Place value explained
The same digit means different things in different columns. That single idea is what makes our number system work and Roman numerals not.
What place value is
A digit has a face value (the 5 in 4,507 is a five) and a place value, which is what it is worth where it sits. In 4,507 that 5 sits in the hundreds column, so it is worth five hundred. The number is the sum of all its place values.
| Digit | Column | Worth | In words |
|---|---|---|---|
| 4 | thousands | 4,000 | four thousand |
| 5 | hundreds | 500 | five hundred |
| 0 | tens | 0 | nothing |
| 7 | ones | 7 | seven |
Every column is ten times the one to its right
| Position | Name | Worth |
|---|---|---|
| 1 | ones | 1 |
| 2 | tens | 10 |
| 3 | hundreds | 100 |
| 4 | thousands | 1,000 |
| 5 | ten thousands | 10,000 |
| 6 | hundred thousands | 100,000 |
| 7 | millions | 1,000,000 |
| 8 | ten millions | 10,000,000 |
| 9 | hundred millions | 100,000,000 |
| 10 | billions | 1,000,000,000 |
The names repeat in threes: ones, tens and hundreds, then the same three again for thousands, then again for millions. That repetition is why numbers are grouped in threes by commas, and why reading a long number is easier than it looks: you are reading the same three-digit pattern over and over with a different scale word after it.
What the zero is doing
In 4,507 there is nothing in the tens column, and the 0 is what says so. Without it the 7 would slide left into the tens and the number would be 457. Here the zero is a placeholder, holding a column open so that the digits either side of it stay where they belong.
This is the thing Roman numerals lack. A numeral is a list of quantities added together, so there is no column for a zero to hold open, and no way to write a number down in a form you can do arithmetic with column by column.
To the right of the point
The pattern continues past the decimal point, dividing by ten instead of multiplying: tenths, hundredths, thousandths. In 0.75 the 7 is worth seven tenths and the 5 is worth five hundredths, which is why the whole thing is seventy-five hundredths. Reading decimals covers that.
Why it matters beyond a worksheet
Rounding, comparing two long numbers, reading a figure aloud, spotting that an amount is out by a factor of ten: all of them are place value. The columns are also why a misplaced comma is such an expensive mistake: it does not change any digit, only which column each digit is read as sitting in.