2.3 Unit Conversion, Scientific Notation, Numerical Precision & Absolute Value

2.3 Unit Conversion, Scientific Notation, Numerical Precision & Absolute Value

Applied measurement, scientific representation of scale, numerical precision, and distance metrics via absolute values represent high-yield topics on the CLEP College Mathematics examination. These mathematical tools bridge theoretical number properties and practical quantitative problem-solving.

---

1. Dimensional Analysis & Multi-Step Unit Conversion

Dimensional analysis (or the factor-label method) utilizes unit conversion factors expressed as ratios equal to $1$ (unity) to systematically cancel undesired units while preserving quantitative value.

$$\text{Given Quantity in Unit A} \times \left( \frac{\text{Conversion Unit B}}{\text{Equivalent Unit A}} \right) = \text{Desired Quantity in Unit B}$$

Metric System Prefixes & Powers of 10

The metric system is a base-10 positional system defined by standard SI prefixes:

US Customary Measurement Standards

Worked Example 1: Multi-Step Rate Conversion

Problem: Convert a highway velocity of $75\text{ miles per hour}$ into $\text{feet per second}$.

$$\text{Rate} = \frac{75\text{ miles}}{1\text{ hour}} \times \left( \frac{5,280\text{ feet}}{1\text{ mile}} \right) \times \left( \frac{1\text{ hour}}{60\text{ minutes}} \right) \times \left( \frac{1\text{ minute}}{60\text{ seconds}} \right)$$ $$\text{Rate} = \frac{75 \times 5,280\text{ feet}}{3,600\text{ seconds}} = \frac{396,000\text{ ft}}{3,600\text{ s}} = 110\text{ ft/s}$$

---

2. Scientific Notation & Exponent Arithmetic

Standard Form Definition

A real number is expressed in scientific notation when written as:

$$a \times 10^k$$

where $a$ is a real decimal coefficient satisfying $1 \le |a| < 10$, and $k$ is an integer exponent ($k \in \mathbb{Z}$).

• Large numbers ($|x| \ge 10$): $k > 0$, representing the number of places the decimal point shifted left ($6,450,000 = 6.45 \times 10^6$).
• Small decimals ($0 < |x| < 1$): $k < 0$, representing the number of places the decimal point shifted right ($0.000382 = 3.82 \times 10^{-4}$).

Arithmetic with Scientific Notation

1. Multiplication: Multiply coefficients and add exponents: $$(a \times 10^m) \times (b \times 10^n) = (a \cdot b) \times 10^{m+n}$$ Re-normalize if $a \cdot b \ge 10$.
2. Division: Divide coefficients and subtract exponents: $$\frac{a \times 10^m}{b \times 10^n} = \left( \frac{a}{b} \right) \times 10^{m-n}$$ Re-normalize if $\frac{a}{b} < 1$.

Worked Example 2: Scientific Notation Division

Problem: Evaluate the expression and write the result in proper scientific notation:

$$\mathcal{Q} = \frac{(4.8 \times 10^7) \times (3.0 \times 10^{-2})}{1.6 \times 10^{-4}}$$

1. Group coefficients and powers of 10: $$\mathcal{Q} = \left( \frac{4.8 \times 3.0}{1.6} \right) \times \left( \frac{10^7 \times 10^{-2}}{10^{-4}} \right)$$
2. Calculate coefficients: $\frac{4.8}{1.6} = 3.0 \implies 3.0 \times 3.0 = 9.0$.
3. Calculate powers of 10: $10^{7 + (-2) - (-4)} = 10^{5 + 4} = 10^9$.
4. Combine: $\mathcal{Q} = 9.0 \times 10^9$.

TI-30XS MultiView [EE] Key Operation

On the official CLEP on-screen TI-30XS MultiView calculator:

• Scientific notation is entered using the dedicated [EE] function via [2nd] followed by the [x⁻¹] key.
• For example, $9.0 \times 10^9$ is entered directly as 9.0EE9.
• Crucial Advantage: The EE notation binds the mantissa and power of $10$ into a single algebraic entity. Entering 1.2 * 10^6 / 3.0 * 10^4 without parentheses causes the calculator to evaluate $(1.2 \times 10^6 / 3.0) \times 10^4 = 4 \times 10^9$. Entering 1.2EE6 / 3.0EE4 guarantees the exact evaluation $\frac{1.2 \times 10^6}{3.0 \times 10^4} = 40$.

