Exact Decimal

An exact decimal is a decimal number that comes to an end after a finite number of digits. In other words, its digits do not continue infinitely. These decimals stop at a certain place value, making them easy to read and work with. Exact decimals, also known as terminating decimals, are decimals that terminate.

Examples of exact decimals include 3.5, 7.24, 12.875, and 2.1717.

Examples for non-exact decimals are 3.3333…, 5.252525…., 3.1415269…., 0.833333…. 

The decimal numbers can be categorized into two types: terminating and non-terminating. These non-terminating decimal numbers are divided into two: repeating non-terminating decimals and non-repeating non-terminating decimals. 

Terminating decimals: Decimal numbers in which the digits after the decimal point come to an end after a certain number of places. For example, dividing 15 by 2 gives 7.5.

Non-terminating decimals: Non-terminating decimals are decimal numbers that continue infinitely without ending. Non-terminating decimal numbers are further divided into two types: repeating and non-repeating decimals.

Repeating non-terminating decimals: These decimal numbers repeat a sequence of digits forever. It keeps repeating the same number or the same pattern. For example, 1 ÷ 3 = 0.33333333…, 2 ÷ 7 = 0.285714…

Non-repeating non-terminating decimals: These decimals continue infinitely without repeating any pattern. For example, the mathematical constant Pi is 3.1415926535…

A fraction produces an exact decimal only when its denominator is made up of the prime factors 2 and 5. That means the denominator must be of the form: \(2^m × 5^n\). 
 

Example 1: Simplifying \({1 \over 8}\)

Here, the denominator is 8 

It can be expressed as 2³ 

So, \({1 \over 8}\)= 0.125
 

Example 2: Simplifying \({1 \over 3}\)

Here, the denominator is 3, which cannot be expressed in the form 2^m or 5ⁿ

So,  \({1 \over 3}\) = 0.333… 

To convert a fraction into an exact decimal, we check whether the denominator contains only the prime factors 2 and 5. To convert fractions to exact decimals, follow these steps: 

• Simplify the fraction by factoring out any common factors from the numerator and denominator.
   
• Verify whether the denominator contains only the primes of 2 or 5.
   
• If so, multiply the numerator and denominator to make the denominator a power of 10. 
   
• Convert to decimal. 
   

Example 1: What is the exact decimal equivalent of \(7\over12\)
 

Here, the denominator12 = 2² × 3

Since it contains a 3, it cannot be converted into a terminating decimal. 

Therefore, \(7\over12\) is a non-exact repeating decimal: 

\(7\over12\) = 0.5833….
 

Example 2: What is the exact decimal equivalent of \(9\over25\)

Here, the denominator 25 = 5²

Making the denominator a power of 10 by multiplying the numerator and denominator by 4: 
 

\({9\over 25} = {(9 × 4)\over (25 × 4) }= {36\over 100}\)
 

Converting to decimal: \(36 \over 100\) = 0.36
 

Irrational numbers cannot be written as fractions of two integers (ab) and have decimals that never end or repeat. Their decimal expansion continues infinitely without any repeating pattern. The characteristics of irrational decimal numbers are as follows: 

• The decimal representation never ends.
   
• The digits do not follow a fixed repeating pattern.
   
• Unlike rational numbers, irrational numbers cannot be written as \(p/q\) (where p and q are integers, q ≠ 0).
   

Here are some examples of irrational numbers and their decimal representation:

• Pi (π) = 3.1415926535… (continues infinitely without repetition)
• Euler’s Number (e) = 2.7182818284…
• Square Root of 2 (2) = 1.4142135623…
• Golden Ratio (φ) = 1.6180339887…

Here are some tips and tricks, which will make it easier for us to identify and represent decimals.

• Understand the concept of decimals thoroughly. A decimal representation is exact if it ends after a few numbers.
   
• Remember that a fraction that is in its simplest form will have an exact decimal representation only if the denominator has no prime factors other than 2 or 5.
   
• Always try to reduce the fraction to its lowest form before checking for exactness.
   
• Keep in mind that while dividing manually, if the remainder becomes 0, it is an exact decimal and if the remainder is repeating, then it is a repeating decimal.
   
• Remember, when we are dealing with money measurement, or weights, we only use exact decimals for precise representation.
   

• Parents can use real-life examples to show children exact decimals through money, measurements, or shopping receipts, making the idea more relatable.
   

• Teachers can introduce factor trees to students to break down denominators into prime factors, quickly testing exactness. 

   

• Teachers can use decimal worksheets or a decimal calculator to help students understand the concept.
   

The exactness of decimal representations is important in many real-life applications where precision matters. Here are some examples showing why the exactness of decimal representation is important. 

• Pharmaceuticals: When measuring medicine dosages, decimal precision is crucial to ensure the correct amount is given. A small error in dosage can lead to ineffective treatment or dangerous side effects.
   
• Banking and finance: Money transactions require exact decimal values to ensure accurate interest calculations, tax computations, and balance updates. Even a small rounding error could lead to significant financial discrepancies. 
   
• GPS and navigation: Decimal precision is necessary in coordinates to ensure accurate location tracking and directions. Even a slight rounding error can misplace a location by several meters.
   
• Science and research: Scientists use exact decimal values in experiments, especially in chemistry and physics, where accurate measurements determine outcomes.
   
• Cooking and recipe making: We often use fractions to mention recipes which convert to exact decimals. We use it for precise measurements and consistency in cooking or baking.