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G1.6 Place Value and Expanded Form

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01 — Explanation

What does a digit's position actually tell me?

Place value is the value a digit contributes based on where it sits in a number — moving one place to the left multiplies a digit's value by 10; moving one place to the right divides it by 10. In $5{,}123.64$ , the digit $5$ is worth $5000$ (thousands place), not just $5$ , because of its position.

02 — Explanation

How do I break a number down into its expanded form?

Write the number as the sum of the value each digit contributes:

5123.64 = 5000 + 100 + 20 + 3 + 0.6 + 0.04

Thousands: $5 \times 1000 = 5000$
Hundreds: $1 \times 100 = 100$
Tens: $2 \times 10 = 20$
Units: $3 \times 1 = 3$
Tenths: $6 \times 0.1 = 0.6$ (or $\frac{6}{10}$ )
Hundredths: $4 \times 0.01 = 0.04$ (or $\frac{4}{100}$ )

Exam Tip

The decimal point itself doesn't have a place value — it's just a marker separating whole numbers from fractional parts. Don't try to assign it a "value" of its own.

03 — Explanation

How do I round a number to a given place value?

Identify the digit in the place value you're rounding to (the rounding digit).
Look at the digit immediately to its right (the deciding digit).
If the deciding digit is $5$ or more, round up (add 1 to the rounding digit); if it's $4$ or less, round down (keep the rounding digit the same). All digits to the right of the rounding digit become zero (or are dropped after a decimal point).

Example: Round 7,581 to the nearest hundred.

The rounding digit is $5$ (hundreds place); the deciding digit is $8$ .
Since $8 \geq 5$ , round up: $5 \to 6$ , and the digits after become zero.
The answer is $7{,}600$ .

Common Mistake

Rounding in stages (e.g. rounding 7,581 to the nearest ten first, getting 7,580, then rounding that to the nearest hundred, getting 7,600 — which happens to work here but fails in general). Always round directly from the original number to avoid compounding errors.

04 — Explanation

How do I compare the size of two numbers using place value?

Compare digits from the left-most (highest) place value first, moving right only when digits are tied. The first place where the digits differ decides which number is bigger, regardless of how many digits follow.

Example: Compare $5218$ and $5204$ .

Thousands: both $5$ — tied.
Hundreds: both $2$ — tied.
Tens: $1$ vs $0$ — not tied, and $1 > 0$ .
So $5218 > 5204$ , without needing to look at the units digit at all.

Exam Tip

This place-value comparison method is far faster and more reliable than converting both numbers to decimals or subtracting them, especially when ordering a list of several numbers at once.

05 — Worked example

Worked Exam-Style Question

A stadium's attendance is recorded as 48,362. (a) Write this number in expanded form. (b) Round it to the nearest thousand.

(a) $48{,}362 = 40000 + 8000 + 300 + 60 + 2$ .
(b) The rounding digit is $8$ (thousands place); the deciding digit is $3$ (hundreds place).
Since $3 < 5$ , round down: the $8$ stays as it is, and everything to its right becomes zero.
The rounded attendance is $48{,}000$ .

Exam Tip

If a question asks you to round to the nearest thousand, your final answer should have exactly three zeros at the end (unless the original number was already smaller) — always check your answer "looks like" the place value you rounded to.