2.1 Whole Numbers, Place Value & Roman Numerals

Whole Numbers & Base-10 Place Value System

Core Concept: Our standard numbering system is a base-10 positional numeral system. The value of any digit depends entirely on its position or 'place' within the number, with each place representing a power of 10. The set of whole numbers, denoted by $\mathbb{W}$, consists of zero and all positive counting numbers: $\mathbb{W} = {0, 1, 2, 3, 4, \dots}$.

1. Understanding Place Value vs. Face Value

To master numerical operations, a fundamental distinction must be drawn between a digit's face value and its place value:

• Face Value: The actual intrinsic value of a single digit regardless of its position. For example, in the number $745$, the face value of the digit $7$ is simply $7$.
• Place Value: The value assigned to a digit based on its position relative to the decimal point. In $745$, the digit $7$ resides in the hundreds place, so its place value is $7 \times 100 = 700$.

2. The Comprehensive Base-10 Place Value Chart

The table below details the positional values from millions down to thousandths:

Step-by-Step Worked Example: Digit Value Extraction

Problem: Consider the number $6,482,915.203$.

1. What is the place value of the digit $8$?
2. What is the value of the digit $3$?

Solution:

1. Locate digit $8$: It is 5 places to the left of the decimal point, corresponding to the ten-thousands place ($10,000$). Thus, its value is $8 \times 10,000 = 80,000$.
2. Locate digit $3$: It is 3 places to the right of the decimal point, corresponding to the thousandths place ($0.001$). Thus, its value is $3 \times 0.001 = 0.003$ or $\frac{3}{1000}$.

---

3. Expressing Numbers in Expanded Form

Numbers can be written in three major formats: Standard Form, Word Form, and Expanded Form.

Formats Comparison Table

When converting a standard number into expanded form, write each non-zero digit multiplied by its place value, joined by addition signs.

---

4. Comparing and Ordering Numbers

To compare two numbers (whole numbers or decimals), align their place values vertically and compare corresponding digits starting from the leftmost (highest-value) position.

Step-by-Step Comparison Rules:

1. Align by Decimal Point: Line up the numbers vertically so that ones line up with ones, tenths with tenths, etc.
2. Compare Left-to-Right: Start at the highest place value position.
3. Identify the First Difference: The number with the larger digit at the first point of difference is the greater number.
4. Use Inequality Symbols:
   • $<$ means less than (e.g., $450 < 480$)
   • $>$ means greater than (e.g., $5.64 > 5.604$)
   • $=$ means equal to (e.g., $0.5 = 0.50$)

Worked Example: Ordering Decimals

Problem: Arrange the following decimals in ascending order (smallest to largest): $5.604, 5.640, 5.064, 5.600$.

Solution:

• Step 1: Compare ones place: All have $5$.
• Step 2: Compare tenths place: $5.064$ has $0$; $5.604, 5.640, 5.600$ all have $6$. Therefore, $5.064$ is the smallest.
• Step 3: Compare hundredths place for remaining numbers:
  • $5.600$ has $0$
  • $5.604$ has $0$
  • $5.640$ has $4$ (largest)
• Step 4: Compare thousandths place for $5.600$ vs $5.604$:
  • $5.600$ has $0$
  • $5.604$ has $4$
• Ascending Order: $5.064 < 5.600 < 5.604 < 5.640$.

Rounding Numbers & Roman Numeral Systems

1. Rules for Rounding Whole Numbers and Decimals

Rounding simplifies numbers to make them easier to work with while keeping their value close to the original. The standard rounding algorithm follows a strict 4-step process:

Rounding Algorithm:

1. Identify the Target Place Value: Underline or locate the digit in the place you are rounding to (e.g., tens, hundreds, thousands, hundredths).
2. Examine the Decider Digit: Look at the digit immediately to the right of the target digit.
3. Apply the Rounding Rule:
   • If the decider digit is $5$ or greater ($5, 6, 7, 8, 9$), round UP: add $1$ to the target digit.
   • If the decider digit is less than $5$ ($0, 1, 2, 3, 4$), round DOWN: keep the target digit unchanged.
4. Adjust Remaining Digits:
   • For whole numbers, replace all digits to the right of the target digit with zeroes.
   • For decimals, drop all digits to the right of the target decimal place.

Step-by-Step Worked Examples

---

2. Roman Numerals: Principles and Conversion Rules

Roman numerals are an ancient non-positional additive-subtractive numeral system developed in ancient Rome. Unlike the base-10 system, Roman numerals do not use place value or a symbol for zero.

The Seven Core Roman Symbols

Four Governing Rules of Roman Numerals

1. Rule of Repetition:
   • The symbols I, X, C, and M may be repeated up to a maximum of three times consecutively to indicate addition (e.g., $\text{III} = 3$, $\text{XXX} = 30$, $\text{CCC} = 300$, $\text{MMM} = 3,000$).
   • The symbols V, L, and D can never be repeated.
2. Rule of Addition:
   • When a smaller or equal symbol is written to the right of a larger symbol, add its value to the larger symbol.
   • Examples: $\text{VI} = 5 + 1 = 6$; $\text{CL} = 100 + 50 = 150$; $\text{LXII} = 50 + 10 + 1 + 1 = 62$.
3. Rule of Subtraction (Subtractive Notation):
   • When a smaller symbol is written to the left of a larger symbol, subtract the smaller from the larger.
   • Subtraction is strictly restricted to six standard pairs:
     • $\text{IV} = 5 - 1 = 4$
     • $\text{IX} = 10 - 1 = 9$
     • $\text{XL} = 50 - 10 = 40$
     • $\text{XC} = 100 - 10 = 90$
     • $\text{CD} = 500 - 100 = 400$
     • $\text{CM} = 1,000 - 100 = 900$
   • Important Constraint: I can only precede V or X. X can only precede L or C. C can only precede D or M. Symbols V, L, and D are never subtracted.
4. Rule of Breakdown:
   • To convert a large Hindu-Arabic number to Roman numerals, decompose the number into expanded form (thousands, hundreds, tens, ones) and translate each part independently.

Comprehensive Conversion Table & Worked Examples

Step-by-Step Conversion Walkthrough

Example 1: Convert Roman Numeral DXLVII to Hindu-Arabic

• Identify segments: $\text{D} = 500$, $\text{XL} = 40$, $\text{V} = 5$, $\text{II} = 2$.
• Sum segments: $500 + 40 + 5 + 2 = 547$.

Example 2: Convert Hindu-Arabic 1,492 to Roman Numeral

• Decompose: $1000 + 400 + 90 + 2$.
• Translate each: $1000 = \text{M}$, $400 = \text{CD}$, $90 = \text{XC}$, $2 = \text{II}$.
• Combine: MCDXCII.