1.1 Rational Number Arithmetic & Signed Numbers

The Real Number System Hierarchy

Quantitative reasoning on the ACCUPLACER begins with a clear understanding of the Real Number System ($\mathbb{R}$). 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 ($\mathbb{Q}$) and Irrational Numbers ($\mathbb{I}$).

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 $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$. In decimal form, rational numbers either terminate (e.g., $\frac{3}{4} = 0.75$) or repeat indefinitely in a periodic pattern (e.g., $\frac{2}{3} = 0.\overline{6}$). The rational numbers contain several nested subsets:

1. Natural (Counting) Numbers ($\mathbb{N}$): $\{1, 2, 3, 4, 5, \dots\}$. These are positive integers used for counting discrete objects.
2. Whole Numbers ($\mathbb{W}$): $\{0, 1, 2, 3, 4, \dots\}$. The natural numbers combined with zero ($0$).
3. Integers ($\mathbb{Z}$): $\{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$. The set of all whole numbers and their negative opposites. Integers contain no fractional or decimal parts.
4. Rational Numbers ($\mathbb{Q}$): All integers (since any integer $z$ can be written as $\frac{z}{1}$), terminating decimals (e.g., $-2.375 = -\frac{19}{8}$), and repeating decimals (e.g., $0.4545\dots = \frac{5}{11}$).

---

2. Irrational Numbers ($\mathbb{I}$)

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 $n$ is not a perfect square, its principal square root $\sqrt{n}$ is irrational (e.g., $\sqrt{2} \approx 1.41421356\dots$, $\sqrt{3}, \sqrt{5}, \sqrt{18} = 3\sqrt{2}$). Conversely, if $n$ is a perfect square, its root is rational (e.g., $\sqrt{49} = 7 = \frac{7}{1} \in \mathbb{Q}$, $\sqrt{\frac{16}{81}} = \frac{4}{9} \in \mathbb{Q}$).
• Transcendental Constants: Universal mathematical constants such as $\pi \approx 3.14159265\dots$ (the ratio of a circle's circumference to its diameter) and Euler's number $e \approx 2.71828182\dots$ are irrational.
• Crucial Exam Trap: The fraction $\frac{22}{7}$ and the decimal $3.14$ are rational approximations of $\pi$, but $\pi$ itself is strictly irrational.

---

Fundamental Arithmetic Properties & Zero Rules

Core Algebraic Properties

---

Special Properties of Zero

1. Zero in Addition & Subtraction: $a + 0 = a$ and $a - 0 = a$.
2. Zero Product Property: For any real number $a$, $a \cdot 0 = 0$. If $a \cdot b = 0$, then at least one factor must be zero ($a = 0$ or $b = 0$).
3. Zero in Numerator ($a \neq 0$): $\frac{0}{a} = 0$. Dividing zero items into $a$ equal groups results in $0$ items per group.
4. Zero in Denominator ($a \neq 0$): $\frac{a}{0} = \text{Undefined}$. Division by zero is mathematically impermissible because no real number $x$ can satisfy $x \cdot 0 = a$ when $a \neq 0$.
5. Indeterminate Form: $\frac{0}{0}$ is indeterminate, because any real number satisfies $x \cdot 0 = 0$.

---

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. $$(+7) + (+5) = +12$$ $$(-8) + (-6) = -(8 + 6) = -14$$
• 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. $$(-15) + (+9) = -(15 - 9) = -6$$ $$(+22) + (-7) = +(22 - 7) = +15$$

---

2. Subtraction of Signed Numbers

Subtraction is formally defined as adding the additive inverse (opposite) of the subtrahend:

$$a - b = a + (-b)$$

• Subtracting a Positive: $8 - 13 = 8 + (-13) = -5$
• Subtracting a Negative (Double Negative Rule): Subtracting a negative number is algebraically identical to adding its positive opposite: $$a - (-b) = a + b$$ $$-12 - (-19) = -12 + 19 = +7$$ $$-4.5 - (-8.2) = -4.5 + 8.2 = 3.7$$

---

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 $- \cdot - = +$). $$(-6) \cdot (-9) = +54$$ $$\frac{-48}{-4} = +12$$
• Unlike Signs: The product or quotient is always negative ($+ \cdot - = -$ and $- \cdot + = -$). $$(-7) \cdot (+8) = -56$$ $$\frac{84}{-12} = -7$$

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. $$(-2) \cdot (-3) \cdot (-4) \cdot (-5) = +120 \quad (4\text{ negative factors} \implies \text{positive})$$
• If the number of negative factors is odd, the final product is negative. $$(-2) \cdot (-3) \cdot (-4) \cdot (+5) = -120 \quad (3\text{ negative factors} \implies \text{negative})$$

---

Order of Operations (PEMDAS / GEMS)

When an arithmetic expression contains multiple operations, you must follow the universally established standard hierarchy:

1. 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 $\frac{\text{Numerator}}{\text{Denominator}}$.
2. E — Exponents & Roots: Evaluate all powers and radicals from left to right.
3. M / D — Multiplication & Division: Perform multiplication and division from left to right in the exact order they appear. Neither takes precedence over the other.
4. A / S — Addition & Subtraction: Perform addition and subtraction from left to right in the exact order they appear.

