Place Value and Rounding Whole Numbers

Where a digit sits decides what it is worth

$$3452$$

Digit	Place	Worth
$3$	thousands	$3000$
$4$	hundreds	$400$
$5$	tens	$50$
$2$	ones	$2$

The same digit means different amounts in different places. The $4$ in $400$ is worth ten times the $4$ in $40$.

Each place is ten times the next

$$1, \quad 10, \quad 100, \quad 1000, \quad 10\,000, \; \ldots$$

We count in tens, so ten ones make one ten, ten tens make one hundred, and ten hundreds make one thousand. Every step left multiplies by ten.

That is the whole design of the number system. It is why moving a digit one place left makes it worth ten times as much, and why multiplying by ten adds a zero.

Three ways to write a number

Form	$3452$ written that way
numerals	$3452$
words	three thousand, four hundred and fifty-two
expanded	$3000 + 400 + 50 + 2$

Expanded form shows what every digit is worth, laid out. It is the clearest way to see that a number is the sum of its places.

A zero holds a place open:

$$3052 = 3000 + 0 + 50 + 2$$

The zero in the hundreds says there are no hundreds. Leaving it out would give $352$, a completely different number, which is why zero is a digit and not merely “nothing”.

Rounding

Rounding replaces a number with a nearby round one. Three steps:

1. Find the place you are rounding to.
2. Look at the digit immediately to its right.
3. If that digit is $5$ or more, round up. Otherwise, leave the place as it is.

Then every digit after the rounding place becomes zero.

Round $3452$ to the nearest hundred.

The hundreds digit is $4$. The digit to its right is $5$, so round up:

$$3452 \rightarrow 3500$$

Round $3452$ to the nearest thousand.

The thousands digit is $3$. To its right is $4$, which is less than $5$:

$$3452 \rightarrow 3000$$

Round $3452$ to the nearest	Look at	Answer
ten	$2$	$3450$
hundred	$5$	$3500$
thousand	$4$	$3000$

Only one digit decides it — the one immediately to the right. The digits further along make no difference.

Why the rule uses 5

Rounding asks which round number the value is closer to.

$3452$ sits between $3400$ and $3500$. The halfway point is $3450$. Anything above the halfway point is nearer the top, and the tens digit alone tells you which side of halfway you are on.

$$3452 > 3450 \quad\Rightarrow\quad \text{nearer } 3500$$

A digit of exactly $5$ lands on the halfway point, belonging to neither side. The convention is to round up, chosen so the rule is consistent rather than because $3450$ is genuinely nearer $3500$.

Why round at all

A rounded number is easier to hold in your head and quicker to work with, and usually as useful.

“About $3500$ people” tells you what you need. ”$3452$ people” is exact and no more informative for most purposes.

It also makes estimating possible. Rounding both numbers before multiplying gives an answer close enough to check the real one against — which is what catches a place-value slip in a long multiplication.

Worked examples

Common mistakes

Practice problems