Study guide
1.1 Rational Number Arithmetic & Signed Numbers
ACCUPLACER Quantitative Reasoning, Algebra, and Statistics Exam
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1 Chapter 1: Rational Numbers & Operations
1.1 Rational Number Arithmetic & Signed Numbers 1.2 Fractions, Decimals, & Mixed Numbers Operations 1.3 Absolute Value, Number Line, & Estimation
2 Chapter 2: Ratios, Proportions & Rates
2.1 Ratios, Unit Rates, & Dimensional Analysis 2.2 Proportions & Direct/Inverse Variation 2.3 Percent Applications, Markup, Discount, & Simple Interest
3 Chapter 3: Exponents & Scientific Notation
3.1 Integer Exponents & Laws of Exponents 3.2 Scientific Notation & Operations
4 Chapter 4: Algebraic Expressions
4.1 Evaluating & Translating Algebraic Expressions 4.2 Simplifying, Combining Like Terms, & Factoring
5 Chapter 5: Linear Equations & Inequalities
5.1 Solving Single-Variable Linear Equations 5.2 Solving & Graphing Linear Inequalities 5.3 Systems of Linear Equations
6 Chapter 6: Linear Applications & Graphing
6.1 Slope & Rate of Change 6.2 Forms of Linear Equations & Graphing 6.3 Linear Modeling & Inequality Graphs
7 Chapter 7: Probability & Set Theory
7.1 Fundamental Probability Concepts 7.2 Set Theory, Operations, & Venn Diagrams
8 Chapter 8: Descriptive Statistics & Data Displays
8.1 Measures of Center: Mean, Median, Mode, & Weighted Average 8.2 Measures of Spread: Range, IQR, Variance, & Standard Deviation 8.3 Data Displays: Histograms, Box Plots, & Scatter Plots
9 Chapter 9: Geometry Concepts & Measurement
9.1 2D Geometry: Perimeter, Area, & Circles 9.2 3D Geometry: Surface Area & Volume of Solids 9.3 Triangles, Angles, & the Pythagorean Theorem
1.1 Rational Number Arithmetic & Signed Numbers
Key Takeaways Explain This Topic AI
The real number system is organized hierarchically: Natural numbers (N) ⊂ Whole numbers (W) ⊂ Integers (Z) ⊂ Rational numbers (Q) ⊂ Real numbers (R), alongside Irrational numbers (I).
Division by zero is mathematically undefined, while zero divided by any non-zero real number is always zero.
In signed arithmetic, subtracting a negative is equivalent to adding its positive opposite (a - (-b) = a + b), and a product of factors is positive if the count of negative factors is even, and negative if odd.
In order of operations, exponentiation applies strictly to its immediate base: (-4)^2 = 16, whereas -4^2 = -(4^2) = -16.
Horizontal fraction bars, radicals, and absolute value bars function as implicit grouping symbols that must be completely simplified before performing outer operations.
Last updated: August 2026
The Real Number System Hierarchy
Quantitative reasoning on the ACCUPLACER begins with a clear understanding of the Real Number System ( R \mathbb{R} R ). Every number tested on the Quantitative Reasoning, Algebra, and Statistics (QAS) exam belongs to a specific mathematical set defined by its structural properties. The real number system is partitioned into two mutually exclusive subsets: Rational Numbers ( Q \mathbb{Q} Q ) and Irrational Numbers ( I \mathbb{I} I ). Real Numbers (R)
- ├── Rational Numbers (Q) (Can be written as a/b, b ≠ 0)
- │ └── Integers (Z) {..., -3, -2, -1, 0, 1, 2, 3, ...}
- │ └── Whole Numbers (W) {0, 1, 2, 3, ...}
- │ └── Natural Numbers (N) {1, 2, 3, ...}
- └── Irrational Numbers (I) (Non-terminating, non-repeating decimals: π, e, √2, √7)
- 1. The Nested Subsets of Rational Numbers
A rational number is any number that can be expressed as the quotient a b \frac{a}{b} b a , where a a a and b b b are integers and b ≠ 0 b \neq 0 b = 0 . In decimal form, rational numbers either terminate (e.g., 3 4 = 0.75 \frac{3}{4} = 0.75 4 3 = 0.75 ) or repeat indefinitely in a periodic pattern (e.g., 2 3 = 0. 6 ‾ \frac{2}{3} = 0.\overline{6} 3 2 = 0. 6 ). The rational numbers contain several nested subsets:
Natural (Counting) Numbers ( N \mathbb{N} N ) : { 1 , 2 , 3 , 4 , 5 , … } \{1, 2, 3, 4, 5, \dots\} { 1 , 2 , 3 , 4 , 5 , … } . These are positive integers used for counting discrete objects.
