Decimal Representation of Irrational Numbers

The decimal representation simply expresses any given number using decimal digits. It depends on whether the digits repeat, terminate, or continue infinitely after the decimal point. Let’s see how decimal numbers are categorized. 

• Terminating decimals: These decimals end after a finite number of digits. For example, 3.25, 0.5, 27.2, etc.
  
   
• Non-terminating decimals: These decimals continue indefinitely without ending. They are further classified into two types. 
  
   
• Recurring decimals: A specific sequence of digits repeats at regular intervals. For example,0.333…, 94346.747474…, 573.636363…, etc.
  
   
• Non-recurring Decimals: These decimals go on infinitely without any repeating pattern. For example, 743.872367346…, 7043927.78687564…, etc.

Real numbers that cannot be simplified into fractions are called irrational numbers, and their decimal expansions are non-terminating and non-repeating.

Since irrational numbers have non-repeating, non-terminating decimal expansions, it is impossible to convert them into fractions. For example, π ≈ 3.14159265… is an irrational number because its decimal form never ends or follows a repeating pattern.

However, for practical calculations, π is often approximated as \(\frac{22}{7} \)or 3.14, even though this is not its exact value.

Irrational numbers have non-repeating, non-terminating decimals, which can be tricky to handle. The following tips and tricks simplify working with decimal representations.
 

• Memorize the common irrationals like π ≈ 3.14159 and e ≈ 2.71828. 
  
   
•  Understand patterns by recognizing decimals that never repeat or terminate.
  
   
• Round to a suitable number of decimal places depending on the required precision.
  
   
• Use fractions or radicals to simplify calculations instead of long decimals.
  
   
• Use calculators or software for accurate decimal expansions in complex problems.

Irrational numbers may seem abstract, but their decimal representations play a crucial role in many real-world applications. From engineering to finance, these numbers help ensure accuracy in various fields.

• Science and Medicine: In the fields of chemistry or physics, there are mentions of values like  'π' 'e’, which belong to the set of irrational numbers. Also, these values are used in MRI or CT scans for analyzing issues.
  
   
• Engineering and Construction: In civil engineering, precise measurements often include irrational numbers. For decimal approximations, these values are applicable.
  
   
• Finance and Economics: In the financial field, compounding interest, or continuous growth calculation, require Euler’s number application. Applications of irrational numbers improve accuracy in financial predictions.
  
   
• Technology and Computing: Computer algorithms, encryption methods, and simulations often involve irrational numbers. They are used for precision in calculations and security applications. 
  
   
• Art and Architecture: Many designs in art and architecture rely on the golden ratio (ϕ), an irrational number approximately equal to 1.618. It is used to create aesthetically pleasing proportions in buildings, paintings, and sculptures, demonstrating the practical use of irrational numbers in creative fields.