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Study Guide

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In a weird way, numbers are like chameleons: they change with their location. Depending on its location, a number can mean hundreds, even millions, or just tenths of one. This is the magic of **place value**.

When alone, each number (or digit) is in the ones place. That means its value is simply the digit itself. For example, 9 stands for 9 ones, and 4 stands for 4 ones. The ones place is pretty straightforward. Kind of like a granny that has given up on manners and tells it like it is. (Older people are the best.) It looks like this:

However, sometimes we have more than 9 things. To write more than 9, we move the 1 to a different place, the tens place, which looks like this:

Consider 25 as an example. The 2 in 25 is in the tens place, making it worth 2 × 10, or 20. That's what makes 25 worth, well, 25, not 7 or 10 or something else.

Let's take it a bit further and change the place of the 1.

Let's consider the number 354. Since the 3 is in the hundreds place, it's worth way more than just 3. It's worth 3 × 100, which is 300. The 5 is in the tens place, making it worth 5 × 10, or, 50. The 4 is in the ones place, and since 4 × 1 = 4, it stays 4. When we add up the value of each digit, we can start to understand why 354 is really worth (3 × 100) + (5 × 10) + (4 × 1) or 300 + 50 + 4.

We can always tell a digit's value based on where it's placed in a number.

Writing out each number as the sum of its values (think: 982 as 9 hundreds + 8 tens + 2) actually has a name: we call it the **expanded form** of the number. Expanded form just expands the number to show the value of each digit in the number. We can do this for 2384 by writing it like this:

(2 × 1000) + (3 × 100) + (8 × 10) + (4 × 1)

We can write 87,721 in expanded form, too:

(8 × 10,000) + (7 × 1000) + (7 × 100) + (2 × 10) + (1 × 1)

If a number is written as 1768 instead of (1 × 1000) + (7 × 100) + (6 × 10) + (8 × 1), then we say the number is in **standard form**, not expanded form.

Let's take a second to notice the commas we've inserted in some of our values. Commas are often used to the left of every three digits to make the number easier to read. We use the commas to separate the hundreds from the thousands, the thousands from the millions, the millions from the billions, and so on. We usually don't use 'em with simple four-digit numbers like 1000, but you'll see some textbooks write it out as 1,000. Either way is fine.

With place value and commas, we can make any digit worth millions, or billions, or even trillions. It's all about attitude—er, place value.

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