---

The Exponent Base Trap: $(-x)^n$ versus $-x^n$

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: $(-a)^n$ The negative sign is part of the base being multiplied repeatedly.

  $$(-4)^2 = (-4) \cdot (-4) = +16$$ $$(-3)^3 = (-3) \cdot (-3) \cdot (-3) = -27$$ $$(-2)^4 = (-2) \cdot (-2) \cdot (-2) \cdot (-2) = +16$$

• Case 2: No Parentheses: $-a^n$ According to the order of operations, exponentiation ($E$) precedes the implicit multiplication by $-1$ or unary negation. Thus, only $a$ is squared, and the negative sign is applied afterward.

  $$-4^2 = -(4^2) = -(4 \cdot 4) = -16$$ $$-3^4 = -(3^4) = -(81) = -81$$

$$\mathbf{(-4)^2 \neq -4^2} \quad (16 \neq -16)$$

---

Step-by-Step Worked Examples

Worked Example 1: Multi-Step Evaluation with Grouping & Exponents

Evaluate the arithmetic expression:

$$\frac{-4^2 + (-3)^3 - 2(-5 + 13)}{|3 - 11| \div 4 + 2^3}$$

Step 1: Simplify the Numerator

• Identify the operations in the numerator: $-4^2 + (-3)^3 - 2(-5 + 13)$.
• Resolve parentheses first: $(-5 + 13) = 8$.
• Evaluate exponents: $$-4^2 = -(4^2) = -16$$ $$(-3)^3 = (-3)(-3)(-3) = -27$$
• Substitute and perform multiplication: $$-16 + (-27) - 2(8) = -16 - 27 - 16$$
• Perform additions/subtractions from left to right: $$-16 - 27 = -43$$ $$-43 - 16 = -59$$ $$\text{Numerator} = -59$$

Step 2: Simplify the Denominator

• Identify the operations in the denominator: $|3 - 11| \div 4 + 2^3$.
• Resolve grouping (absolute value bar): $|3 - 11| = |-8| = 8$.
• Evaluate exponent: $2^3 = 8$.
• Perform division before addition: $8 \div 4 = 2$.
• Add: $2 + 8 = 10$. $$\text{Denominator} = 10$$

Step 3: Compute the Final Quotient

$$\frac{\text{Numerator}}{\text{Denominator}} = \frac{-59}{10} = -5.9$$

---

Worked Example 2: Real-World Signed Arithmetic (Atmospheric Research Station)

An alpine meteorological station at an elevation of $3,200\text{ meters}$ records an initial ambient temperature of $-18.4^\circ\text{C}$ at 3:00 AM. Over the next $5\text{ hours}$, the sun causes the temperature to rise at a constant rate of $3.2^\circ\text{C}$ per hour. At 8:00 AM, a severe blizzard front moves across the ridge, causing the temperature to drop by $14.9^\circ\text{C}$ over the next $3\text{ hours}$. What is the recorded temperature at 11:00 AM?

Step 1: Calculate the total temperature increase from 3:00 AM to 8:00 AM

$$\Delta T_1 = 5\text{ hours} \times (+3.2^\circ\text{C/hour}) = +16.0^\circ\text{C}$$

Step 2: Calculate the intermediate temperature at 8:00 AM

$$T_{\text{8 AM}} = T_{\text{initial}} + \Delta T_1 = -18.4 + 16.0 = -2.4^\circ\text{C}$$

Step 3: Apply the cold front temperature drop from 8:00 AM to 11:00 AM

$$T_{\text{11 AM}} = T_{\text{8 AM}} - 14.9 = -2.4 - 14.9 = -2.4 + (-14.9) = -17.3^\circ\text{C}$$

The temperature at 11:00 AM is $-17.3^\circ\text{C}$.

---

Common Pitfalls & ACCUPLACER Exam Traps

1. Evaluating Multiplication Before Division Unconditionally: In PEMDAS, multiplication and division have equal priority. In the expression $24 \div 6 \cdot 2$, you must evaluate left-to-right: $24 \div 6 = 4$, then $4 \cdot 2 = 8$. Evaluating $6 \cdot 2 = 12$ first to get $24 \div 12 = 2$ is an incorrect order violation.
2. Confusing Subtraction of a Negative with Simple Subtraction: Writing $-8 - (-5)$ as $-8 - 5 = -13$ is a severe error. The double negative converts to addition: $-8 + 5 = -3$.
3. Misclassifying Radicals as Irrational: Always check whether the radicand is a perfect square. $\sqrt{144} = 12$ is an integer and rational; only non-perfect square radicals like $\sqrt{140}$ are irrational.
4. Omitting Parentheses When Substituting Negative Values: If evaluating $x^2 - 3x$ for $x = -4$, write $(-4)^2 - 3(-4) = 16 + 12 = 28$. Omitting parentheses often leads students to write $-4^2 - 3(-4) = -16 + 12 = -4$, producing the incorrect answer.