Integer Place Value

In 300, the 3 sits in the hundreds column, so it represents 3 × 100 = 300. In 30, the 3 sits in the tens column, so it represents 3 × 10 = 30. Since 300 is ten times larger than 30, the same digit is worth ten times more simply because of its position.

This is the key idea of place value: the position of a digit determines its value, not the digit itself. Two identical digits can represent completely different amounts depending on where they sit.

Compare column by column, starting from the left. Thousands: both have 4. Hundreds: both have 0. Tens: 4050 has a 5, but 4005 has a 0. Since 5 tens (50) is greater than 0 tens (0), 4050 > 4005 — we don’t even need to check the units.

The zero in the hundreds column of both numbers is doing important work as a placeholder: it ensures the 4 stays in the thousands column. Meanwhile, the position of the 5 makes all the difference — fifty is not the same as five.

Students often translate each part of the words separately: “six thousand” becomes 6000 and “thirty-seven” becomes 37, then they join them together to get 600037. But our place value system doesn’t work by concatenation — each digit occupies exactly one column. “Six thousand” means the digit 6 in the thousands column, giving 6 _ _ _. “Thirty-seven” fills the tens and units: 6 _ 3 7. The hundreds column is empty, so a zero holds the place: 6037.

A quick check confirms it: 600037 is a six-digit number — that’s six hundred thousand, nowhere near six thousand! Partitioning correctly: 6000 + 30 + 7 = 6037, a four-digit number, as expected.

The standard (canonical) partition breaks each digit into its column value: 4 thousands + 7 hundreds + 3 tens + 2 units = 4000 + 700 + 30 + 2. But 4700 + 32 is equally valid: 4700 represents 47 hundreds, and 32 represents 3 tens and 2 units. Combining: 4700 + 32 = 4732. ✔

Place value allows flexible grouping because adjacent columns can be read together. You could also write 4732 = 4600 + 132, or 4732 = 4730 + 2. This flexibility is essential for mental calculation and column methods — for instance, when “borrowing” in subtraction, you are regrouping from one partition to another.

Place value is a base-10 system. Because the thousands column is immediately to the left of the hundreds column, it is exactly ten times larger. Therefore, ten hundreds must be bundled together to make one thousand.

This multiplicative reasoning underpins how we carry and regroup across columns. It works at every boundary: ten units make a ten, ten tens make a hundred, and ten hundreds make a thousand.

Example: 1253

Another: 3050

Creative: 9050 — a clean four-digit number where the only 5 is exactly where it needs to be, avoiding the confusion of repeated digits.

Trap: 5432 — a student might pick this thinking “it has a 5 so the 5 is worth 50.” But the 5 here is in the thousands column, so it’s worth 5000, not 50. The digit must be in the tens column to be worth 50.

Example: 3200

Another: 2750

Creative: 2500 — right on the boundary. By the convention of rounding 5 up, 2500 rounds up to 3000. Or 3499 — the very largest number that still rounds down to 3000.

Trap: 3500 — a student might think this rounds to 3000 because it “starts with 3.” But the hundreds digit is 5, which means it rounds up to 4000, not down to 3000.

Example: 139 and 913 — same digits {1, 3, 9}. Since 913 > 2 × 139 = 278, it’s more than double. ✔

Another: 189 and 918 — 918 > 2 × 189 = 378. ✔

Creative: 109 and 910 — 910 is more than eight times 109. This works because moving a large digit from the units to the hundreds column has a dramatic effect.

Trap: 234 and 243 — these use the same three digits, but 243 is only 9 more than 234, and 2 × 234 = 468, so 243 doesn’t come close to double. Swapping digits in the tens and units columns barely changes the number — you need the hundreds digit to change for a big effect.

Example: −300

Another: −345

Creative: −1300 — the hundreds digit is still 3 even though the number is in the thousands. Or −399 — as close to −400 as possible while keeping a 3 in the hundreds.

Trap: −30 — a student might think the 3 is in the hundreds column because “minus thirty sounds like a big number.” But the 3 is in the tens column, and −30 is actually greater than −100 (closer to zero), so it doesn’t satisfy “less than −100” either.

