Number and Place Value - Expanded, Standard, and Word Forms

The Place Value System: Numbers are organized into periods (like the 'Ones' period and the 'Thousands' period). Visually, imagine a Place Value Chart with columns from right to left: Ones, Tens, Hundreds, Thousands, Ten Thousands, and Hundred Thousands. Each position to the left is exactly $10$ times larger than the one to its right.

Standard Form: This is the default way we write numbers using digits. Commas are used to separate groups of three digits (periods) to make large numbers easier to read. For example, $450,231$ uses digits and a comma to represent four hundred fifty thousand, two hundred thirty-one.

Word Form: This involves writing the number using words exactly as you would say it out loud. When writing numbers between $21$ and $99$, always use a hyphen (e.g., $45$ is 'forty-five'). For example, $2,315$ is 'two thousand, three hundred fifteen'.

Expanded Form: This method breaks a number apart to show the specific value of each digit as an addition sentence. Visually, you can think of this like 'stretching' the number out. For $5,284$, the expanded form is $5,000 + 200 + 80 + 4$.

Digit vs. Value: A 'digit' is a single numeral ($0, 1, 2, 3, 4, 5, 6, 7, 8,$ or $9$). The 'value' is how much that digit is worth based on its position. In the number $7,321$, the digit is $3$, but because it is in the hundreds place, its value is $300$.

The Role of Zero as a Placeholder: Zero is used to hold a place when there is no value for that specific column. Visually, in a number like $1,052$, the $0$ in the hundreds column acts as a bridge, ensuring the $1$ stays in the thousands place. Without the zero, the number would incorrectly look like $152$.

Comparing and Ordering: Numbers can be compared by looking at the highest place value first. We use the symbols $>$ (greater than), $<$ (less than), and $=$ (equal to). Visually, the open end of the symbol always points toward the larger number, like an alligator's mouth: $12,400 > 12,399$.