Rational Numbers and Fractions

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.

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 \(\mathbb{Q}\).

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 \(6/7\).

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.

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:

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:

1. Check the type of fraction if they are like fractions or not.

2. Convert the given fraction to a like fraction, if they are not. Skip to step 4 if they are like fractions.

3. 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.

4. Add/subtract the numerator while keeping the denominator as it is.

5. 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:

1. Convert all the denominators to positive numbers if anyone of them is negative.

2. Use LCM to make all the denominators as same, if they are different.

3. Add/subtract the integers in the numerator by following the rules of integers. And keep the denominator as it is.

4. 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:

1. Multiply the numerators of all the given numbers with each other.

2. Multiply the denominators for all numbers with each other.

3. Simplify the obtained fraction/rational number.

Division of rational numbers and fractions

The division of rational numbers is as follows:

1. Take the reciprocal of the second fraction/rational number by interchanging the numerator with the denominator and vice versa.

2. Change the division sign to multiplication after performing reciprocal.

3. 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.