What makes a number mean what it does? Is it simply the digits in the number? For example, let's consider the number 521.
If we switch the digits around, we might get 125. Definitely not the same number. So, it's not just the digits that make a number; it's the digits and their positions within the number. A digit's position in a number and its resulting value is called place value.
The place value chart for our number system looks like this:
If we place our original number, 521, into this chart, it now looks like this:
Since the digit 5 is in the hundreds place, it means that we have five hundreds, or 500, of whatever we're talking about. Since the two is in the tens place, it means two tens, or 20. And since one is in the ones place, it means one one, or 1.
When we write the number 521, what that number really means is that we have the total of 500 + 20 + 1. We've expanded the number to show the value of each of its digits. When we expand a number to show the value of each digit, we're writing that number in expanded form.
Let's try it with a different, bigger number: 1,234,567. We'll start by placing the number's digits in the place value chart.
Now, let's use the chart to determine the value of each digit.
- Since the 1 is in the millions place, it means one million, or 1,000,000
- Since the 2 is in the hundred thousands place, it means two hundred thousands, or 200,000
- Since the 3 is in the ten thousands place, it means three ten thousands, or 30,000
- Since the 4 is in the thousands place, it means four thousands, or 4,000
- Since the 5 is in the hundreds place, it means five hundreds, or 500
- Since the 6 is in the tens place, it means six tens, or 60
- Since the 7 is in the ones place, it means seven ones, or 7
If we write the number out in expanded form, showing the value of each digit, we would write it as 1,000,000 + 200,000 + 30,000 + 4,000 + 500 + 60 + 7. And that's all there is to it!
That's all simple enough, but what if there's a zero in a number? Let's look at 4,803. We know from our previous work that the 4 really means 4,000, the 8 means 800, and the 3 means 3. If we were to use strict expanded form, and use every digit, our number would look like this: 4,000 + 800 + 0 + 3.
But the zero doesn't affect the number at all because it doesn't add any value to it. When we're writing numbers in expanded form, we don't have to worry about the zeroes. This means we can shorten the expanded number to 4,000 + 800 + 3.
Expanded Form to Standard Form
Converting numbers the other way, from expanded to standard form, is simple. All we need to do is add the given values. For example, if we have the expanded number 1,000 + 400 + 60 + 5, we can treat it as a long addition problem:
So, 1,000 + 400 + 60 + 5 is the same number as 1,465.
A digit's position in a number and its resulting value is called place value. When we expand a number to show the value of each digit, we're writing that number in expanded form.
Writing numbers in expanded form just means that we're showing the value of each digit in the number. With the exception of zeroes, the larger the number, the longer its expanded form will be. This form is handy because it shows what a number really means by defining each of its digits.
Standard Form vs. Expanded Form
|Standard Form||Expanded Form|
When you are finished, you should be able to:
- Convert a number from standard form to expanded form and vice versa
- State the place value of a given digit in a number
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What is Expanded Form in Math? - Definition & Examples Quiz
Instructions: Choose an answer and click 'Next'. You will receive your score and answers at the end.
In the number 45,902, what is the value of the 5?
Practice Questions on Expanded Form
1. Write the following number in expanded form: 23,465.
2. Write in expanded form: 5,065.
3. Convert to standard form: 40000 + 5000 + 700 + 60 + 8.
4. Convert to standard form: 8000 + 700 + 5.
1. Considering the place value of each digit with 2 being in the ten-thousands place, 3 in the thousands place, 4 in the hundreds place, 6 in the tens place and 5 in the ones place, we get
23,465 = 20,000 + 3,000 + 400 + 60 + 5 as the expanded form.
2. Considering the place value of each digit with 5 in the thousands place, 0 in the hundreds place, 6 in the tens place, and 5 in the ones place, we get 5,065 = 5,000 + 60 + 5 as the expanded form.
3. To convert to standard form, we simply look at the place values of the numbers and add them to obtain 40,000 + 5,000 + 700 + 60 + 8 = 45,768 as the standard form of the number. Really we just need to line up the numbers starting with the highest order down to the lowest order.
4. To convert to standard form, we simply look at the place values of the numbers and add them to obtain 8,000 + 700 + 5 = 8,705 as the standard form of the number. Really we just need to line up the numbers starting with the highest order down to the lowest order. Notice there is a zero in the tens place as there is no tens term in the expanded form.
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