Numbers up to 10,000 - Forming Numbers with Given Digits

Understanding Place Value: Every 4-digit number consists of four places: Thousands ($Th$), Hundreds ($H$), Tens ($T$), and Ones ($O$). Imagine a grid with four columns where the leftmost column represents the largest value ($1,000s$) and the rightmost represents the smallest ($1s$). For example, the number $4,532$ has $4$ in the $Th$ place and $2$ in the $O$ place.

Forming the Greatest Number: To create the largest possible number from a set of digits, arrange them in descending order (from biggest to smallest). If given digits are $3, 9, 1, 7$, we place the largest digit $9$ in the thousands place, followed by $7$ in the hundreds, $3$ in the tens, and $1$ in the ones to get $9,731$. Visualise this as filling the 'Thousands' bucket first with the heaviest value.

Forming the Smallest Number: To create the smallest number, arrange the digits in ascending order (from smallest to biggest). For digits $5, 2, 8, 4$, the smallest combination is $2,458$. This ensures the smallest values are placed in the higher place-value positions.

The Zero Rule: A 4-digit number cannot start with $0$ because it would then become a 3-digit number (e.g., $0521$ is actually $521$). If one of the given digits is $0$, place the smallest non-zero digit in the Thousands place and then put $0$ in the Hundreds place. For digits $6, 0, 4, 9$, the smallest number is $4,069$ rather than $0,469$.

Digit Repetition: If you are asked to form a 4-digit number using only 3 digits, you must repeat one digit. To make the greatest number, repeat the largest digit (e.g., using $2, 5, 8$ to make $8,852$). To make the smallest number, repeat the smallest digit (e.g., using $2, 5, 8$ to make $2,258$).

Face Value vs. Place Value: The 'Face Value' is the digit itself, while the 'Place Value' depends on its position. In the number $7,245$, the face value of $7$ is simply $7$, but its place value is $7 \times 1000 = 7000$. Visualise the digit $7$ sitting in a chair labeled 'Thousands'.

Successor and Predecessor: The successor is the number that comes just after ($+1$), and the predecessor is the number that comes just before ($-1$). For the largest 4-digit number $9,999$, the successor would be a 5-digit number, $10,000$.