Whole Numbers ( W \mathbb{W} W ) : { 0 , 1 , 2 , 3 , 4 , … } \{0, 1, 2, 3, 4, \dots\} { 0 , 1 , 2 , 3 , 4 , … } . The natural numbers combined with zero ( 0 0 0 ).
Integers ( Z \mathbb{Z} Z ) : { … , − 3 , − 2 , − 1 , 0 , 1 , 2 , 3 , … } \{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\} { … , − 3 , − 2 , − 1 , 0 , 1 , 2 , 3 , … } . The set of all whole numbers and their negative opposites. Integers contain no fractional or decimal parts.
Rational Numbers ( Q \mathbb{Q} Q ) : All integers (since any integer z z z can be written as z 1 \frac{z}{1} 1 z ), terminating decimals (e.g., − 2.375 = − 19 8 -2.375 = -\frac{19}{8} − 2.375 = − 8 19 ), and repeating decimals (e.g., 0.4545 ⋯ = 5 11 0.4545\dots = \frac{5}{11} 0.4545 ⋯ = 11 5 ).
2. Irrational Numbers ( I \mathbb{I} I )
An irrational number cannot be written as a ratio of two integers. When expressed in decimal form, irrational numbers are non-terminating and non-repeating decimals.
Radicals of Non-Perfect Squares : If a positive integer n n n is not a perfect square, its principal square root n \sqrt{n} n is irrational (e.g., 2 ≈ 1.41421356 … \sqrt{2} \approx 1.41421356\dots 2 ≈ 1.41421356 … , 3 , 5 , 18 = 3 2 \sqrt{3}, \sqrt{5}, \sqrt{18} = 3\sqrt{2} 3 , 5 , 18 = 3 2 ). Conversely, if n n n is a perfect square, its root is rational (e.g., 49 = 7 = 7 1 ∈ Q \sqrt{49} = 7 = \frac{7}{1} \in \mathbb{Q} 49 = 7 = 1 7 ∈ Q , 16 81 = 4 9 ∈ Q \sqrt{\frac{16}{81}} = \frac{4}{9} \in \mathbb{Q} 81 16 = 9 4 ∈ Q ).
Transcendental Constants : Universal mathematical constants such as π ≈ 3.14159265 … \pi \approx 3.14159265\dots π ≈ 3.14159265 … (the ratio of a circle's circumference to its diameter) and Euler's number e ≈ 2.71828182 … e \approx 2.71828182\dots e ≈ 2.71828182 … are irrational.
Crucial Exam Trap : The fraction 22 7 \frac{22}{7} 7 22 and the decimal 3.14 3.14 3.14 are rational approximations of π \pi π , but π \pi π itself is strictly irrational.
- Fundamental Arithmetic Properties & Zero Rules
- Core Algebraic Properties
- Property Addition Formulation Multiplication Formulation
- Commutative a + b = b + a a + b = b + a a + b = b + a a ⋅ b = b ⋅ a a \cdot b = b \cdot a a ⋅ b = b ⋅ a
Associative ( a + b ) + c = a + ( b + c ) (a + b) + c = a + (b + c) ( a + b ) + c = a + ( b + c ) ( a ⋅ b ) ⋅ c = a ⋅ ( b ⋅ c ) (a \cdot b) \cdot c = a \cdot (b \cdot c) ( a ⋅ b ) ⋅ c = a ⋅ ( b ⋅ c )
Identity a + 0 = a a + 0 = a a + 0 = a ( 0 0 0 is additive identity) a ⋅ 1 = a a \cdot 1 = a a ⋅ 1 = a ( 1 1 1 is multiplicative identity)
Inverse a + ( − a ) = 0 a + (-a) = 0 a + ( − a ) = 0 (Additive inverse) a ⋅ 1 a = 1 a \cdot \frac{1}{a} = 1 a ⋅ a 1 = 1 ( a ≠ 0 a \neq 0 a = 0 , Multiplicative inverse/reciprocal)
Distributive a ( b + c ) = a b + a c a(b + c) = ab + ac a ( b + c ) = ab + a c a ( b − c ) = a b − a c a(b - c) = ab - ac a ( b − c ) = ab − a c
Special Properties of Zero
Zero in Addition & Subtraction : a + 0 = a a + 0 = a a + 0 = a and a − 0 = a a - 0 = a a − 0 = a .
