Fractions

Comparing numbers allows us to evaluate quantities quickly. When comparing two natural numbers, apply two infallible rules:

1. Rule of Digit Count: The number with the greater number of digits is always strictly larger. For instance, $10,002$ (5 digits) is greater than $9,998$ (4 digits): $$10,002 > 9,998$$
2. Rule of Highest Place Value: If two numbers have the identical number of digits, compare their digits starting from the extreme left (the highest place value). The number with the greater leftmost digit is larger. If the leftmost digits are equal, compare the next digit to the right, and proceed sequentially until a difference is discovered.
   Example: Compare $75,642$ and $75,810$. Both have 5 digits. The ten-thousands digits ($7$) and thousands digits ($5$) match. Looking at the hundreds place: $6 < 8$, hence $$75,810 > 75,642$$

Forming the Greatest and Smallest Numbers from Digits

• Greatest Number: Arrange the given digits in descending order (largest to smallest).
  Given digits: $4, 7, 1, 9, 3 \to \mathbf{97,431}$.
• Smallest Number: Arrange the given digits in ascending order (smallest to largest).
  Given digits: $4, 7, 1, 9, 3 \to \mathbf{13,479}$.
• The Zero Constraint: If $0$ is among the digits, it cannot occupy the extreme leftmost position, because a leading zero reduces the digit count ($0357$ is a 3-digit number, not 4-digit). Hence, place the smallest non-zero digit first, followed immediately by $0$.
  Given digits: $6, 0, 8, 3 \to \text{Smallest 4-digit number} = \mathbf{3,068}$.

What happens when the position of digits in a number changes? Consider the number $395$. If we interchange the hundreds digit ($3$) and ones digit ($5$), we get $593$. Shifting digits completely changes the place value of the digits and drastically alters the magnitude of the number.

The Mathematical Progression of Base 10

In our base-10 Hindu-Arabic decimal system, adding $1$ to the greatest number of any given digit count produces the smallest number of the next digit count:

Expanded Form of Large Numbers

Every number can be written as the sum of the products of its digits and their respective place values:

The Indian system organizes large numbers into four primary periods: Ones, Thousands, Lakhs, and Crores.

Comma Placement Rules in the Indian System

1. The first comma is placed after 3 digits from the right (marking the end of the Ones period: Units, Tens, Hundreds).
2. Subsequent commas are placed after every 2 digits (Thousands period, Lakhs period, Crores period).

Writing & Reading: With commas: $54,32,09,816$.
In Words: Fifty-four crore thirty-two lakh nine thousand eight hundred sixteen.

The International Place Value System is utilized globally in science, trade, and international finance. It partitions numbers into periods of three digits each: Ones, Thousands, Millions, and Billions.

Comma Placement Rules in the International System

Commas are placed consistently after every 3 digits from the right: Units, Thousands, Millions, and Billions.

Writing & Reading: With commas: $543,209,816$.
In Words: Five hundred forty-three million two hundred nine thousand eight hundred sixteen.

BSEB and competitive scholarship examinations frequently test interconversions between Indian and International systems. The underlying mathematical bridges are:

Real-life calculations require converting between metric units of measurement. In the metric system, prefixes indicate powers of ten:

• Kilo (k): $1,000$ times ($10^3$)
• Hecto (h): $100$ times ($10^2$)
• Deca (da): $10$ times ($10^1$)
• Deci (d): $\frac{1}{10}$ of base ($10^{-1}$)
• Centi (c): $\frac{1}{100}$ of base ($10^{-2}$)
• Milli (m): $\frac{1}{1,000}$ of base ($10^{-3}$)

Essential Conversion Relationships

Rule of Conversion:
• Higher unit to lower unit $\to$ Multiply (e.g., $4.5\text{ km} = 4.5 \times 1,000 = 4,500\text{ m}$).
• Lower unit to higher unit $\to$ Divide (e.g., $2,500\text{ g} = \frac{2500}{1000} = 2.5\text{ kg}$).

Estimation means finding an approximate value close enough to the exact calculation to make rapid mental sense and verify computed answers.

Rounding Rules by Place Value

• Rounding to the Nearest Ten: Examine the digit in the ones place.
  • If ones digit is $0, 1, 2, 3, 4$ ($< 5$), round down (replace ones digit with $0$).
  • If ones digit is $5, 6, 7, 8, 9$ ($\ge 5$), round up (add $1$ to tens digit and replace ones digit with $0$).
  Examples: $73 \to 70$; $75 \to 80$; $286 \to 290$.
• Rounding to the Nearest Hundred: Examine the digit in the tens place.
  • If tens digit $< 5$, round down.
  • If tens digit $\ge 5$, round up (add $1$ to hundreds digit, replace tens and ones digits with $00$).
  Examples: $418 \to 400$; $782 \to 800$; $5,350 \to 5,400$.
• Rounding to the Nearest Thousand: Examine the digit in the hundreds place.
  Examples: $2,499 \to 2,000$; $2,500 \to 3,000$.

The General Rule of Estimation

When calculating sums, differences, or products, the General Rule states: Round off each number to its greatest (highest) place value before performing the arithmetic operation.

Example: Estimate the product $578 \times 161$.
• Highest place of $578$ is hundreds $\to$ round off to $600$.
• Highest place of $161$ is hundreds $\to$ round off to $200$.
• Estimated product $= 600 \times 200 = \mathbf{120,000}$.

Brackets help avoid confusion in arithmetic expressions. For example, $6 \times 10 + 2$ could be interpreted as $(6 \times 10) + 2 = 62$ or $6 \times (10 + 2) = 72$. Brackets define precise precedence.

Roman Numerals: The 7 Classical Symbols

The 4 Golden Rules of Roman Numerals

1. Repetition Rule: If a symbol is repeated, its value is added. Only $\text{I, X, C, M}$ can be repeated up to a maximum of three consecutive times ($\text{XXX} = 30$; $\text{CCCC}$ is invalid). $\mathbf{V, L, D}$ are never repeated.
2. Addition Rule: A symbol of smaller value written to the right of a symbol of greater value is added to it:
   $\text{VI} = 5 + 1 = 6$; $\text{XII} = 10 + 2 = 12$; $\text{LX} = 50 + 10 = 60$; $\text{CL} = 100 + 50 = 150$.
3. Subtraction Rule: A symbol of smaller value written to the left of a symbol of greater value is subtracted from it:
   $\text{IV} = 5 - 1 = 4$; $\text{IX} = 10 - 1 = 9$; $\text{XL} = 50 - 10 = 40$; $\text{XC} = 100 - 10 = 90$; $\text{CD} = 500 - 100 = 400$; $\text{CM} = 1000 - 100 = 900$.
4. Subtraction Restrictions:
   • $\mathbf{V, L, D}$ are never subtracted.
   • Symbol $\text{I}$ can be subtracted from $\text{V}$ and $\text{X}$ only.
   • Symbol $\text{X}$ can be subtracted from $\text{L}$ and $\text{C}$ only.
   • Symbol $\text{C}$ can be subtracted from $\text{D}$ and $\text{M}$ only.