How To Write Numbers In Expanded Form

It is tempting to say that the digits in a number are what define its value, but if you write 25 and 52 – using the same digits but in different places – you get two different values. Learning to write numbers in expanded form is an easy way to remember the importance of each digit's placement, or its place value, in a number.

Try counting up from zero: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 are all pretty straightforward, but once you get to 10, something changes. You now have two digits in the number – the 1 and the 0. Each digit occupies a "slot" or place in the final number, and each place has a different value. The slot on the left represents tens, and the digit 1 in that slot tells you that you have one 10. The slot on the right represents ones – the same numbers you started counting with – and the zero in that slot tells you that you don't have any extra 1s.

If you keep counting, you'll notice that the digits in the ones column change first. The next number is 11. If you take it apart to its component place values, which is known as decomposing the number, you'll see that there is a 1 in the tens slot and a 1 in the ones slots. So you have one 10 and one 1. The next number is 12, which still has a 1 in the tens slot, but now there's a 2 in the ones slot. Keep counting for long enough, and you'll reach 19, then 20. Notice that now the number in the tens slot has increased to 2, but the ones slot has reset to zero. This pattern continues as you count up. The number in the ones slot keeps increasing until it hits 9; then the tens value goes up, and the ones value resets to zero.

You can decompose any number at all, even large ones. Consider the number 392. It has three digits, so you have a new slot or place value to deal with in a larger number. You're already familiar with the ones place, which remains on the far right of the number; in this case, you have two 1s. The tens place is still the next column to the left. There's a 9 there, so you have nine 10s. The next column to the left is called the hundreds column, and there's a 3 there, so you have three 100s.

Expanded form is a specific way of writing the digits of a number that you've broken into each of its component place values. To write numbers in expanded form, you link each digit in the number to its place value with a multiplication sign. Consider the example of 392. Reading the numbers from left to right, you start with the biggest slot, the hundreds place, which has a 3 in it. You have

\(3 × 100 = 300\)

The next slot to the right is the tens place, and there's a 9 in it. You have

\(9 × 10 = 90\)

There's a 2 in the ones place, so you have

\(2 × 1 = 2\)

There are three pieces to this number: 300, 90 and 2. Connect those pieces with addition signs, and you have the number in expanded form:

\(300 + 90 + 2 = 392\)

There's no limit to how big or small a number you can write in expanded form. You just have to know the value of each place or slot in the number. Perhaps you've already noticed this pattern: The place values start with ones on the right, then for each slot you move to the left, the value is multiplied by 10. The next slot on the left is tens is hundreds, and the place after that is thousands, followed by 10 thousands and so on.

You can even write decimals in expanded form, as long as you understand how those place values work. When you have a decimal point, the slot just to the right of the decimal is the tenths slot, the slot to the right of that is the hundredths slot and so on. If you have the number 0.231, there's a 2 in the tenths slot, a 3 in the hundredths slot and a 1 in the thousandths slot. You can write that number in expanded form by multiplying each digit by its place value, then adding them together:

\(2 × 0.1 = 0.2, \,3 × 0.01 = 0.03 \text{ and }1 × 0.001 = 0.001\)

The final step is to connect the results with addition signs:

\(0.2 + 0.03 + 0.001\)

Let's write another number in expanded form. Consider 457.2. When you multiply each digit by its place value, you have

\(4 × 100 = 400, \, 5 × 10 = 50, \, 7 × 1 = 7 \text{ and }2 × 0.1 = 0.2\)

Put an addition sign between each component, and you have the number in expanded form:

\(400 + 50 + 7 + 0.2\)

You can always check your work by adding the components of the number together, which is called composing the number or writing it in standard form. When you do the addition and add

\(400 + 50 + 7 + 0.2 = 457.2\)

you end up with the original number.