Zero Product Property : For any real number a a a , a ⋅ 0 = 0 a \cdot 0 = 0 a ⋅ 0 = 0 . If a ⋅ b = 0 a \cdot b = 0 a ⋅ b = 0 , then at least one factor must be zero ( a = 0 a = 0 a = 0 or b = 0 b = 0 b = 0 ).
Zero in Numerator ( a ≠ 0 a \neq 0 a = 0 ) : 0 a = 0 \frac{0}{a} = 0 a 0 = 0 . Dividing zero items into a a a equal groups results in 0 0 0 items per group.
Zero in Denominator ( a ≠ 0 a \neq 0 a = 0 ) : a 0 = Undefined \frac{a}{0} = \text{Undefined} 0 a = Undefined . Division by zero is mathematically impermissible because no real number x x x can satisfy x ⋅ 0 = a x \cdot 0 = a x ⋅ 0 = a when a ≠ 0 a \neq 0 a = 0 .
Indeterminate Form : 0 0 \frac{0}{0} 0 0 is indeterminate, because any real number satisfies x ⋅ 0 = 0 x \cdot 0 = 0 x ⋅ 0 = 0 .
Signed Arithmetic Operations
Mastery of positive and negative numbers is essential across every domain of the ACCUPLACER exam.
1. Addition of Signed Numbers
Same Signs : Add the absolute values (magnitudes) of the numbers and attach the common sign. ( + 7 ) + ( + 5 ) = + 12 (+7) + (+5) = +12 ( + 7 ) + ( + 5 ) = + 12 ( − 8 ) + ( − 6 ) = − ( 8 + 6 ) = − 14 (-8) + (-6) = -(8 + 6) = -14 ( − 8 ) + ( − 6 ) = − ( 8 + 6 ) = − 14
Different Signs : Find the difference between the absolute values (larger magnitude minus smaller magnitude) and attach the sign of the number with the larger absolute value. ( − 15 ) + ( + 9 ) = − ( 15 − 9 ) = − 6 (-15) + (+9) = -(15 - 9) = -6 ( − 15 ) + ( + 9 ) = − ( 15 − 9 ) = − 6 ( + 22 ) + ( − 7 ) = + ( 22 − 7 ) = + 15 (+22) + (-7) = +(22 - 7) = +15 ( + 22 ) + ( − 7 ) = + ( 22 − 7 ) = + 15
2. Subtraction of Signed Numbers
Subtraction is formally defined as adding the additive inverse (opposite) of the subtrahend: a − b = a + ( − b ) a - b = a + (-b) a − b = a + ( − b )
Subtracting a Positive : 8 − 13 = 8 + ( − 13 ) = − 5 8 - 13 = 8 + (-13) = -5 8 − 13 = 8 + ( − 13 ) = − 5
Subtracting a Negative (Double Negative Rule) : Subtracting a negative number is algebraically identical to adding its positive opposite: a − ( − b ) = a + b a - (-b) = a + b a − ( − b ) = a + b − 12 − ( − 19 ) = − 12 + 19 = + 7 -12 - (-19) = -12 + 19 = +7 − 12 − ( − 19 ) = − 12 + 19 = + 7 − 4.5 − ( − 8.2 ) = − 4.5 + 8.2 = 3.7 -4.5 - (-8.2) = -4.5 + 8.2 = 3.7 − 4.5 − ( − 8.2 ) = − 4.5 + 8.2 = 3.7
3. Multiplication and Division of Signed Numbers
When multiplying or dividing two real numbers:
Like Signs : The product or quotient is always positive ( + ⋅ + = + + \cdot + = + + ⋅ + = + and − ⋅ − = + - \cdot - = + − ⋅ − = + ). ( − 6 ) ⋅ ( − 9 ) = + 54 (-6) \cdot (-9) = +54 ( − 6 ) ⋅ ( − 9 ) = + 54 − 48 − 4 = + 12 \frac{-48}{-4} = +12 − 4 − 48 = + 12
