3.3 Integers, Absolute Value, & Signed Rational Number Operations

3.3 Integers, Absolute Value, & Signed Rational Number Operations

In mathematics and workplace applications, numbers often have direction as well as magnitude. Elevations exist above and below sea level, temperatures rise above and drop below freezing, and financial balance sheets record positive credits and negative debits. Understanding signed integers, rational numbers, and absolute value on the TABE 13&14 Mathematics test ensures you can evaluate algebraic expressions and solve real-world problems accurately.

---

1. The Real Number System: Integers and Rational Numbers

The real number system is organized into a hierarchy of subsets:

1. Natural Numbers (Counting Numbers, $\mathbb{N}$): ${1, 2, 3, 4, 5, \dots}$
2. Whole Numbers ($\mathbb{W}$): ${0, 1, 2, 3, 4, \dots}$ (Natural numbers including $0$).
3. Integers ($\mathbb{Z}$): ${\dots, -3, -2, -1, 0, 1, 2, 3, \dots}$ (Whole numbers and their negative opposites).
4. Rational Numbers ($\mathbb{Q}$): Any number that can be expressed as the quotient $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. Rational numbers include all integers (since $n = \frac{n}{1}$), terminating decimals (e.g., $0.75 = \frac{3}{4}$), and repeating decimals (e.g., $0.\overline{3} = \frac{1}{3}$).

+-------------------------------------------------------------------------+
|                        Rational Numbers (Q)                             |
|   -3/4,   0.25,   -12.8,   5/9,   0.333...                              |
|  +-------------------------------------------------------------------+  |
|  |                         Integers (Z)                              |  |
|  |   -15,   -7,   -1                                                 |  |
|  |  +-------------------------------------------------------------+  |  |
|  |  |                      Whole Numbers (W)                      |  |  |
|  |  |   0                                                         |  |  |
|  |  |  +-------------------------------------------------------+  |  |  |
|  |  |  |                 Natural Numbers (N)                   |  |  |  |
|  |  |  |   1,   2,   3,   4,   100,   ...                      |  |  |  |
|  |  |  +-------------------------------------------------------+  |  |  |
|  |  +-------------------------------------------------------------+  |  |
|  +-------------------------------------------------------------------+  |
+-------------------------------------------------------------------------+

The Directed Number Line

The number line is an infinite continuous horizontal axis where:

• Origin ($0$): Neutral center that is neither positive nor negative.
• Positive Numbers ($x > 0$): Located to the right of $0$, increasing in value as you move rightward.
• Negative Numbers ($x < 0$): Located to the left of $0$, decreasing in value as you move leftward.
• Comparison Rule: For any two numbers $a$ and $b$, $a < b$ if $a$ lies to the left of $b$ on the number line (e.g., $-7 < -2$ because $-7$ is further left than $-2$).

---

2. Absolute Value as Geometric Distance

The absolute value of a real number $x$, denoted by $|x|$, is defined as the geometric distance along the number line between $x$ and the origin ($0$).

$\text{Piecewise Definition: } |x| = \begin{cases} x & \text{if } x \ge 0 \\ -x & \text{if } x < 0 \end{cases}$

Key Properties of Absolute Value:

1. Non-Negativity: Distance is never negative: $|x| \ge 0$ for all $x \in \mathbb{R}$.
2. Opposites: A number and its opposite have identical absolute values: $|-a| = |a|$. (e.g., $|-9| = 9$ and $|9| = 9$).
3. Distance Between Two Points: The distance between any two points $a$ and $b$ on a number line is given by $d = |a - b| = |b - a|$.
4. Grouping Precedence: In the order of operations (PEMDAS), absolute value bars act like parentheses. You must evaluate the expression inside the bars completely before taking the absolute value.

Worked Example: Absolute Value in Complex Expressions

Problem: Evaluate the expression $|-24 + 9| - 2 \times |3 - 11|$.

1. Evaluate inside first absolute value: $-24 + 9 = -15 \implies |-15| = 15$.
2. Evaluate inside second absolute value: $3 - 11 = -8 \implies |-8| = 8$.
3. Substitute values back into the expression: $15 - 2 \times 8$.
4. Perform multiplication before subtraction: $2 \times 8 = 16$.
5. Subtract: $15 - 16 = -1$.

