Terminating Decimal

Decimals represent a fraction or part of a whole number. These decimals are of two types, terminating decimals and non-terminating decimals. Terminating decimals are those decimal numbers where the values after the decimal point come to an end. For example, if we divide \(\frac{1}{8} \) we get 0.125, which is a terminating decimal because it ends after two decimal places.

Non-terminating decimals are those decimal numbers that do not end after some decimal value.

These decimals are further divided into two categories:

• Non-terminating recurring, where we get the same value repeating continuously. For example, \(\frac{1}{3} \)
   
• Non-terminating non-recurring decimals, where the decimal is not ending, and it does not have a repeating value as well. For example, the value of \(π\).

You can determine whether a decimal number is terminating by checking the following conditions: 
 

• The terminating decimal will always have finite numbers after the decimal point. That means the number will stop after some digits.
   
• All terminating decimals are rational numbers. 
   
• Fractions that have denominators in the form \(2m \times 5n \), If m and n are non-negative integers, the decimal terminates.
   
• If the denominator of a fraction is in the form of 2^m × 5ⁿ, where m and n are positive integers, then it has a terminating decimal expansion.

Identifying terminating decimals can be easy if you know the right tricks. Here are some simple tips to help you quickly recognize and remember them.

• Check the denominator: A fraction \(\frac{p}{q} \) has a terminating decimal only if the denominator contains only the prime factors 2 and/or 5 (e.g., \(\frac{1}{8} = 0.125, \quad \frac{1}{25} = 0.04 \)). 
   
• Divide and observe: If a fraction converts into a decimal that stops after a few digits, it is terminating. If it keeps going with a pattern, it is a non-terminating recurring decimal. 
   
• Use prime factorization: Factorize each denominator and check whether it contains 2 and/or 5 as its factors. If yes, then it is terminating, otherwise it is non-terminating. 
   
• Memorize common examples: Knowing simple terminating decimals can help you recognize patterns. For example, \(\frac{1}{2}, \frac{3}{8}, \frac{1}{4}, \text{etc.} \)
   
• Practice with division: Convert fractions to decimals using long division. The long division reveals whether the decimal terminates or repeats, to find whether the decimal number is terminating or non-terminating. 

Terminating decimals are not just used in math class, they play an important role in everyday life. From money and measurements to science and sports, these decimals help us make accurate calculations in various real-world situations.
 

• Money and currency: Since most countries limit currency values to two decimal places, terminating decimals, like $4.75. $89.5 are commonly used in payments and financial transactions.
   
• Measurements: We represent length, weight, and volumes frequently by terminating decimals such as 2.5 meters, 1.75 liters, 3.2 kilograms respectively.
   
• Time calculation: Time is sometimes represented in decimal forms, such as 1.5 hours (1 hour 30 minutes) or 2.75 hours (2 hours 45 minutes), to schedule, work hours, or travel durations.
   
• In medical and science: In experiments, we use terminating decimals of the required solution or a chemical to give a precise by product.
   
• Engineering and construction: We use terminating decimals in dimensions and specifications for construction or manufacturing.