Mathematics › 🎯 Grade 4 › Place Value: The Architecture of Number

Place Value: The Architecture of Number

Every digit in a number has a job, and its job depends only on where it stands.

Grade 4Elementary

Place Value: The Architecture of Number — illustration

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The big idea: The same digit can mean wildly different amounts depending on its position in a number — a 5 might be worth five, fifty, or five thousand. Every place in a number is worth exactly ten times the place to its right, so a number is really a kind of stacked, ten-times-bigger code. Once you can read that code, you can build numbers apart (expanded form), compare them without guessing, and round them to a friendlier neighbor.

🎯 By the end, you'll be able to

Identify the value of a digit based on its place in a multi-digit number
Write a number in expanded form as a sum of place values
Compare multi-digit numbers by checking digits from left to right
Round a number to a given place value

📎 You should already know

Reading and writing whole numbers up to the thousands
Basic addition facts

One digit, many jobs

Look at the digit 5 in three different numbers: 5, 50, and 5,000. It is exactly the same shape each time, yet it means five, fifty, and five thousand. The digit never changes; what changes is where it stands.

That is the whole idea behind place value: every position in a number has a name and a worth, and a digit's true value is the digit multiplied by the worth of its spot.

🔑 Each place is ten times the one beside it

Moving one spot to the left in a number multiplies the value of that spot by 10. Ones, tens, hundreds, thousands: each is exactly ten copies of the one before it. That is why we call it a base-ten system.

\[ 3{,}482 = 3\times1000 + 4\times100 + 8\times10 + 2\times1 \]

Every digit's true value is the digit times the worth of its place: 3 thousands, 4 hundreds, 8 tens, and 2 ones.

🎮 Zoom the Number Line LIVE

Zoom in and out across place values — ones, tens, hundreds, thousands — and watch each digit's value scale by 10.

Standard form and expanded form

Writing 5,206 the normal way is called standard form. Writing it as 5,000 + 200 + 6 is called expanded form: it shows exactly which place values add up to make the number.

Notice there is no tens term in that sum. The 0 in the tens spot is not skipped: it is a placeholder that keeps every other digit in its correct spot. Without it, 5,206 would collapse into 526, a completely different number.

📝 Worked example: Write 5,206 in expanded form.

5 is in the thousands place, worth 5,000.
2 is in the hundreds place, worth 200.
0 is in the tens place, worth 0 (a placeholder).
6 is in the ones place, worth 6.

✓ 5,206 = 5,000 + 200 + 6 (the tens term drops out because it is 0).

Comparing numbers: start from the left

To compare two multi-digit numbers, first check whether they have the same number of digits. A number with more digits is always greater, no matter what its first digit looks like: 1,000 beats 950, even though 9 looks like a big first digit, because 950 only has three digits.

When two numbers have the same number of digits, compare them place by place starting from the left (the biggest place). The first place where the digits differ decides which number is greater.

📝 Worked example: Which is greater, 3,268 or 3,286?

Thousands: 3 = 3, tied so keep going.
Hundreds: 2 = 2, tied so keep going.
Tens: 6 versus 8. They differ here, so this place decides it.

✓ 3,286 is greater, because 8 tens beats 6 tens once every place to the left is tied.

⚠️ More digits always wins first

Do not compare digit by digit until you have checked the length of each number. A four-digit number is always bigger than a three-digit number, however large that three-digit number looks: 1,000 > 950, full stop.

Rounding: finding a friendlier neighbor

Sometimes an exact number is more detail than you need, and it helps to round it to the nearest ten, hundred, or thousand. To round to a place, look at the digit just to its right. If that digit is 5 or more, round the target place up by one; if it is less than 5, leave the target place as it is. Every place to the right of the rounding spot becomes 0.

📝 Worked example: Round 4,368 to the nearest hundred.

The hundreds digit is 3.
Look at the digit to its right, the tens digit: 6.
6 is 5 or more, so round the hundreds digit up from 3 to 4, and replace everything after it with zeros.

✓ 4,368 rounds to 4,400.

Check your understanding

1. What is the value of the digit 7 in the number 3,762?

In 3,762 the 7 sits in the hundreds place, so its value is 7 x 100 = 700.

2. Which expanded form matches 4,050?

4,050 has 4 thousands, 0 hundreds, 5 tens, and 0 ones, so it expands to 4,000 + 50.

3. Which number is greater: 5,238 or 5,283?

Thousands and hundreds match (5 and 2). The tens digits differ: 8 beats 3, so 5,283 is greater.

4. To round 2,847 to the nearest hundred, which digit decides the answer?

You round the hundreds place by looking at the digit right after it, which is the tens digit: 4.

5. Round 2,847 to the nearest hundred.

The tens digit is 4, which is less than 5, so the hundreds digit stays at 8, giving 2,800.

✅ Key takeaways

A digit's true value is the digit times the worth of its place; the same digit means different amounts depending on where it stands.
Each place value is 10 times the place to its right, which is why we call it a base-ten system.
Expanded form shows a number as a sum of its place values, e.g. 5,206 = 5,000 + 200 + 6.
Compare multi-digit numbers by digit count first, then place by place from the left, stopping at the first difference.
To round, check the digit to the right of the target place: 5 or more rounds up, less than 5 stays the same.