Unlike Signs : The product or quotient is always negative ( + ⋅ − = − + \cdot - = - + ⋅ − = − and − ⋅ + = − - \cdot + = - − ⋅ + = − ). ( − 7 ) ⋅ ( + 8 ) = − 56 (-7) \cdot (+8) = -56 ( − 7 ) ⋅ ( + 8 ) = − 56 84 − 12 = − 7 \frac{84}{-12} = -7 − 12 84 = − 7
The Multi-Factor Sign Rule
For a product of non-zero factors:
If the number of negative factors is even , the final product is positive . ( − 2 ) ⋅ ( − 3 ) ⋅ ( − 4 ) ⋅ ( − 5 ) = + 120 ( 4 negative factors ⟹ positive ) (-2) \cdot (-3) \cdot (-4) \cdot (-5) = +120 \quad (4\text{ negative factors} \implies \text{positive}) ( − 2 ) ⋅ ( − 3 ) ⋅ ( − 4 ) ⋅ ( − 5 ) = + 120 ( 4 negative factors ⟹ positive )
If the number of negative factors is odd , the final product is negative . ( − 2 ) ⋅ ( − 3 ) ⋅ ( − 4 ) ⋅ ( + 5 ) = − 120 ( 3 negative factors ⟹ negative ) (-2) \cdot (-3) \cdot (-4) \cdot (+5) = -120 \quad (3\text{ negative factors} \implies \text{negative}) ( − 2 ) ⋅ ( − 3 ) ⋅ ( − 4 ) ⋅ ( + 5 ) = − 120 ( 3 negative factors ⟹ negative )
Order of Operations (PEMDAS / GEMS)
When an arithmetic expression contains multiple operations, you must follow the universally established standard hierarchy:
G / P — Grouping Symbols / Parentheses : Evaluate expressions inside innermost grouping symbols first: parentheses ( ) ( ) ( ) , brackets [ ] [ ] [ ] , braces { } \{ \} { } , absolute value bars ∣ … ∣ | \dots | ∣ … ∣ , radical signs … \sqrt{\dots} … , and horizontal fraction bars Numerator Denominator \frac{\text{Numerator}}{\text{Denominator}} Denominator Numerator .
E — Exponents & Roots : Evaluate all powers and radicals from left to right.
M / D — Multiplication & Division : Perform multiplication and division from left to right in the exact order they appear. Neither takes precedence over the other.
A / S — Addition & Subtraction : Perform addition and subtraction from left to right in the exact order they appear.
The Exponent Base Trap: ( − x ) n (-x)^n ( − x ) n versus − x n -x^n − x n
One of the most frequent point-loss traps on the ACCUPLACER test is failing to identify the exact base to which an exponent applies:
Case 1: Base Enclosed in Parentheses: ( − a ) n (-a)^n ( − a ) n The negative sign is part of the base being multiplied repeatedly. ( − 4 ) 2 = ( − 4 ) ⋅ ( − 4 ) = + 16 (-4)^2 = (-4) \cdot (-4) = +16 ( − 4 ) 2 = ( − 4 ) ⋅ ( − 4 ) = + 16 ( − 3 ) 3 = ( − 3 ) ⋅ ( − 3 ) ⋅ ( − 3 ) = − 27 (-3)^3 = (-3) \cdot (-3) \cdot (-3) = -27 ( − 3 ) 3 = ( − 3 ) ⋅ ( − 3 ) ⋅ ( − 3 ) = − 27 ( − 2 ) 4 = ( − 2 ) ⋅ ( − 2 ) ⋅ ( − 2 ) ⋅ ( − 2 ) = + 16 (-2)^4 = (-2) \cdot (-2) \cdot (-2) \cdot (-2) = +16 ( − 2 ) 4 = ( − 2 ) ⋅ ( − 2 ) ⋅ ( − 2 ) ⋅ ( − 2 ) = + 16