For positive whole numbers this is always true: 342 has more digits than 56, and 342 > 56. A three-digit positive integer is always bigger than a two-digit positive integer because the smallest three-digit number (100) is already larger than the largest two-digit number (99).

But once we allow negative numbers it breaks down: −1000 has four digits but is far smaller than 5, which has one. This also breaks down when we introduce decimals: 4.123 has four digits, but it is much smaller than 56, which only has two. So the number of digits only tells you the size when both numbers are positive whole numbers.

If the two digits are different and you swap them, the value changes because each digit moves into a column with a different value. For example, swapping the 1 and 3 in 132 gives 312 — a completely different number.

But if the two digits you swap are identical, the number stays the same. Swap the two 1s in 121 and you still get 121. Similarly, swapping the two 3s in 3003 has no effect. So it depends on which digits you swap.

This is the fundamental rule of our base-10 place value system. Each column is worth exactly 10 times the column to its right: units × 10 = tens, tens × 10 = hundreds, hundreds × 10 = thousands, and so on.

So a 7 in the tens column is worth 70, but moved one column left to the hundreds it’s worth 700 — exactly ten times more. This works for every digit and every pair of adjacent columns, including zero (0 × 10 = 0). It’s why we call it a “base-10” number system.

Writing 007, 050, or 00482 doesn’t change anything — the number is still 7, 50, or 482. Leading zeros don’t create a new place-value column; they simply fill columns that were already there with a value of zero. That’s why we don’t normally write them.

Students sometimes confuse leading and trailing zeros because both involve “putting a zero next to the number.” But the two behave very differently: adding a zero to the end of a whole number (5 → 50) makes it ten times larger, while adding a zero to the front (5 → 05) has no effect at all. The asymmetry comes from how our number system grows: new, higher-value columns extend to the left, so digits on the left matter — empty columns on the left don’t.

The student is treating the 0 as if it physically doesn’t exist, causing them to “squish” the remaining digits together. By pulling the 4 and 5 out of their respective columns, they have destroyed the place value of the number.

While a zero does mean “none of this value,” it is performing a crucial job as a placeholder in the tens column. The 4 is in the hundreds column (400), and the 5 is in the units column (5). They cannot be merged into “45” just because the tens column is empty.

The student falsely believes the rule “zeros make a number bigger”. While adding a zero to the end of a number multiplies it by 10 (e.g., 3 becomes 30), a zero on its own simply holds an empty column. It has no magical enlarging power.

To compare two numbers, we must look at their highest-value columns. Both are three-digit numbers, so we compare the hundreds columns. 498 has 4 hundreds, while 300 only has 3 hundreds. Because 400 is greater than 300, 498 is the larger number — the zeros are entirely irrelevant here.

The student is confusing face value with place value. The face value of the digit is indeed 9, but the question asks for its value within the number. In 7294, the 9 sits in the tens column, so its place value is 9 × 10 = 90, not 9.

To see why 9 can’t be right, consider: if the 9 in 7294 were worth only 9, then 7294 would equal 7000 + 200 + 9 + 4 = 7213 — a completely different number. The position of the digit multiplies its value, and that multiplication is what place value is all about.

The student correctly placed the 7 in the thousands column but then made a misguided decision about zero placement. They put 0 in the hundreds column “to keep it four digits,” but this wastes the highest available column. The zero should go in the lowest possible column, not near the top.

The correct answer is 7420. To make the largest number, place the digits in descending order: 7 thousands, 4 hundreds, 2 tens, 0 units. The student’s 7042 is 378 less than 7420 because the 4 ended up in the tens column (worth 40) instead of the hundreds (worth 400).

The student treated the tens column in isolation and failed to regroup across the boundary. When 9 tens and 1 ten combine, they make 10 tens, which is exactly one hundred.

That new hundred must be passed to the hundreds column. But the hundreds column also has a 9, so 9 hundreds + 1 hundred = 10 hundreds, which makes one thousand. That new thousand pushes the thousands column from 3 to 4. The correct cascading exchange results in 4005.