120 LearnersLast updated on July 25th, 2025
Exactness of Decimal Representations
The exactness of decimal representation refers to whether a number has a finite decimal (ending after a few digits) or an infinite decimal (repeating or non-repeating). This concept is important in mathematics, especially in rational numbers, real numbers, and fractions.
What is the Exactness of Decimal Representations?
Decimal numbers can be classified based on whether they come to an end or continue infinitely. That is, whether decimals stop after a certain number of digits, while others continue infinitely. The exactness of the decimal indicates the type of decimal number we are working with. There are two types of decimal numbers: terminating and non-terminating. Let’s understand the difference.
What Are the Types of Decimals?
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, 15 ÷ 2 = 7.5
Non-terminating decimals: Those decimal numbers that continue infinitely without ending. Non-terminating decimals are, then again, divided into two types: repeating and non-repeating.
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 = 0.285714285714…
Non-repeating Non-terminating Decimals: These decimal numbers go on infinitely without ever forming a repeating pattern. For example, the mathematical constant Pi is 3.1415926535…
Exactness of Decimal Representations of Irrational Numbers
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. These are the following characteristics of irrational decimal numbers.
- The decimal representation never ends.
- The digits do not follow a fixed repeating pattern.
- Unlike rational numbers, irrational numbers cannot be written as pq (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 () = 1.6180339887…
Real Life Applications of Exactness of Decimal Representations
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.
Common Mistakes of Exactness of Decimal Representations
When working with decimals, students often make mistakes that might confuse terminating, repeating, and non-repeating decimals. Understanding these errors and how to avoid them helps improve accuracy in identifying different types of decimals.
Mistake 1
Believing that all irrational numbers have similar decimal patterns
Different irrational numbers have unique decimal expansions. Some, like π, seem random, while others, like the golden ratio ( 1.6180339887…), follow patterns but never repeat.
Mistake 2
Assuming that a fraction always has a terminating decimal
Check the denominator of the fraction (after simplifying). If it has only 2s and/or 5s as prime factors, it will terminate. Otherwise, it will repeat.
Mistake 3
Misidentifying a repeating decimal as a terminating decimal
Look carefully at the decimal expansion. If a pattern of digits repeats indefinitely, it is a repeating decimal, not a terminating one. For example, 16 = 0.1666…is repeating, not terminating.
Mistake 4
Thinking that all non-terminating decimals are irrational
Repeating non-terminating decimals (e.g., 0.3 or 13) are rational because they can be written as fractions. Only non-repeating, non-terminating decimals are irrational (e.g., π, 2).
Mistake 5
Confusing rounding with exact representation
A round decimal (e.g, π 3.14) is not an exact value. Always distinguish between an approximation and an exact representation when working with decimals.
Solved Examples for Exactness of Decimal Representations
Problem 1
Lily has $5.75, and she buys a toy for $2.50. How much money does she have left?
3.25
Explanation
Subtract dollars and cents carefully.
5 dollars minus 2 dollars = 3 dollars
75 cents minus 50 cents = 25 cents
Thus, Lily has $3.25 left.
Problem 2
A recipe needs 2.5 cups of flour, but Alex accidentally adds only 2.25 cups. How much more flour does he need to add?
0.25 cups
Explanation
To find the missing amount, we must subtract 2.25 from 2.5.
Then converting to fractions:
2.5 = 212 = 2.50
2.25 stays the same
2.50 – 2.25 = 0.25 (¼ cup)
Thus, Alex needs to add ¼ cup more.
Problem 3
In the morning, the temperature was 18.6° C, and in the afternoon, it rose to 22.3° C. How much did the temperature increase?
22.3 – 18.6 = 3.7° C
Explanation
Subtract the morning temperature from the afternoon temperature.
22.3 – 18.6 = 3.7° C
So, the temperature rose by 3.7 °C.
FAQs Related to Exactness of Decimal Representations
1.Can all numbers be represented exactly as decimals?
2.Why do GPS systems need precise decimals?
3.Why is exact decimal representation important in money calculations?
4.How can children in Vietnam use numbers in everyday life to understand Exactness of Decimal Representations?
5.What are some fun ways kids in Vietnam can practice Exactness of Decimal Representations with numbers?
6.What role do numbers and Exactness of Decimal Representations play in helping children in Vietnam develop problem-solving skills?
7.How can families in Vietnam create number-rich environments to improve Exactness of Decimal Representations skills?
Hiralee Lalitkumar Makwana
About the Author
Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.
Fun Fact
: She loves to read number jokes and games.
