Difference Between Fraction and Rational Number

Fractions and rational numbers are two mathematical concepts that are often used interchangeably. Fraction and Rational Number are not alike, and there are significant differences between them.

Here, we will have a look at the differences between fractions and rational numbers.

1. Definition

Fraction: Fraction is a number that represents a portion of a whole. Fractions are written as a ratio of two different integers, where the numerator represents the part and the denominator represents the whole.

Rational number: Rational number is a number that can be stated as the ratio of two different integers. Rational Number can be written in the form of a fraction, but it can also be denoted in a form of decimal or in form of a percentage.

2. Range of Numbers

Fraction: Fractions can represent any part of a whole, from the smallest fraction (1/∞) to the largest fraction (∞/1). It has to be noted that the set of fractions does not include all real numbers.

Rational Number: Rational numbers include all fractions as well as many other numbers that cannot be expressed as fractions. All real numbers that can be expressed as a ratio of two integers are included in the set of rational numbers.

3. Types of Fractions and Rational Numbers

Fraction: There are several types of fractions- Proper fractions, improper fractions, and mixed numbers.

1. Proper fraction is a fraction that represents where the numerator is smaller than the denominator
2. Improper fractions are fractions that represents where the numerator is larger than the denominator
3. Mixed number is a combination of a whole number and a proper fraction.

Rational Number: Rational numbers can be classified as either terminating or non-terminating.

1. Terminating rational numbers are those that have a fixed number of digits after the decimal point.
2. Non-terminating rational numbers are those that have an immeasurable number of digits after the decimal point.

4. Operations

Fraction: Fractions can be added, subtracted, multiplied, and divided. To add or subtract fractions, the denominators should be same, and if you have to multiply or divide fractions, you simply have to multiply or divide the numerators and denominators separately.

Rational Number: Rational numbers can also be added, subtracted, multiplied, and divided. In some cases, it may be necessary to alter irrational numbers to fractions before performing these operations.

5. Simplification

Fraction: With the use of GCF- Greatest Common Factor, fractions can be simplified by dividing the numerator and denominator. Abridging a fraction does not change its value but makes it easier to work with.

Rational Number: Rational numbers can also be simplified by reducing them to their lowest terms, whereas, irrational numbers cannot be simplified in this way.

6. Representation

Fraction: Fractions can be signified as a ratio, as a mixed number, or as a decimal. However, when represented as a decimal, the decimal may be either terminating or non-terminating.

Rational Number: Rational numbers can be represented as a fraction, as a decimal, or as a percent. When represented as a decimal, the decimal may be either terminating or non-terminating.

7. Irrational Numbers

Fraction: Fractions cannot represent irrational numbers, as they are numbers that cannot be showcased as the ratio of two integers. Samples of irrational numbers are- √2, π, and e.

Rational number: Rational numbers include all the numbers that can be represented as the ratio of two integers. However, they do not include irrational numbers.

8. Use in Everyday Life

Fraction: Fractions are commonly used in everyday life to represent parts of a whole, such as when cooking or measuring ingredients.

Rational number: Rational numbers are also used in everyday life, for example when calculating interest rates, or when converting between units of measurement.

9. Prime Factorization

Fraction: Fractions can be simplified by dividing the numerator and denominator with their greatest common factor which is known as 'GCF'. GCF is found by prime factorizing the numbers. Prime factorization is the process of finding the prime factors of a number. Prime Factors of numbers are the factors that are only divisible by 1 and themselves.

Rational number: Rational numbers can also be simplified by dividing the numerator and denominator with their GCF. Prime factorization can be used to find the GCF of two numbers, which can then be used to simplify the rational number.

10. Equivalent Fractions

Fraction: Equivalent fractions are fractions that represent the same part of a whole. For example, 2/4 and 1/2 are equivalent fractions, as they both denote half of a whole value.

Rational number: Rational numbers can also be equivalent, as long as they have the same value. For example, 0.5 and 1/2 are equivalent rational numbers, as they both represent half of a whole value.

