Here is a small miracle hiding in plain sight: with only ten little symbols, you can write any number that has ever existed or ever will. The secret is not the symbols themselves — it is where you put them. That idea is called place value, and once you see it, numbers stop being a wall of digits and start making sense.
Ten symbols, and that's all
To write numbers, we use exactly ten symbols, called digits:
\[ 0 \quad 1 \quad 2 \quad 3 \quad 4 \quad 5 \quad 6 \quad 7 \quad 8 \quad 9 \]
There is no single symbol for "ten." When we count past nine, we don't invent a new shape — we run out of digits and start a fresh column to the left. That fresh column is the whole trick. A number like 352 is not three separate ideas glued together; it is a set of instructions about how many of each size of group you have.
Position decides value
Reading from the right, each spot a digit can sit in is a place, and each place has a name:
- the ones place (worth 1 each),
- the tens place (worth 10 each),
- the hundreds place (worth 100 each),
- the thousands place (worth 1000 each), and so on.
The same digit means different amounts depending on where it stands. In 352, the 3 is in the hundreds place, so it means \( 3 \times 100 = 300 \), not just 3. So the number is really
\[ 3 \times 100 \;+\; 5 \times 10 \;+\; 2 \times 1 \;=\; 300 + 50 + 2. \]
In words a digit's value is the digit multiplied by the value of its place. The digit tells you "how many," and the place tells you "how big."
Why it's called base ten
Move one place to the left and the value grows ten times. Move one place to the right and it shrinks to a tenth. Ones, then tens, then hundreds, then thousands — each step is \( \times 10 \) of the step before it:
\[ 1 \;\xrightarrow{\;\times 10\;}\; 10 \;\xrightarrow{\;\times 10\;}\; 100 \;\xrightarrow{\;\times 10\;}\; 1000 \]
Because every jump is a factor of ten, we call our system base ten (or decimal). It is almost certainly base ten because we have ten fingers — a counting tool we were all born holding.
The quiet hero: zero
Zero does something no other digit does — it holds a place open. Look at 4,072. There are no hundreds in it, but we cannot simply skip that column, or we would write "472," a completely different number. The 0 says "this place is empty, but it still counts." It keeps the 4 sitting firmly in the thousands place. Without zero as a placeholder, place value would collapse.
Expanded form: pulling a number apart
Expanded form just means writing a number as the sum of what each digit is worth. It makes place value visible. Let's work two examples carefully.
- Label the places from the right: 2 is ones, 7 is tens, 0 is hundreds, 4 is thousands.
- Multiply each digit by its place value: \( 4 \times 1000 = 4000 \), \( 0 \times 100 = 0 \), \( 7 \times 10 = 70 \), \( 2 \times 1 = 2 \).
- Drop the zero term (it adds nothing) and write the sum: \( 4000 + 70 + 2 \).
- Check by adding it back up: \( 4000 + 70 + 2 = 4072 \). It matches.
So the expanded form is \( 4000 + 70 + 2 \). Read aloud, the number is "four thousand seventy-two."
- Both numbers have the same number of digits, so line them up by place and compare from the left (the biggest places first).
- Thousands place: both have \( 3 \). A tie, so move one place right.
- Hundreds place: the first has \( 1 \) (worth 100), the second has \( 8 \) (worth 800). Since \( 800 > 100 \), the second number is already ahead — and no smaller places can ever overturn that gap.
- Therefore \( 3180 < 3810 \).
Comparing left-to-right works because one bigger place outweighs everything to its right combined. The hundreds settled it before we even looked at the tens or ones.
Tip — why it matters Place value is the reason 50 is worth far more than 5: the digit 5 simply moved one place to the left. It is also why money works (a coin's value depends on its column of cents, dimes, dollars) and why the metric system is so easy — every unit is ten times the next. Master this one idea and arithmetic, decimals, and big numbers all get easier.
Reading and writing multi-digit numbers
To read a long number, group its digits in threes from the right and name each group: ones, thousands, millions, and so on. To write a number from spoken words, set up the places first and drop each digit into its column — using 0 wherever a place is empty. If you can name the place of every digit, you can read and write any whole number, no matter how long.
Practice
Try each one yourself, then reveal the full solution.
1. In the number 6,540, what is the value of the digit 5 (not just the digit itself)?
2. Write 2,305 in expanded form.
3. Order these three numbers from smallest to largest: 870, 708, 780.