Number System - Place Value and Face Value

Face Value: The face value of a digit is the value of the digit itself, regardless of its position in a number. For example, in the number $8,542$, the face value of $5$ is simply $5$. Visually, if you imagine a number on a digit card, the face value is the actual number written on that card.

Place Value: This is the numerical value that a digit holds based on its position in a number. It is calculated by multiplying the digit's face value by the value of its place (e.g., Ones, Tens, Hundreds). In the number $8,542$, the digit $5$ is in the hundreds place, so its place value is $5 \\times 100 = 500$.

Indian Place Value System: This system organizes numbers into periods called Crores, Lakhs, Thousands, and Ones. Commas are used to separate these periods: the first comma is placed after the hundreds place (three digits from the right), and subsequent commas are placed after every two digits. For example, $75,23,10,409$ is read as Seventy-five Crore, Twenty-three Lakh, Ten Thousand, Four Hundred Nine.

International Place Value System: This system organizes numbers into periods of Billions, Millions, Thousands, and Ones. Unlike the Indian system, commas are placed after every three digits from the right. For example, $752,310,409$ is read as Seven Hundred Fifty-two Million, Three Hundred Ten Thousand, Four Hundred Nine. Visually, each period in a place value chart always contains three columns: Hundreds, Tens, and Units of that specific period.

Expanded Form: Writing a number in expanded form means expressing it as the sum of the place values of all its digits. For the number $64,302$, the expanded form is $60,000 + 4,000 + 300 + 0 + 2$. Visually, this is like 'stretching' the number out to see the individual value contributed by each digit's position.

The Role of Zero: Zero ($0$) acts as a placeholder in the number system. While the face value and place value of $0$ are always $0$, its presence is crucial to maintain the correct positions of other digits. For example, in the number $507$, the $0$ ensures that the $5$ remains in the hundreds place rather than shifting to the tens place.

Successor and Predecessor: The Successor is the number that comes immediately after a given number ($+ 1$), and the Predecessor is the number that comes immediately before a given number ($- 1$). On a horizontal number line, the Successor is located one unit to the right, and the Predecessor is one unit to the left.