You are studying with your friend Max for the examination and both of you started to discuss how long you guys studied. Max said both of you studied for \(3.5\) hours, but you believe that you both studied for \(3\) and half an hour \((3\frac{1}{2})\). But which one of you is correct about the time? If both of you studied for the same amount then why the difference in the number? Well, both \(3.5\) and \(3\frac{1}{2}\) are the same. They are different forms of numbers. One is in the rational number form and the other is in the fraction form.
In this section, we will understand the concept of rational numbers and fractions and how they are different.
Meaning of Rational numbers and Fractions
Rational numbers and fractional numbers are two mathematical concepts that seem very closely related and most often are used interchangeably. However, this section will introduce the meaning of rational numbers and fractions, and how different they are described.
Rational numbers are a type of real number that can be written as the ratio of two integers. They are expressed in the form \(\frac{p}{q}\).
Note that \(p\) and \(q\) are both integers and \(q\) is non-zero. Examples of rational numbers are \(12, 10/12, 3/10,\) and \(0.5\). The set of rational numbers is always denoted by \(\mathbb{Q}\).
Integers are the set of all positive, negative numbers, and zero. They do not contain any decimal or fraction form.
Fractions are numbers that define a part or portion of any whole quantity written as the ratio of whole numbers. They are in the form \(a / b\) where \(a\) is the numerator and \(b\) is the denominator.
Note that, both the numerator and denominator are whole numbers and the denominator cannot be zero. Examples of fractions are \(10/32, 12/10, 4/23,\) and \(6/7\).
Whole numbers are a collection of natural numbers and zero. That is all the positive integers with zero without any decimal, fraction, or negative part.
Though both fractions and rational numbers look alike, they are not the same every time. As rational numbers have integers and contain negative numbers, they cannot be considered fractions. As fractions do not include negative numbers.
Rational number
As defined above, a rational number is a type of real number that is expressed in the form \(p / q\), where \(p\) and \(q\) are integers and not equal to \(0\). Simply, it can be said that any fraction with a non-zero denominator is a rational number.
The standard form of rational numbers is expressed as when it has no common factors aside from one between the dividend and divisor and therefore the divisor is positive. So, the standard form of fractions is achieved by simplifying fractions. For example, \(3/9\) is an example of a rational number, but it can be divisible furthermore to have \(1/3\). In the form \(1/3\), it is considered the standard form since the number is no longer divisible by any number other than \(1\) and itself.
Remember that the number which you divide is called the dividend, and the number by which you perform the division is called the divisor.
Fractions
The mathematical concept of fractions is used to describe parts of a whole. When we take a slice of pizza from the whole, for example, we say that we have a fraction of it. Fractions are numbers given in the form \(\frac{a}{b}\) where \(a\) and \(b\) are whole numbers and \(b\) is not equal to \(0\).
The fraction with numerator \(0\) is \(0\), but the fraction with \(0\) as denominator is undefined. A fraction is denoted by the simple "/" in the form \(a/b\) where \(a\), the upper number is called a numerator, and \(b\), the lower number is called the denominator. Examples of fractions are \(12/20, 5/6,\) and \(50/100\).
Fig. 1. Fraction of the parts of a circle.
Differences between rational numbers and fractions
There are significant differences between rational numbers and fractions, and they are as follows:
| Rational Numbers | Fraction |
| 1. Rational numbers are in the form \(p/q\), where \(p\) and \(q\) are integers. | 1. Rational numbers are in the form \(a/b\), where \(a\) and \(b\) are whole numbers. |
| 2. Rational number have the positive and negative form of numbers. | 2. Fraction only have the positive form of numbers. |
| 3. Rational numbers cannot be considered as fractions. | 3. Fraction can be considered as rational number. |
| 4. Examples - \(-3/7, 1/3, 12\) | 4. Examples - \(4/5, 1\frac{2}{5}, 23/6\) |
Types of rational numbers
There are several types of rational numbers, and they are as follows;
Integers, for example, \(-3, 5,\) and \(4\).
Fractions of the form \(p / q\) where \(p\) and \(q\) are integers, For Example, \(1/2\).
Numbers that do not have infinite decimals, for example, \(1/4\) of \(0.25\). They are also called terminating decimals.
Numbers that have infinite decimals, for example, \(1/3\) of \(0.333....., 1.222.....\), etc. They are known as non-terminating decimals.
Whole numbers.
Types of fractions
There are quite a number of types of fractions. If we want to use the numerator and denominator as a basis, they can be considered as three; a proper fraction, mixed fraction, and improper fraction. However, these are further subdivided into different types.
Proper fraction
Proper fractions are ones where the numerators are smaller than the denominator (numerator < denominator). For example, \(3/4\) is a proper fraction.
Mixed fraction
Mixed fractions are combinations of integers and proper fractions. They are written as \(a\frac{m}{n}\) where \(a\) is the integer. An example of a mixed fraction is \(1\frac{2}{5}\).
Improper fraction
Improper fractions, in contrast to proper fractions, are ones whose denominator is smaller than the numerators. (denominator < numerator). An example is \(4/3\).