---

3. Numerical Precision, Estimation & Significant Digits

Rules for Determining Significant Figures

1. All non-zero digits are significant: $48.7$ has $3$ sig figs.
2. Captive zeros (between non-zero digits) are significant: $5,008$ has $4$ sig figs.
3. Leading zeros (placeholders to the left) are NEVER significant: $0.0042$ has $2$ sig figs ($4$ and $2$).
4. Trailing zeros to the right of a decimal point are significant: $72.00$ has $4$ sig figs.
5. Trailing zeros in whole numbers without decimals are ambiguous: $4,500$ has $2$ sig figs unless written as $4.500 \times 10^3$ ($4$ sig figs) or $4500.$ ($4$ sig figs).

Rounding Standards

• Identify the rounding place value.
• If the immediate subsequent digit is $\ge 5$, round up by adding $1$ to the target digit.
• If the immediate subsequent digit is $< 5$, leave the target digit unchanged.

---

4. Absolute Value: Definition, Geometry & Equations

Formal Algebraic Definition

The absolute value of a real number $x$, denoted $|x|$, is defined piecewise as:

$$|x| = \begin{cases} x & \text{if } x \ge 0 \\[4pt] -x & \text{if } x < 0 \end{cases}$$

Geometric Interpretation on the Number Line

Geometrically, $|x|$ represents the undirected distance between the point $x$ and the origin $0$ on the real number line. More generally, $|a - b|$ represents the distance between points $a$ and $b$.

<---|-------|-------|-------|-------|-------|-------|--->
   -3      -2      -1       0       1       2       3    
    |<--------------------->| (Distance = |-3| = 3)      
                            |<------------->| (Distance = |2| = 2)

Fundamental Properties of Absolute Value

• Non-negativity: $|a| \ge 0$ for all $a \in \mathbb{R}$, with $|a| = 0 \iff a = 0$.
• Symmetry / Negation: $|-a| = |a|$.
• Multiplication Rule: $|a \cdot b| = |a| \cdot |b|$.
• Division Rule: $\left| \frac{a}{b} \right| = \frac{|a|}{|b|}$ for $b \neq 0$.
• The Triangle Inequality: $|a + b| \le |a| + |b|$ for all $a, b \in \mathbb{R}$. (Equality $|a + b| = |a| + |b|$ holds if and only if $a$ and $b$ have the same sign or at least one is zero).

Solving Linear Absolute Value Equations

To solve an equation of the form $|ax + b| = c$:

1. If $c > 0$: The equation splits into a two-part disjunction: $$ax + b = c \quad \text{OR} \quad ax + b = -c$$
2. If $c = 0$: Exactly one unique solution: $ax + b = 0 \implies x = -\frac{b}{a}$.
3. If $c < 0$: An absolute value can never be negative. The solution set is strictly empty ($\emptyset$).

Worked Example 3: Solving Absolute Value Equations

Problem: Find all real solutions to the equation $3|2x - 9| - 7 = 14$.

1. Isolate the absolute value expression: $$3|2x - 9| = 14 + 7 = 21 \implies |2x - 9| = \frac{21}{3} = 7$$
2. Split into two linear branches: $$\begin{aligned} 2x - 9 &= 7 & \text{or} & & 2x - 9 &= -7 \\[4pt] 2x &= 16 & & & 2x &= 2 \\[4pt] x &= 8 & & & x &= 1 \end{aligned}$$
3. Verify solutions in original equation:
   • For $x = 8$: $3|2(8) - 9| - 7 = 3|16 - 9| - 7 = 3(7) - 7 = 21 - 7 = 14$ (Valid).
   • For $x = 1$: $3|2(1) - 9| - 7 = 3|2 - 9| - 7 = 3|-7| - 7 = 3(7) - 7 = 14$ (Valid).
4. The complete solution set is $\{1, 8\}$, with a sum of solutions equal to $1 + 8 = 9$.