Session 1: What are Decimals? – Definition, Place Value Chart, Expansion, Types, Conversion and Interactives

During the Vedic period (6th and 7th centuries BC), motivated by astronomy, the use of a numerical system and basic mathematical operations developed in northern India. Hindu cosmology required the mastery of very large numbers, so used a set of 10 symbols. This system was introduced to Middle Eastern mathematicians, who later introduced the system to Europe. This is the origin of the Hindu Arabic number system and the decimal system used today. 
Hindu-Arabic numerals, set of ten symbols—1, 2, 3, 4, 5, 6, 7, 8, 9, 0—represent the decimal number system. 

Decimal Place Value Chart

In Latin, ‘Decimus’ means tenth, hence the word decimal. Therefore, the decimal system has ten as its base and is sometimes called a base-10 system. This number system is used to express the whole number and decimal fraction( fractions with denominators 10,100,1000 and so on ) together. Here, the whole number is separated from the decimal fraction by inserting a “.” called a decimal point. The decimal point form the centre of the number value system.

Decimal Numbers

A decimal number consists of the whole number part and the decimal fraction, or simply the decimal part. These two parts are separated from each other by the decimal point.

The value of a digit in a decimal number depends on its position or place in the number. Each place is ten times the value of the next place to its right.

Decimals in Expanded Form

We can stretch out a decimal to see how much value each digit of the decimal is worth. This is called expanded form. In the decimal number system, the place value of a digit decreases ten times from left to right and increases ten times from right to left. 

1) Express 987.65 in expanded form. 

We know, 

Since \(\frac{1}{10}\), \(\frac{1}{100}\) can be named as 0.1 and 0.01, an expanded form using decimals can be shown as:

Similarly, 

Reading and Writing Decimal

There are two ways to read a decimal number. The first way is to read the whole number, then “point”, then read the decimal parts separately. No matter how many digits are after that point, we read each one of them independently. For example, we read 58.43 as fifty-eight point four three.

The second way is to read the whole number part followed by “and” then to read the decimal part in the same way as we read whole numbers but followed by the place value of the last digit in the chart. For example, we read 7.812 as seven and eight hundred twelve thousandths.

How to Read Decimal Numbers?

To read decimal numbers, we use the following steps

Step 1: Read the whole number part.

Step 2: Read the decimal point as ”point”

Step 3: Read the number to the right of the decimal point independently.

                                                          or

Step 1: Read the whole number part.

Step 2: Read the decimal point as ”and”

Step 3: Read the number to the right of the decimal point and name the place of the last digit.

Equivalent Decimals

Equivalent decimals are decimal numbers that have the same value. For example, 0.7 and 0.70 are equivalent decimals and can be represented pictorially as follows. 

7 tenths take the same amount of space as 70 hundredths. 

⇒ 7 tenths =70 hundredths

                \(\frac{7}{10}\)=\(\frac{70}{100}\)

Thus, adding any number of zeros after the extreme right digit in the decimal part of a decimal number does not change the value. 

Like Decimals and Unlike Decimals

The decimal numbers having an equal number of decimal places are called like decimal numbers. For example:

i. 3.7, 162.9 and 45.0 are like decimals because all these numbers have one decimal place. 

ii. 3.98, 896.27 and 16.90 are like decimals because all these numbers have two decimal places.

iii. 0.286, 89.028 and 179.900 are like decimals because all these numbers have three decimal places.

The decimal numbers having a different number of decimal places are called unlike decimal numbers. For example:

The decimal numbers 2.3, 896.70, 10.529 are unlike decimal numbers because these numbers have different numbers of decimal places. 

How to Convert Unlike Decimals to Like Decimals?

Let us consider unlike decimal numbers 25.9 and 180.98. 

The number of decimal places in 25.9=1

and the number of decimal places in 180.98=2

To convert 25.9 and 180.98 to like decimal numbers, we should have two decimal places in 25.9. 

We know that any extra zero(s) written at the end of a decimal number does not change its value. Therefore, the number which is equal to 25.9 and has two decimal places will be 25.90.

Thus, 25.90 and 180.98 are like decimals.

Note: Any number of zeros may be put to the extreme right of the decimal part of a decimal without changing the decimal value. 

 Thus, 0.7=0.70=0.700 etc. 

DID YOU KNOW? The Dewey Decimal Classification (DDC), also called Dewey Decimal System, is the world’s most widely used way to organise library collections. It is a system for organising the contents of a library based on dividing all knowledge into ten groups, with each group assigned 100 numbers. These ten main groups are subdivided repeatedly to provide more specific subject groups.  The three-digit whole number designates a general category. The numbers after the decimal point indicate more specific topics. For instance, to find a book on exponents, you’d look for the call number 513.21. All natural sciences and mathematics books have a “5” in the hundreds place. A “1” in the tens place corresponds to mathematics. “3” in the ones place is arithmetic. A “2” in the tenth place corresponds to arithmetic operations. “1” in the hundredth place is exponents.

What are Decimals? – Examples

Example 1

Write the following in decimal notations and expanded forms. 

i. Five hundred forty-six thousandths

ii. Seventy-eight and thirteen ten-thousandths

 

i. Five hundred forty-six thousandths=\(\frac{546}{1000}\)=0.546.

   Expanded form=

 ii. Seventy-eight and thirteen ten-thousandths=

   Expanded form=

                            

Example 2

Write the following numbers in standard form as decimals. Also, write them in words in both ways. 

i. 

ii. 

 i. 

   In words, 

      a. Six thousand fifty-three point zero four eight. 

      b. Six thousand fifty-three and forty-eight thousandths. 

 

ii. 

   In words, 

      a. Five hundred two point nine zero five eight. 

      b. Five hundred two and nine thousand fifty-eight thousandths. 

 Remember this! 

• Decimals are an extension of our number system. 
• Decimal fractions are fractions whose denominators are 10, 100, 1000 etc. 
• A decimal has two parts: the whole number part and the decimal part. 
• Decimals having the same number of decimal places are called like decimals; otherwise, they are known as, unlike decimals.