Number and Place Value - Comparing and Ordering Large Numbers

Place Value Positions: To compare large numbers, you must understand the value of each digit. For a six-digit number, the positions from left to right are Hundred Thousands ($100,000$), Ten Thousands ($10,000$), Thousands ($1,000$), Hundreds ($100$), Tens ($10$), and Ones ($1$). Imagine a place value chart where each column to the left is $10$ times larger than the column to its right.

Counting Digits Rule: The simplest way to compare two numbers is to count their digits. A number with more digits is always greater than a number with fewer digits. For example, $10,234$ (5 digits) is automatically larger than $9,988$ (4 digits) because it reaches the Ten Thousands place while the other only reaches the Thousands place.

Comparing Left to Right: When two numbers have the same number of digits, compare them digit by digit starting from the highest place value (the far left). If the digits in the highest place are the same, move one position to the right and compare those digits. Continue this process until you find a difference.

Comparison Symbols: We use specific symbols to show the relationship between numbers. The 'Greater Than' symbol ($>$) points its open side toward the larger number. The 'Less Than' symbol ($<$) points its vertex toward the smaller number. The 'Equal To' symbol ($=$) is used when the numbers have the exact same value in every place.

Ascending Order: This means arranging numbers from the smallest to the largest value. Visualise this as climbing up a staircase, where each step represents a larger number ($1,234 \rightarrow 5,678 \rightarrow 10,111$).

Descending Order: This means arranging numbers from the largest to the smallest value. Visualise this as walking down a staircase, starting from the highest value and moving down to the lowest ($50,000 \rightarrow 40,000 \rightarrow 30,000$).

Number Line Representation: On a horizontal number line, numbers increase as you move from left to right. If number $A$ is to the right of number $B$, then $A > B$. This visual tool helps in identifying which large numbers are closer to certain milestones, such as the nearest $10,000$ or $100,000$.