in both directions without end.
3.Angle measures are in degrees and are assumed to be positive and less than or equal to 360 degrees.
4.When a square, circle, polygon, or other closed geometric figure is described in words but not shown, the figure is assumed to enclose a convex region. It is also assumed that such a closed geometric figure is not just a single point or a line segment.
5.The phrase area of a rectangle means the area of the region enclosed by the rectangle. The same terminology applies to circles, triangles, and other closed figures.
6.The distance between a point and a line is the length of the perpendicular line segment from the point to the line, which is the shortest distance between the point and the line. Similarly, the distance between two parallel lines is the distance between a point on one line and the other line.
7.In a geometric context, the phrase similar triangles (or other figures) means that the figures have the same shape. See the Geometry section of the Math Review for further explanation of the terms similar and congruent.
gmatclubot
Mathematical Conventions for the Quantitative Reasoning [#permalink]
01 May 2018, 08:12
Moderators:
|
|
||