Case 2: No Parentheses: − a n -a^n − a n According to the order of operations, exponentiation ( E E E ) precedes the implicit multiplication by − 1 -1 − 1 or unary negation. Thus, only a a a is squared, and the negative sign is applied afterward. − 4 2 = − ( 4 2 ) = − ( 4 ⋅ 4 ) = − 16 -4^2 = -(4^2) = -(4 \cdot 4) = -16 − 4 2 = − ( 4 2 ) = − ( 4 ⋅ 4 ) = − 16 − 3 4 = − ( 3 4 ) = − ( 81 ) = − 81 -3^4 = -(3^4) = -(81) = -81 − 3 4 = − ( 3 4 ) = − ( 81 ) = − 81 ( − 4 ) 2 ≠ − 4 2 ( 16 ≠ − 16 ) \mathbf{(-4)^2 \neq -4^2} \quad (16 \neq -16) ( − 4 ) 2 = − 4 2 ( 16 = − 16 )
- Step-by-Step Worked Examples
- Worked Example 1: Multi-Step Evaluation with Grouping & Exponents
Evaluate the arithmetic expression: − 4 2 + ( − 3 ) 3 − 2 ( − 5 + 13 ) ∣ 3 − 11 ∣ ÷ 4 + 2 3 \frac{-4^2 + (-3)^3 - 2(-5 + 13)}{|3 - 11| \div 4 + 2^3} ∣3 − 11∣ ÷ 4 + 2 3 − 4 2 + ( − 3 ) 3 − 2 ( − 5 + 13 )
Step 1: Simplify the Numerator
Identify the operations in the numerator: − 4 2 + ( − 3 ) 3 − 2 ( − 5 + 13 ) -4^2 + (-3)^3 - 2(-5 + 13) − 4 2 + ( − 3 ) 3 − 2 ( − 5 + 13 ) .
Resolve parentheses first: ( − 5 + 13 ) = 8 (-5 + 13) = 8 ( − 5 + 13 ) = 8 .
Evaluate exponents: − 4 2 = − ( 4 2 ) = − 16 -4^2 = -(4^2) = -16 − 4 2 = − ( 4 2 ) = − 16 ( − 3 ) 3 = ( − 3 ) ( − 3 ) ( − 3 ) = − 27 (-3)^3 = (-3)(-3)(-3) = -27 ( − 3 ) 3 = ( − 3 ) ( − 3 ) ( − 3 ) = − 27
Substitute and perform multiplication: − 16 + ( − 27 ) − 2 ( 8 ) = − 16 − 27 − 16 -16 + (-27) - 2(8) = -16 - 27 - 16 − 16 + ( − 27 ) − 2 ( 8 ) = − 16 − 27 − 16
Perform additions/subtractions from left to right: − 16 − 27 = − 43 -16 - 27 = -43 − 16 − 27 = − 43 − 43 − 16 = − 59 -43 - 16 = -59 − 43 − 16 = − 59 Numerator = − 59 \text{Numerator} = -59 Numerator = − 59
Step 2: Simplify the Denominator
Identify the operations in the denominator: ∣ 3 − 11 ∣ ÷ 4 + 2 3 |3 - 11| \div 4 + 2^3 ∣3 − 11∣ ÷ 4 + 2 3 .
Resolve grouping (absolute value bar): ∣ 3 − 11 ∣ = ∣ − 8 ∣ = 8 |3 - 11| = |-8| = 8 ∣3 − 11∣ = ∣ − 8∣ = 8 .
Evaluate exponent: 2 3 = 8 2^3 = 8 2 3 = 8 .
Perform division before addition: 8 ÷ 4 = 2 8 \div 4 = 2 8 ÷ 4 = 2 .
Add: 2 + 8 = 10 2 + 8 = 10 2 + 8 = 10 . Denominator = 10 \text{Denominator} = 10 Denominator = 10
Step 3: Compute the Final Quotient Numerator Denominator = − 59 10 = − 5.9 \frac{\text{Numerator}}{\text{Denominator}} = \frac{-59}{10} = -5.9 Denominator Numerator = 10 − 59 = − 5.9
Worked Example 2: Real-World Signed Arithmetic (Atmospheric Research Station)
An alpine meteorological station at an elevation of 3 , 200 meters 3,200\text{ meters} 3 , 200 meters records an initial ambient temperature of − 18.4 ∘ C -18.4^\circ\text{C} − 18. 4 ∘ C at 3:00 AM. Over the next 5 hours 5\text{ hours} 5 hours , the sun causes the temperature to rise at a constant rate of 3.2 ∘ C 3.2^\circ\text{C} 3. 2 ∘ C per hour. At 8:00 AM, a severe blizzard front moves across the ridge, causing the temperature to drop by 14.9 ∘ C 14.9^\circ\text{C} 14. 9 ∘ C over the next 3 hours 3\text{ hours} 3 hours . What is the recorded temperature at 11:00 AM?
