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Expanded Notation for Decimals

Instructor: Thomas Higginbotham

Tom has taught math / science at secondary & post-secondary, and a K-12 school administrator. He has a B.S. in Biology and a PhD in Curriculum & Instruction.

Using expanded notation to express numbers in standard form can be a helpful way to decompose a number into its component parts for ease of use in mental math and other applications. In this lesson, learn how to write numbers in expanded notation form.

The Different Two's

Not all 2's are created equally. The value of the '2' digit in the number 412 is much different than the value of the '2' digit in the number 24,891. Sometimes it can be difficult to conceptualize numbers. Expanded notation helps us break numbers down into their component parts to make complex concepts a little easier to understand. This skill of deconstruction of numbers can be helpful throughout our math lives in support of mental math.

Place Value in Base Ten

We have learned from elementary school onward, that a digit's place relative to the decimal point determines its value. The row second from bottom in the chart shows that the digit immediately left of the decimal point is in the 'ones' place, two to the left, the tens, three to the left, the hundreds, etc. To the right of the decimal point come tenths, hundredths, and thousandths, etc. For the number 412, the hundreds have a value of 4, the tens have a value of 1, and the ones have a value of 2.

A Place Value Chart
Place Value Chart

Powers of Ten

We also learned early in our math schooling the powers of ten. For example, ten raised to the second power (which means ten times itself twice, or 10 x 10) is 100. Ten raised to the third power (ten times itself three times, or 10 x 10 x 10) is 1,000. On the other side of things, ten to the zero power equals one, ten to the negative first equals 0.1, ten to the negative second power equals 0.01, etc. These values are indicated in the row third from bottom in the chart.

Multiples of Powers of Ten

One of the easier multiplications is multiplying by an even ten, hundred, or other power of ten. For example, 6 x 100 = 600, 4 x 10 = 40, and 7 x 1,000 = 7,000. We can leverage the ease of these math facts in expanded notation.

Integers in Expanded Notation

Let's consider the number 412. For this number, the ones value is two, the tens value is one, and the hundreds value is four. If we want to break this number down, we could say that 412 is equal to:

(4 x 100) + (1 x 10) + (2 x 1)

Another way to show this is: (4 x 10^2) + (1 x 10^1) + (2 x 10^0)

This is expanded notation, and you can see how it helps us really express the number's value in more accessible terms.

How might this be helpful? You want to buy gold watch that costs $412, but you only multiples of tens in your wallet (one-, ten-, and hundred-dollar bills). Using this expanded notation for 412, we would use four $100, one $10, and two $1 dollar bills to pay for the watch.

Decimals in Expanded Notation

Integers are straightforward, as demonstrated above. So are decimal values with the addition of one little trick. In the decimal value 0.638, the tenths value is 6, the hundredths value is 3, and the thousandths value is 8. Breaking this down, we could say that that 0.638 is equal to:

(6 x 0.1) + (3 x 0.01) + (8 x 0.001)

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