Expanded form of a number – how to break a large number into a sum of places

Expanded form of a number

Expanded form (sometimes also called positional notation or place-value decomposition) lets you see the real value each digit carries in a number.

Where it comes from

When you write the number 45,237, each of its digits hides a value that depends on its place:

  • 4 is at the ten thousands place → its value is 4 × 10,000 = 40,000
  • 5 is at the thousands place → 5 × 1,000 = 5,000
  • 2 is at the hundreds place → 2 × 100 = 200
  • 3 is at the tens place → 3 × 10 = 30
  • 7 is at the ones place → 7 × 1 = 7

When we add all these values together, we get the same number back:

And that is the expanded form of the number 45,237.

The general procedure

  1. Find out how many digits the number has and which places appear in it (see Digit at a given place).
  2. For each non-zero digit, write its value as the product of the digit and the value of its place.
  3. Add the individual values together with +.
  4. Start with the largest place and end with the ones.

Example 1 – a number without zeros

Problem: Write the number 368,524 in expanded form.

Example 2 – a number with zeros

Problem: Write the number 40,207 in expanded form.

We can see that the digits at the thousands and tens places are 0. Zero places are left out of the expanded form – they carry no value.

Example 3 – a larger number

Problem: Write the number 2,075,083 in expanded form.

First let's identify the places:

  • millions: 2
  • hundred thousands: 0
  • ten thousands: 7
  • thousands: 5
  • hundreds: 0
  • tens: 8
  • ones: 3

We leave out the zero places. What remains are the millions, ten thousands, thousands, tens, and ones:

Why the expanded form is useful

  • It shows you why a number looks the way it does – every digit has a clear value.
  • It makes adding and subtracting large numbers easier – once you know that 40,000 + 5,000 is 45,000, adding 45,237 + 3,000 is almost automatic.
  • It prepares you for decimals, where the same rule also applies to decimal places (tenths, hundredths…).

Practice it