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Difference Between A Fraction and A Rational Number

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What are Fraction and Rational Numbers: Introduction

To explain fraction and rational number: Fractions represent numbers as parts of a whole or divisions, while rational numbers encompass fractions, whole numbers, and integers. Rational numbers are expressed as fractions or can be in the form of decimals, both finite and recurring. Understanding fractions and rational numbers is essential in various mathematical operations and real-life applications involving parts, divisions, and relationships between quantities.


Defining Fractions

Fractions are numerical expressions used to represent parts of a whole or divisions of quantities. Here are some related points about fractions:


  • Numerator and Denominator: A fraction consists of two parts: a numerator and a denominator. The denominator is the total number of equal parts that make up the whole, whereas the numerator is the number of equal parts that are being taken into account or tallied.

  • Proper and Improper Fractions: Fractions can be classified as proper or improper. A fraction is called a proper fraction if the numerator is smaller than the denominator. For example, 2/5 is a proper fraction. A fraction is called an improper fraction if the numerator is equal to or greater than the denominator. For example, 7/4 is an improper fraction.

  • Equivalent Fractions: Equivalent fractions represent the same value, even though they may have different numerators and denominators. They have different symbols but represent the same part-to-whole relationship. For example, 3/5, 6/10, and 15/25 are all equivalent fractions.

  • Mixed Numbers: The combination of a whole number and a proper fraction is known as mixed numbers. They are useful for representing quantities that are greater than one. 

  • Operations with Fractions: Fractions can be operated on using various mathematical operations, including addition, subtraction, multiplication, and division. The operations involve manipulating the numerators and denominators while maintaining the relationship between the parts and the whole.

  • Applications: Fractions are used in everyday situations such as cooking, measurements, ratios, proportions, and comparisons. They are also essential in mathematical fields like geometry, algebra, and statistics.


Understanding fractions and their properties is crucial for developing mathematical fluency, solving problems involving parts and divisions, and effectively communicating and interpreting numerical relationships.


Defining Rational Numbers 

Rational numbers are numbers that can be expressed as fractions, where the numerator and denominator are both integers (whole numbers) and the denominator is not zero. Here are some related points about rational numbers:


  • Fractions and Integers: Rational numbers include both fractions and integers. Fractions represent parts of a whole, while integers represent whole numbers without fractional or decimal parts. For example, (-2), 3/4, and 7 are all rational numbers.

  • Terminating and Recurring Decimals: Rational numbers can be expressed as decimals, which can either terminate (end) or recur (repeat a pattern). Terminating decimals have a finite number of digits after the decimal point, such as 0.75. Recurring decimals have an infinitely repeating pattern, such as 0.333..., which can be represented as 1/3.

  • Operations with Rational Numbers: The common mathematical operations of addition, subtraction, multiplication, and division can be performed on rational numbers. The operations involve manipulating the numerators and denominators while preserving the properties of rational numbers.

  • Ordering Rational Numbers: Rational numbers can be compared and ordered on a number line. Larger rational numbers have numerators that are greater or denominators that are smaller, while smaller rational numbers have numerators that are smaller or denominators that are greater.

  • The density of Rational Numbers: Rational numbers are densely distributed on the number line, meaning that between any two rational numbers, there exists an infinite number of other rational numbers. This property makes rational numbers useful for precise measurements and calculations.

  • Applications: Rational numbers are used in a wide range of applications, including measurements, finance, engineering, and scientific calculations. They provide a versatile and accurate way of representing quantities and making calculations.


Understanding rational numbers is fundamental in mathematics, as they encompass fractions, integers, and decimal representations. They allow for precise and flexible mathematical operations and provide a foundation for more advanced mathematical concepts.


Fraction and Rational Number Differences

Category

Fractions

Rational Numbers

Representation

Fractions are represented as a ratio of two integers, that are numerator and denominator, separated by a fraction bar.

rational numbers encompass both fractions and other numbers that can be expressed as a fraction, including whole numbers and integers.

Inclusivity

Every fraction is a rational number. Fractions are a specific subset of rational numbers.

Not every rational number is a fraction. Rational numbers encompass a broader range of numbers, including fractions, whole numbers, and integers.

Decimal Representation

Fractions can have decimal representations that are either terminating (ending) or recurring (repeating a pattern).

Rational numbers can have both terminating and recurring decimal representations, as well as non-repeating decimals.

Mathematical Operations

While adding or subtracting fractions, the denominators need to be made equal. Then the standard addition or subtraction takes place on numerators. 

While adding or subtracting rational numbers, the numbers are simply added or subtracted by using the standard mathematical operations of addition, and subtraction.

Example

Examples of fractions can be given by 1/2, 3/4, 8/5, etc.

Examples of rational numbers can be given by 1 (integer), 3/5 (fraction), 2.25 (decimal), (-7) (negative), etc.

Applications

Fractions are used in everyday situations such as cooking, measurements, ratios, proportions, and comparisons. They are also essential in mathematical fields like geometry, algebra, and statistics.

Rational numbers are used in a wide range of applications, including measurements, finance, engineering, and scientific calculations. They provide a versatile and accurate way of representing quantities and making calculations.


Summary

From this article, it can be concluded that fractions are numerical expressions used to represent parts of a whole or divisions of quantities, whereas rational numbers are numbers that can be expressed as fractions, where the numerator and denominator are both integers (whole numbers) and the denominator is not zero. Fractions are a specific subset of rational numbers. Not every rational number is a fraction, but every fraction is surely a rational number. Rational numbers encompass a broader range of numbers, including fractions, whole numbers, and integers. The article also discusses the characteristics of fraction and rational number and their diverse applications in practical contexts. It touches upon their usage in measurement, cooking, finance, and various mathematical operations such as addition, subtraction, multiplication, and division.

Last updated date: 26th Sep 2023
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FAQs on Difference Between A Fraction and A Rational Number

1. Are the fractions and the rational numbers the same?

No. Not every rational number is a fraction, but every fraction is surely a rational number. Fractions are a specific subset of rational numbers.

2. What are the two types of fractions?

Fractions can be classified as proper or improper. A fraction is called a proper fraction if the numerator is smaller than the denominator. For example, 2/5 is a proper fraction. A fraction is called an improper fraction if the numerator is equal to or greater than the denominator. For example, 7/3 is an improper fraction.

3. How rational numbers can be represented as decimals?

Rational numbers can be expressed as decimals, which can either terminate (end) or recur (repeat a pattern). Terminating decimals have a finite number of digits after the decimal point, such as 0.75. Recurring decimals have an infinitely repeating pattern, such as 0.333..., which can be represented as 1/3.

4. Write the smallest and the largest rational numbers.

The smallest and largest rational numbers do not exist as specific values because rational numbers extend infinitely in both positive and negative directions on the number line.

5. What are some applications of rational numbers?

Rational numbers are used in a wide range of applications, including measurements, finance, engineering, and scientific calculations. They provide a versatile and accurate way of representing quantities and making calculations.