---

3. Addition of Signed Numbers

When adding signed numbers, use the sign relationship rules:

Rule 1: Same Signs (Both Positive or Both Negative)

• Procedure: Add the absolute values (magnitudes) of the numbers and attach the common sign to the sum.
• $(+8) + (+14) = +(8 + 14) = +22$
• $(-9) + (-13) = -(9 + 13) = -22$

Rule 2: Different Signs (One Positive and One Negative)

• Procedure: Subtract the smaller absolute value from the larger absolute value. The final sum takes the sign of the number with the larger absolute value.
• $(-18) + (+11) = -(18 - 11) = -7$ (since $|-18| > |11|$, sign is negative)
• $(+25) + (-14) = +(25 - 14) = +11$ (since $|25| > |-14|$, sign is positive)

---

4. Subtraction of Signed Numbers: The Additive Inverse

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

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

To subtract any signed number, use the Keep-Change-Change rule:

1. Keep the first number ($a$) with its original sign.
2. Change the subtraction sign ($-$) to an addition sign ($+$).
3. Change the second number ($b$) to its opposite (invert its sign).
4. Follow the standard integer addition rules.

Worked Example: Multi-Step Subtraction

Problem: Simplify $-15 - (-22) - 18 + (-7)$.

1. Convert subtractions to additions of opposites: $-15 + 22 + (-18) + (-7)$
2. Group positive and negative terms:
   • Positives: $22$
   • Negatives: $(-15) + (-18) + (-7) = -40$
3. Combine: $22 + (-40) = -(40 - 22) = -18$.

---

5. Multiplication and Division of Signed Numbers

Multiplication and division follow identical sign rules:

Two-Factor Sign Rules:

• Same Signs $\implies$ Positive Result: $(+) \times (+) = (+) \qquad \text{and} \qquad (-) \times (-) = (+)$ $(+) \div (+) = (+) \qquad \text{and} \qquad (-) \div (-) = (+)$
• Different Signs $\implies$ Negative Result: $(+) \times (-) = (-) \qquad \text{and} \qquad (-) \times (+) = (-)$ $(+) \div (-) = (-) \qquad \text{and} \qquad (-) \div (+) = (-)$

General Rule for Multiple Factors (Counting Negatives):

When multiplying or dividing a chain of non-zero numbers:

• If the count of negative signs is even ($0, 2, 4, 6, \dots$), the result is positive.
• If the count of negative signs is odd ($1, 3, 5, 7, \dots$), the result is negative.

$\text{Example: } (-2) \times (-3) \times (-4) \times (-5) = +120 \quad (4 \text{ negatives} \implies \text{positive})$ $\text{Example: } (-2) \times (+3) \times (-4) \times (-5) = -120 \quad (3 \text{ negatives} \implies \text{negative})$

---

6. Signed Rational Numbers: Fractions and Decimals

Negative Fractions Placement

A negative sign in a fraction can be written in front of the fraction, in the numerator, or in the denominator without changing its value:

$-\frac{a}{b} = \frac{-a}{b} = \frac{a}{-b} \quad (b \neq 0)$

Worked Example 1: Operating with Signed Fractions

Problem: Compute $-\frac{3}{4} - \left(-\frac{5}{6}\right)$.

1. Convert subtraction to addition of opposite: $-\frac{3}{4} + \frac{5}{6}$
2. Find common denominator $\text{LCD}(4, 6) = 12$: $-\frac{9}{12} + \frac{10}{12} = \frac{-9 + 10}{12} = \frac{1}{12}$

Worked Example 2: Operating with Signed Decimals

Problem: Evaluate $-14.75 - (-8.40) + (-3.65)$.

1. Convert subtraction to addition: $-14.75 + 8.40 + (-3.65)$.
2. Combine negative terms: $(-14.75) + (-3.65) = -18.40$.
3. Add positive term: $-18.40 + 8.40 = -10.00$.

---

7. Real-World Applications: Elevation, Temperature, and Finance

Elevation Differential Example:

Death Valley's Badwater Basin is at an elevation of $-282\text{ feet}$, while the nearby Telescope Peak reaches $+11,049\text{ feet}$. What is the vertical difference in elevation between them?

$\text{Difference} = 11,049 - (-282) = 11,049 + 282 = 11,331\text{ feet}$

---

TABE Exam Traps & Common Errors

[!CAUTION] Trap 1: The Double Negative Confusion Subtraction of a negative is NOT negative! Two adjacent negative signs without a number between them cancel to positive: $a - (-b) = a + b$.

[!WARNING] Trap 2: Treating Absolute Value Bars as Mere Parentheses Absolute value bars do NOT merely hold terms. A negative outside the absolute value remains negative: $-|-7| = -(7) = -7$. However, $|-7| = +7$.

[!NOTE] Trap 3: Exponents on Negative Bases Be careful with parentheses! $(-4)^2 = (-4) \times (-4) = +16$, whereas $-4^2 = -(4 \times 4) = -16$ because the exponent applies only to $4$, not the negative sign.