How to Write Decimals in Expanded Form with Clear Examples
A decimal number is a way to represent fractions. In fraction numbers, we represent them in the form of \[\frac{p}{q}\] , where q is not equal to zero. Now from here when we move one step further by dividing p by q by the long division method, then we get a decimal number. For example: suppose we have a fraction \[\frac{9}{5}\]. when we divide it, we’ll get the below result.
[Image will be Uploaded Soon]
Observe then we got 1.80. The dot (.) is the decimal representation and it means that after this all numbers are partitioned.
The Expanded Form of Decimal Numbers
Before diving into expansion first we need to understand the place value system. The place value refers to the value at a place in a number. According to the Indian numeral system, starting from right to left we count places as unit place, tens place, hundreds of places, thousand places, ten thousand, lakh then ten lakh and so on. On the other hand, according to the International numeral system, we start counting from right to left as once, tens, hundreds, thousands, ten thousand, hundred thousand, million then ten million and so on. Later we’ll see the expanded form of 50 in the section.
Decimal Expansion Definition
Now to expand any number we need to divide all the place values by their place number. For example, suppose we want to expand 254 then observe that 4 will be at a unit place, 5 at tens and 2 hundred places. Now we’ll multiply these place values with their place numbers.
4 multiplied by 1, 5 multiplied by 10 and 2 multiplied by hundred. And in the end, we’ll add them up.
As shown below:
254= 2 Hundreds + 5 tens + 4 ones
⇒ 254 = 2 x 100 + 5 x 10 + 4 x 1
This is the required expansion on the given number.
How to Write Decimals in Expanded Form?
Now if we have any decimal number, then instead of multiplying we’ll divide the place value with their place number for all numbers written after the decimal place. Let’s take an example of 3.482. Its expansion will be -
3.482 = 3 ones + 4 ones of tenth + 8 ones of hundredth + 2 ones of thousandth
⇒ 3.482 = 3 x 1 + \[\frac{4}{10}\] + \[\frac{8}{100}\] + \[\frac{2}{1000}\]
It required the expansion of a given decimal. We can also solve it one step further.
⇒ 3.482 = 3 + 0.4 + 0.08 + 0.002
This expansion is called the expanded form of decimal numbers.
Applications of Expanded Form with Decimals
The scope of decimal numbers is within the areas where accuracy and precision are required. The concept of expanded form is beneficial in comprehending the worth of a number also because of the numeric value of a quantity. Even in our real-life something is too precious like gold or silver, we used to take it’s measurement 3 or sometimes 4 places after the decimal.
Did you know?
For Sand Reckoner, mathematician Archimedes invented a decimal positional system to represent large numbers that were multiples of 10.
In Indus Valley civilization, weight used in the ratio of 1⁄2, 1⁄5, 1⁄10, 1⁄20 and multiples of 10 for trade purposes.
In ancient China, (during 305 BC) for mathematical operations like multiplication, Rod calculus used bamboo strips with a decimal system.
Decimal Expansion Example
1. How to Write Decimals in an Expanded Form of the Following Decimal Numbers.?
a. 3479.105
⇒ 3 thousand + 4 hundred + 7 ten + 9 one + 1 tenth + 0 hundredth + 5 thousandth
⇒ 3000 + 400 + 70 + 9 + \[\frac{1}{10}\] + \[\frac{0}{100}\] + \[\frac{5}{1000}\]
b. 7833.45
⇒ 7 thousand + 8 hundred + 3 ten + 3 one + 4 tenth + 5 hundredth
⇒ 7000 + 800 + 30 + 3 + \[\frac{4}{10}\] + \[\frac{5}{100}\]
c. 21.1097
⇒ 2 ten + 1 one + 1 tenth + 0 hundredth + 9 thousandth + 7 ten thousandth
⇒ 20 + 1 + \[\frac{1}{10}\] + \[\frac{0}{100}\] + \[\frac{9}{1000}\] + \[\frac{7}{10000}\]
2. Write the Following as a Decimal Fraction.
a. 6 hundred + 9 ten + 8 tenth + 4 thousandth
⇒ 690.804
b. 9 hundred + 4 ten + 3 tenth + 4 hundredth
⇒ 940.34
FAQs on Expanded Form with Decimals Made Easy: Definition, Uses & Examples
1. What is the expanded form with decimals?
The expanded form with decimals is a method of writing a decimal number by breaking it down into the sum of the place values of each of its digits. This includes the whole number part and the fractional part (the digits after the decimal point). For example, the number 12.3 is the sum of 1 ten, 2 ones, and 3 tenths.
2. How do you write the decimal 3.405 in expanded form?
To write 3.405 in expanded form, you identify the place value of each digit. The '3' is in the ones place, '4' is in the tenths place, '0' is in the hundredths place, and '5' is in the thousandths place. Therefore, the expanded form is 3 + 0.4 + 0.005. Alternatively, using fractions, it can be written as 3 + 4/10 + 0/100 + 5/1000.
3. What are the different ways to represent the expanded form of a decimal number like 45.67?
There are three common ways to represent the expanded form of a decimal, which helps in understanding the number's composition. For the number 45.67, these are:
- Place Value Form: (4 × 10) + (5 × 1) + (6 × 1/10) + (7 × 1/100)
- Decimal Form: 40 + 5 + 0.6 + 0.07
- Fraction Form: 40 + 5 + 6/10 + 7/100
4. What is the importance of writing decimals in expanded form?
Writing decimals in expanded form is important because it clearly shows the value of each digit in a number. This strengthens a student's understanding of the place value system, which is crucial for accurately comparing, ordering, adding, and subtracting decimal numbers. It builds a strong foundation for more advanced mathematical concepts.
5. How does the place value chart help in writing the expanded form of decimals?
A place value chart visually organises each digit according to its position relative to the decimal point. When you place a number like 28.19 in the chart, you can easily see that '2' is in the tens place (value 20), '8' is in the ones place (value 8), '1' is in the tenths place (value 0.1), and '9' is in the hundredths place (value 0.09). Adding these values together (20 + 8 + 0.1 + 0.09) directly gives you the expanded form.
6. What is a common mistake when writing the expanded form of a decimal with a zero, such as 60.08?
A common mistake is incorrectly handling the zeros. In 60.08, the '0' in the ones place and the '0' in the tenths place act as placeholders. Students might skip them, which is conceptually fine in the final sum but can lead to errors. The correct expanded form is 60 + 0 + 0.0 + 0.08, or simply 60 + 0.08. The key is to recognise that the '8' is in the hundredths place, not the tenths place, because the zero holds the tenths position.
7. How does the expanded form of a whole number differ from that of a decimal number?
The main difference lies in the types of place values used. The expanded form of a whole number (e.g., 352 = 300 + 50 + 2) only involves sums of whole numbers based on powers of 10 (100s, 10s, 1s). The expanded form of a decimal number (e.g., 3.52 = 3 + 0.5 + 0.02) includes both a whole number part and a fractional part, where digits after the decimal point are represented as tenths, hundredths, thousandths, and so on.