Like fractions
Like fractions are fractions that have the same denominators. An example is \(1/7\) and \(4/7\). Both fractions here have the same denominator as \(7\), hence are considered like fractions.
Unlike fractions
These types of fractions are in contrast to like fractions. They have different values in their denominator. An example of unlike fraction is \(11/13\) and \(4/9\).
Equivalent fraction
Two fractions can be considered equivalent when after simplifying by performing multiplication or division. they both give the same. For example, \(2/3\) and \(4/6\) are equivalent fractions because \(4/6\) can be further simplified to be \(2/3\).
Unit fractions
When a fraction's numerator is equal to \(1\), it is known as a unit fraction. Examples are \(1/2\) (one-half of a whole) and \(1/4\) (one-fourth of a whole).
Adding and Subtracting rational numbers and fractions
Denominator plays a major role in adding and multiplying rational numbers and fractions. Let's see how we can perform the operation of adding and subtracting on rational numbers and fractions.
Addition and subtraction of fractions
The given steps should be followed to add and subtract fractions:
Check the type of fraction if they are like fractions or not.
Convert the given fraction to a like fraction, if they are not. Skip to step 4 if they are like fractions.
Find the least common multiple for the given denominators. Then multiply with the equivalent number to all the given denominators to obtain the same value of the denominator.
Add/subtract the numerator while keeping the denominator as it is.
If possible, reduce the obtained fractions.
Addition and subtraction of rational numbers
We follow the same process for fractions to add and subtract rational numbers:
Convert all the denominators to positive numbers if anyone of them is negative.
Use LCM to make all the denominators as same, if they are different.
Add/subtract the integers in the numerator by following the rules of integers. And keep the denominator as it is.
Simplify the rational number.
Multiplication and division of rational numbers and fractions
The multiplication and division of rational numbers and fractions follow the same rule with the addition of rules of integers for rational numbers.
Multiplication of Rational numbers and fractions
The multiplication of Rational numbers and fractions follows the given rules:
Multiply the numerators of all the given numbers with each other.
Multiply the denominators for all numbers with each other.
Simplify the obtained fraction/rational number.
Remember when dealing with rational numbers always follow the positive and negative sign multiplication rules.
Division of rational numbers and fractions
The division of rational numbers is as follows:
Take the reciprocal of the second fraction/rational number by interchanging the numerator with the denominator and vice versa.
Change the division sign to multiplication after performing reciprocal.
Follow the rules of multiplying for rational numbers and fractions as above.
Rational numbers and fractions example
Let us see some rational number and fraction examples.
Identify the types of fractions.
1. \(\frac{6}{7}, \frac{4}{7}\)
2. \(2\frac{3}{8}\)
3. \(\frac{12}{9}\)
Solution:
1. Like fraction - As both the denominators are the same.
2. Mixed fraction - It has a fraction with a whole number.
3. Improper fraction - Numerator is greater than the denominator.
Divide the rational number \(\frac{-3}{10}\) by \(\frac{7}{5}\).
Solution:
Here we need to perform
\[\frac{-3}{10}\div\frac{7}{5}\]
Step 1- Take reciprocal of \(\frac{7}{5}\)
\[\Rightarrow \frac{5}{7}\]
Step 2 - Changing the division sign to multiplication after getting the reciprocal we have,
\[\frac{-3}{10}\times \frac{5}{7}\]
Step 3 - Now we multiply the numerator of both the numbers to each other and same with the denominators.
\begin{align} \frac{-3}{10}\times \frac{5}{7} &= \frac{-3\times 5}{10\times 7} \\ &=\frac{-15}{70} \\ \end{align}
Step 4 - Now we reduce the obtained rational number.
\[\Rightarrow \frac{-15}{70} = \frac{-3}{14}\]
Calculate the given fraction.
\[\frac{1}{3}+\frac{4}{5}\]
Solution:
Step 1 - Here both the fractions are not like fractions. So, we convert it to like fractions.
Step 2 - We consider LCM of denominators \(3\) and \(5\) to convert the given fractions to like fractions.
Hence, the LCM of \(3\) and \(5\) is \(15\).
Step 3 - Now we multiply \(5\) to \(\frac{1}{3}\) to make the denominator \(15\). Similarly, we multiply \(3\) to \(\frac{4}{5}\) and add both the numerators we get,
\begin{align} \Rightarrow \frac{1\times 5}{3\times 5} + \frac{4\times 3}{5\times 3} &= \frac{5}{15}+\frac{12}{15} \\ &=\frac{5+12}{15} \\ &=\frac{17}{15} \\ \end{align}
Here, \(\frac{17}{15}\) is already in the reduced form.
Rational Numbers and Fractions - Key takeaways
- A rational number is a type of real number that is expressed in the form \(p / q\), where \(p\) and \(q\) are integers and not equal to \(0\).
- Fractions are numbers given in the form \(a / b\) where \(a\) and \(b\) are whole numbers and \(b\) is not equal to \(0\).
- All fractions are rational numbers while not all rational numbers are fractions.
Rational numbers involve numbers having either terminating or non-terminating decimals.
Different types of fractions are - proper fraction, improper fraction, mixed fraction, like fraction, unlike fraction, equivalent fraction, and unit fraction.
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