Number System - Indian and International System of Numeration

Face Value and Place Value: The face value of a digit is the digit itself, regardless of its position in a number. For example, in the number $4,582$, the face value of $5$ is $5$. The place value is the product of the face value and the value of its place. For the same number, the place value of $5$ is $5 \times 100 = 500$ because it is in the hundreds place.

Indian System of Numeration: This system groups digits into periods of 'Ones', 'Thousands', 'Lakhs', and 'Crores'. Visually, commas are placed after the first three digits from the right and then after every two digits. For example, $987654321$ is written as $98,76,54,321$. The places are: Ones, Tens, Hundreds (Ones period); Thousands, Ten-Thousands (Thousands period); Lakhs, Ten-Lakhs (Lakhs period); Crores, Ten-Crores (Crores period).

International System of Numeration: This system groups digits into periods of 'Ones', 'Thousands', and 'Millions'. Visually, commas are placed after every three digits from the right. For example, $987654321$ is written as $987,654,321$. Each period has three places: Ones, Tens, Hundreds. For instance, the Thousands period contains Thousands, Ten-Thousands, and Hundred-Thousands.

Comparison of Systems: It is important to identify how the two systems overlap. The first five places (Ones through Ten-Thousands) are identical in both. After that, $1 \text{ Lakh} = 100,000$ (One Hundred Thousand), $10 \text{ Lakhs} = 1 \text{ Million}$, $1 \text{ Crore} = 10 \text{ Millions}$, and $10 \text{ Crores} = 100 \text{ Millions}$.

Expanded Form: This represents a number as the sum of the place values of each of its digits. For example, the expanded form of $7,34,512$ is $7,00,000 + 30,000 + 4,000 + 500 + 10 + 2$. This allows students to see the weight of each digit based on its position in the place-value chart.

Successor and Predecessor: The successor of a number is the number that comes immediately after it, calculated by adding $1$. The predecessor is the number that comes immediately before it, calculated by subtracting $1$. For example, for the number $9,99,999$, the successor is $10,00,000$ and the predecessor is $9,99,998$.

Number Names: When writing number names, we read all the digits in the same period together followed by the period name (except for the ones period). In the Indian system, $5,02,10,000$ is 'Five crore two lakh ten thousand'. In the International system, $50,210,000$ is 'Fifty million two hundred ten thousand'.