## Introduction to Decimal Number

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.

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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

1. Question: Why Expanded Form of Decimals Required in Mathematics?

Answer: To understand a number better, we need an expansion of it, which is given by expanded form. The value of each digit based on its place value can be understood from the expanded form. Expanded form of a number is a crystal clear understanding of it which is much needed in the beginning stage of learning numbers.

2. Question: In What Case, Expanded Form Only Involves Multiples of 10.

Answer: Yes. This is possible when the given number doesn't involve decimal (.) at all. Observe that we used to write the number after decimal by dividing by 10 or multiples of 10 and numbers before multiple by multiplying with 10 or multiples of 10. So, if a number doesn’t have any digit after the decimal, then in the expansion we don’t need to divide the place value with 10 or it’s multiples.