Number System - Knowing our Numbers: Estimation and Roman Numerals

The Indian System of Numeration uses periods to group large numbers: Ones, Thousands, Lakhs, and Crores. Commas are placed after the first three digits from the right (hundreds place) and then after every two digits. For example, $7,34,50,619$ is read as Seven crore, thirty-four lakh, fifty thousand, six hundred nineteen. Visualize a place-value chart where columns are grouped in sets of 2, except for the rightmost group of 3.

The International System of Numeration groups digits into periods of three: Ones, Thousands, and Millions. Commas are placed every three digits from the right. For example, $73,450,619$ is read as Seventy-three million, four hundred fifty thousand, six hundred nineteen. Visualize a chart where every period (Millions, Thousands, Ones) contains exactly three sub-columns (Hundreds, Tens, Units).

Estimation (Rounding Off) is the process of finding a number close enough to the exact value to make calculations easier. To round to the nearest ten, look at the ones digit: if it is $5$ or more, round up; if less than $5$, round down. Visualize a number line where $48$ is closer to $50$ than $40$, so it 'slides' toward $50$.

Estimating Outcomes involves rounding numbers before performing operations like addition, subtraction, or multiplication. The General Rule is to round each factor to its greatest place. For example, to estimate $5,290 + 17,986$, round them to the nearest thousand to get $5,000 + 18,000 = 23,000$.

Roman Numerals use seven letters to represent numbers: $I=1, V=5, X=10, L=50, C=100, D=500, M=1000$. There is no symbol for zero. Imagine a sequence of symbols where the position determines whether you add or subtract values.

Rules for Writing Roman Numerals: (1) Repetition of a symbol means addition (e.g., $III = 3$), but symbols can be repeated a maximum of three times. (2) $V, L, D$ are never repeated. (3) A smaller symbol to the right of a larger one is added ($XI = 10+1=11$). (4) A smaller symbol to the left of a larger one is subtracted ($IX = 10-1=9$).

Subtraction Constraints in Roman Numerals: The symbol $I$ can be subtracted only from $V$ and $X$. The symbol $X$ can be subtracted from $L, M$, and $C$. The symbol $C$ can be subtracted from $D$ and $M$. The symbols $V, L$, and $D$ are never subtracted.

Brackets help in simplifying expressions by clearly defining the order of operations. Using brackets like $102 \times 103 = (100 + 2) \times (100 + 3)$ allows for the application of distributive laws to make mental math easier.