11. Recurring Decimals

Fraction: When a fraction is converted to a decimal, it may result in a recurring decimal, which is a decimal that repeats indefinitely. For example, 4/3 = 1.3333…

Rational number: Rational numbers can also result in indefinite decimals. For example, 1/7 = 0.142857142857…, where the digits 142857 repeat indefinitely.

12. Real Numbers

Fraction: Fractions are a subcategory of the real numbers, which include all numbers that can be represented on a number line.

Rational number: Rational numbers are also a subcategory of the real numbers, which include all numbers that can be expressed as the ratio of two integers.

13. Irrational Numbers

Fraction: Fractions can't represent irrational numbers, which are numbers that cannot be showcased as the ratio of two integers. Examples of irrational numbers include √2, π, and e.

Rational number: Rational numbers do not comprise of irrational numbers. However, irrational numbers can be approached by using rational numbers.

14. Decimal Expansion

Fraction: When a fraction is converted to a decimal, the decimal expansion may terminate, as in the case of 1/4 = 0.25, or it may be non-terminating and non-repeating, as in the case of 1/3 = 0.3333…

Rational number: Rational numbers can also have terminating or non-terminating decimal expansions. For example, 3/5 = 0.6, which is a terminating decimal, while 1/7 = 0.142857142857…, which is a non-terminating, recurring decimal.

15. Comparing Numbers

Fraction: Fractions can be compared by finding a common denominator and comparing the numerators. For example, to compare 2/5 and 3/4, we can find a common denominator of 20 and rewrite the fractions as 8/20 and 15/20, respectively. Since 15/20 is greater than 8/20, we can conclude that 3/4 is greater than 2/5.

Rational number: Rational numbers can also be compared by finding a common denominator or by converting them to a common form, such as a decimal or percent.

16. Absolute Value

Fraction: Fractions do not have an absolute value, as they can be positive or negative.

Rational number: Rational numbers have an absolute value, which is the distance of the number from zero on the number line. The absolute value of a rational number is always positive.

17. Mixed Numbers

Fraction: A mixed number is a mixture of a whole number and a proper fraction. For example, 2 1/2 is a mixed number, which represents 2 whole units and 1/2 of a unit.

Rational number: Rational numbers can also be expressed as mixed numbers. To turn a rational number into a mixed number, divide the numerator by the denominator and write the quotient as the whole number, and write the remainder over the denominator as the proper fraction. For example, to convert 7/3 to a mixed number, we divide 7 by 3 to get 2 with a remainder of 1, so 7/3 can be written as the mixed number 2 1/3.

18. Addition and Subtraction

To simply add or subtract fractions with the same denominator, you have to add or subtract the numerators and write the result over the shared denominator. For example, to add 1/4 and 3/4, we add the numerators to get 4/4, which simplifies to 1. To add or subtract fractions with different denominators, find a as a ratio of two integers, while a rational number is a number that can be expressed as the ratio of two integers.

While fractions and rational numbers can be used interchangeably in some cases, they are not always the same and have some differences in how they are expressed and calculated.

Understanding the differences between fractions and rational numbers is important for various mathematical concepts and applications. Both fractions and rational numbers are used in algebra, geometry, and other fields of mathematics, and they are also used in everyday life, such as in cooking, shopping, and calculating percentages

Overall, here are the key differences between fractions and rational numbers:

• Fractions are a representation of a part of a whole, while rational numbers are numbers that can be expressed as a ratio of two integers.
• Fractions are written as a numerator over a denominator, while the rational numbers can be written as decimals or fractions.
• Fractions can be expressed as mixed numbers, while rational numbers can also be expressed as complex fractions.
• Fractions can be added, subtracted, multiplied, and divided in the same way as rational numbers, but the methods may vary slightly.
• Rational numbers are a broader concept that includes fractions, but not all rational numbers are fractions.

By understanding the differences between fractions and rational numbers, one can better understand and work with these numbers in various mathematical contexts.