Step 1: Calculate the total temperature increase from 3:00 AM to 8:00 AM Δ T 1 = 5 hours × ( + 3.2 ∘ C/hour ) = + 16.0 ∘ C \Delta T_1 = 5\text{ hours} \times (+3.2^\circ\text{C/hour}) = +16.0^\circ\text{C} Δ T 1 = 5 hours × ( + 3. 2 ∘ C/hour ) = + 16. 0 ∘ C
Step 2: Calculate the intermediate temperature at 8:00 AM T 8 AM = T initial + Δ T 1 = − 18.4 + 16.0 = − 2.4 ∘ C T_{\text{8 AM}} = T_{\text{initial}} + \Delta T_1 = -18.4 + 16.0 = -2.4^\circ\text{C} T 8 AM = T initial + Δ T 1 = − 18.4 + 16.0 = − 2. 4 ∘ C
Step 3: Apply the cold front temperature drop from 8:00 AM to 11:00 AM T 11 AM = T 8 AM − 14.9 = − 2.4 − 14.9 = − 2.4 + ( − 14.9 ) = − 17.3 ∘ C T_{\text{11 AM}} = T_{\text{8 AM}} - 14.9 = -2.4 - 14.9 = -2.4 + (-14.9) = -17.3^\circ\text{C} T 11 AM = T 8 AM − 14.9 = − 2.4 − 14.9 = − 2.4 + ( − 14.9 ) = − 17. 3 ∘ C
The temperature at 11:00 AM is − 17.3 ∘ C -17.3^\circ\text{C} − 17. 3 ∘ C .
Common Pitfalls & ACCUPLACER Exam Traps
Evaluating Multiplication Before Division Unconditionally : In PEMDAS, multiplication and division have equal priority. In the expression 24 ÷ 6 ⋅ 2 24 \div 6 \cdot 2 24 ÷ 6 ⋅ 2 , you must evaluate left-to-right: 24 ÷ 6 = 4 24 \div 6 = 4 24 ÷ 6 = 4 , then 4 ⋅ 2 = 8 4 \cdot 2 = 8 4 ⋅ 2 = 8 . Evaluating 6 ⋅ 2 = 12 6 \cdot 2 = 12 6 ⋅ 2 = 12 first to get 24 ÷ 12 = 2 24 \div 12 = 2 24 ÷ 12 = 2 is an incorrect order violation.
Confusing Subtraction of a Negative with Simple Subtraction : Writing − 8 − ( − 5 ) -8 - (-5) − 8 − ( − 5 ) as − 8 − 5 = − 13 -8 - 5 = -13 − 8 − 5 = − 13 is a severe error. The double negative converts to addition: − 8 + 5 = − 3 -8 + 5 = -3 − 8 + 5 = − 3 .
Misclassifying Radicals as Irrational : Always check whether the radicand is a perfect square. 144 = 12 \sqrt{144} = 12 144 = 12 is an integer and rational; only non-perfect square radicals like 140 \sqrt{140} 140 are irrational.
Omitting Parentheses When Substituting Negative Values : If evaluating x 2 − 3 x x^2 - 3x x 2 − 3 x for x = − 4 x = -4 x = − 4 , write ( − 4 ) 2 − 3 ( − 4 ) = 16 + 12 = 28 (-4)^2 - 3(-4) = 16 + 12 = 28 ( − 4 ) 2 − 3 ( − 4 ) = 16 + 12 = 28 . Omitting parentheses often leads students to write − 4 2 − 3 ( − 4 ) = − 16 + 12 = − 4 -4^2 - 3(-4) = -16 + 12 = -4 − 4 2 − 3 ( − 4 ) = − 16 + 12 = − 4 , producing the incorrect answer.