A Basic Guide to Decimals

Decimals are one of the most basic and fundamental concept in mathematics. In a number line, decimals are placed between whole numbers.  

Decimals are an alternate way of representation for expressing fractions. This set of numbers separates integer and fractional parts with a decimal point. For example, you bought a cake, and it costs 5 dollars and 25 cents.

So, using decimals, we can precisely define the price of the cake as $5.25. In the above example, a decimal point separates 5 which is a whole number and 25 which is a fractional part. 

The history of decimals spans thousands of years with a fascinating journey. Several civilizations and cultures contributed to the development of decimals. During the 16th century, decimal numbers began to appear more in mathematical works.

In 1614, John Napier introduced decimal point notation through his work “Mirifici Logarithmorum Canonis Descriptio”.

In 1585, a mathematician, Simon Stevin published a book named “Decimal Arithmetic” and his book promoted decimal fractions.

With the contribution of zero from the Indian mathematician and astronomer, Aryabhata, the concept of zero has greatly affected the modern decimal notation.  
 

Decimals are a set of numbers that express both whole numbers and fractional parts. They play a vital role in everyday calculations to scientific operations. Several properties that make decimals a convenient way to represent numbers are listed below:

• Each digit in a decimal number has a unique place value depending on its position. Digits on the left of the decimal points have values of units, tens, hundreds, and so on. Digits on the right have values such as tenths, hundredths, thousandths, and so on. For example, 30.567; in this the value on the left are the whole numbers which has 3 in the tens place and 0 in the units place, and 5 is in tenths place, 6 is in hundredths place and 7 is in thousandths place
   

• Before adding or subtracting numbers with decimals, the decimals should be aligned by the decimal point. For example, if we add 13.56 + 9.5, 23.06 = 13.56 + 9.50 
   

• When we multiply decimals, we multiply them as integers, and after we get the result, we have to see where the decimal point was there in the question and then put the decimal point there. For instance, 3.5 × 1.4 = 4.9. 35 × 14 = 490. Here, we consider it as a whole number and then multiply it. Finally, place the decimal point in the result. 
  
   
• When we divide numbers with decimals, the decimal point in the dividend and divisor are adjusted to make it a whole number, to make it a whole number we multiply the decimal point numbers by 100. For example, 4.0 ÷ 0.2. To convert the decimal point number given in the question, we multiply by 100:. 4.0 x 100 = 40 and 0.2 x 100 = 2. 40 ÷ 2 = 20. Therefore, 4.0 ÷ 0.2 = 20. 
  
   
• There are some decimals that lie between two decimals. For example, between 0.1 and 0.2, there are some decimals like 0.15, 0.17 and so on.

Decimals are classified mainly into three categories to enhance our understanding of the concept. The categorization depends on the types of numbers that come after the decimal point. Terminating decimals, non-terminating decimals, and recurring decimals are the three main categories of decimals. 

i) Terminating Decimals

After the decimal point, terminating decimals have a finite number of digits. These digits end or terminate after a certain point, and they do not repeat. Also, terminating decimals are easy to convert into fractions. The result is the decimals from the divisions that have no remainder. 
For instance, 
0.25 is a terminating decimal. 
0.25 can be written as,
0.25 = 25/100 = 1/4 

ii) Non-Terminating Decimals

Non-terminating decimals have infinite numbers of digits after the decimal point. Sometimes, non-terminating decimals repeat in a pattern. 

For example, 1.677
0.5555… is a non-terminating decimal. 
The two types of non-terminating decimals are recurring decimals and non-recurring decimals. The above example is a recurring decimal. 

iii) Recurring Decimals

After the decimal point, recurring decimals follow a pattern of repeated digits. Recurring decimals belong to a type of non-terminating decimals. 
1.6777... is also an example of recurring decimals. Here, the repeating digit is 7 and it continues infinitely. These repeated decimals can be written in fractions. 

For example, 
1/7 = 0.142857142857…
 1/3 = 0.3333…
 

Decimals provide precise results for calculations related to daily life and academic concepts. In advanced mathematics, understanding topics such as algebra, percentages, and ratios requires proper mastery of decimals. Decimals are important to measure, calculate and solve problems and get a precise result.

Students who have a good comprehension of decimals can focus on scientific engineering, technical work, computer programming, and coding. 
 

Identifying the types of decimals helps students solve mathematical calculations more effectively. Pure decimals and mixed decimals are the two most common types of decimals. 

i) Pure Decimals

The decimals which have digits only after the decimal point are known as pure decimals. These decimals do not have a whole number, just the fractional part. The value of pure decimals will be less than 1. Generally, to denote fractions, we use pure decimals. 

For example, 
0.5
0.25
0.0089
These are some examples of pure decimals. 

ii) Mixed Decimals

 Mixed decimals refer to the decimals which have an integer part and a fractional part. A decimal point separates these two parts. The left side of the decimal point has integers, and the other side is the fractional. 

For instance, 
3.67,
11.42, and
540.9

These are mixed decimals, which contain both integers and fractional parts. 
 

To solve mathematical problems effectively, we need to understand some tips and tricks. Here are some useful tips and tricks for kids to learn more about decimals. 

• Always remember the place values of decimal digits. The values of digits, after the decimal point, are such as tenths, hundredths, thousandths, and so on. 

• Align the decimal points when we add or subtract the numbers.

• When we multiply, overlook the decimal point and multiply it as a whole number. After that, apply the point to the result.

• If some numbers have fewer place values, add zeroes to them to make calculations straightforward. 
   

In our daily life, we constantly use decimals for various purposes. For example, if we want to measure our weight, the accurate measurement will be in decimals. Likewise, the real-world applications of decimals are countless.

Financial Transactions:

Prices in shops and banks are written in decimal form (e.g., $8.55).
Interest rates for loans are often in decimals (e.g., 1.7%).

Measurement of Weight and Length:

Body weight is measured in decimals (e.g., 65.4 kg).
Objects' lengths are measured using decimals (e.g., 2.75 meters).

Distance Calculation:

Distances are measured with decimal precision (e.g., 5.8 km).

Scientific Applications:

Scientific calculations use decimals for accuracy.

Time Measurement:

Decimal values are used in time calculations (e.g., an athlete completes a race in 7.65 seconds)