Expanded Form of Multi-Digit Numbers

Expanded form is a way of writing a multi-digit number so that every digit's true value is shown separately, connected by addition. Instead of writing a number the usual way (called standard form), expanded form breaks it apart digit by digit based on where that digit sits in the number.

For example, the number \(4325\) means 4 thousands, 3 hundreds, 2 tens, and 5 ones. In expanded form, that becomes:

\( 4325 = 4000 + 300 + 20 + 5 \)

Notice that each part is the digit multiplied by its place value: the 4 is worth 4000 because it sits in the thousands place, the 3 is worth 300 because it sits in the hundreds place, and so on. Before working through expanded form, it helps to be comfortable with identifying the place value of a digit, since expanded form is really just place value written out as a sum.

Every digit in a multi-digit number has two things attached to it: its face value (the digit itself, 0 through 9) and its place value (how much that position is worth, such as ones, tens, hundreds, or thousands). Expanded form multiplies the two together for each digit and adds the results.

A place value table can make this easier to see, since it lines up each digit under the correct heading before you write the expanded sum.

Follow these steps for any multi-digit number:

1. Look at each digit and note which place it occupies (ones, tens, hundreds, thousands, ten thousands, and so on).
2. Multiply the digit by the value of its place.
3. Write each result as a separate term.
4. Join all the terms with addition signs, ordered from the largest place value to the smallest.

If a digit is 0, its term is worth nothing, so it is usually left out of the expanded form rather than written as "+ 0". For example:

\( 5062 = 5000 + 60 + 2 \)

The hundreds digit here is 0, so no hundreds term appears in the sum.

The same idea extends to numbers with more digits. Once a number reaches the ten thousands or hundred thousands place, there are simply more terms to add.

\( 273859 = 200000 + 70000 + 3000 + 800 + 50 + 9 \)

Each digit still keeps its own place value: the 2 is in the hundred thousands place, the 7 is in the ten thousands place, the 3 is in the thousands place, the 8 is in the hundreds place, the 5 is in the tens place, and the 9 is in the ones place. Understanding how digits shift value as they move positions is also the key skill behind learning to convert between place values of multi-digit numbers.

You may also see the term "expanded notation," which writes each term as a multiplication instead of the already-multiplied value:

\( 4325 = (4 \times 1000) + (3 \times 100) + (2 \times 10) + (5 \times 1) \)

This shows exactly where each value comes from, while plain expanded form just shows the multiplied results (4000, 300, 20, 5). Both describe the same number, and both rely on the same place value reasoning.

Write \(30748\) in expanded form.

• 3 is in the ten thousands place: \(3 \times 10000 = 30000\)
• 0 is in the thousands place: contributes nothing
• 7 is in the hundreds place: \(7 \times 100 = 700\)
• 4 is in the tens place: \(4 \times 10 = 40\)
• 8 is in the ones place: \(8 \times 1 = 8\)

Putting the nonzero terms together: \( 30748 = 30000 + 700 + 40 + 8 \)

Watch out for these errors when writing expanded form:

• Forgetting that a zero digit means that place contributes nothing to the sum.
• Mixing up the order of digits, which changes their place value entirely.
• Writing the face value alone instead of multiplying it by the place value (writing "3" instead of "300" for a hundreds digit).

Practicing with a reading and writing digits exercise alongside expanded form can help reinforce how the digits, their names, and their values all connect.

To check an expanded form answer, simply add all the terms back together. The sum should equal the original standard form number. For \( 30000 + 700 + 40 + 8 \), adding gives \(30748\), confirming the expanded form is correct.

Related lessons

• Math 3