Rational Numbers

Rational numbers are those numbers that can be expressed as a quotient (division) of two integers. In other words, rational numbers are fractions of the form a/b where a and b are integers and b is not zero. Some examples are: 1/2, 3/4, -5/6, 29/3, 10/1, -7/2.

An important characteristic of rational numbers is that their decimal representation is either terminating or repeating, which means there is a group of digits that repeats. All natural numbers and integers are also rational numbers.

Why Do Rational Numbers Arise?

Natural numbers and integers are insufficient to represent relationships between a part and a whole. For example, if we divide a bar into three equal parts, there is no integer that can represent one of those parts. From a mathematical standpoint, equations like 3x=1 cannot be solved with integers.

Due to these limitations, the concept of fractions arises. Fractions are expressions of the form *\dfrac{a}{b},* where a and b are integers and b is not zero. The number a is called the numerator, and it represents the number of equal parts taken from a whole. The number b is called the denominator, and it represents the number of equal parts into which a whole is divided.

These new numbers allow us to express the relationships between a part and a whole. For example:

• If a chocolate bar is divided into three equal parts, each part represents 1/3 of the total.
• If there are 10 candies to be shared among 5 people, each person will receive 10/5 of the total, which is equal to 10÷5=2 candies per person.
• A day has 24 hours, so 18 hours represent 3/4 of a day. The day has been divided into 4 parts of 6 hours each.
• The equation *3x=1* has the fraction *x=\dfrac{1}{3},* as its solution, because *3\cdot \dfrac{1}{3}=1.*

The Set of Rational Numbers

Every integer is a fraction with a denominator of 1. For example: 3=3/1, -4=-4/1, 0=0/1. Therefore, all integers are also rational numbers. Additionally, all rational numbers are real numbers.

Other examples of rational numbers are: *\dfrac{5}{8},-\dfrac{100}{5},\dfrac{40}{97},-\dfrac{1}{7}, \dfrac{2}{5}, -\dfrac{20}{14}.*

Properties of Rational Numbers

The set Q has the following properties and characteristics:

1. It is infinite: there is an unlimited number of rational numbers. New fractions can always be generated.
2. It has no first or last element: there is no rational number that is smaller than all others, nor one that is the largest.
3. It is a dense set: between any two rational numbers, there are infinitely many other rational numbers. This also implies that, unlike integers, there are no successors or predecessors.
4. It is not a continuous or complete set: rational numbers do not cover the entire number line; there are "gaps" or "empty spaces" on the line that are not occupied by rational numbers. These gaps are filled by irrational numbers, such as *\sqrt{2}* or *\pi,* which cannot be represented as fractions.
5. It is an ordered set: Rational numbers are totally ordered, which means we can always determine if one fraction is larger than, smaller than, or equal to another. Specifically, for any two fractions *\dfrac{a}{b}* and *\dfrac{c}{d},* it holds that *\dfrac{a}{b}<\dfrac{c}{d}* if and only if *ad<bc.*
6. Decimal representation: every rational number can be expressed as a terminating decimal or a repeating decimal.

Decimal Representation

As mentioned, every rational number can be expressed in decimal form. To do this, you divide the numerator by the denominator. For example, *\dfrac{3}{10}=0.3;~\dfrac{9}{4}=2.25.* Numbers with a finite number of decimal places are called terminating decimals.

There are other rational numbers whose decimal representation is not exact but repeating; they have an infinite number of decimal places and are called repeating decimals. For example, *\dfrac{1}{3}* is represented in decimal form as *0.3333...,* with the digit 3 repeating, and *\dfrac{47}{11}* can be represented as *4.2727272...,* where the digits 2 and 7 repeat in that order. The digit or group of digits that repeats is called the period.

Within repeating decimals, we find two groups: 

• Pure repeating decimals: those where the repeating part begins immediately after the decimal point. Example: *1.3333333…* 
• Mixed repeating decimals: those that have some non-repeating digits after the decimal point, which constitute the so-called anteperiod. Example: *1.06666…* (the 0 is not part of the period).

A more compact way to write repeating decimals without using ellipses is to place a bar over the repeating digits. For example:

*\dfrac{1}{3}=0.3333...=0.\overline{3}*

*\dfrac{2}{3}=0.6666...=0.\overline{6}*

*\dfrac{16}{15}=1.06666...=1.0\overline{6}*

We can therefore define four types of rational numbers based on their decimal representation: integers (those with a denominator of 1), terminating decimals, pure repeating decimals, and mixed repeating decimals.

Representation on the Number Line

Rational numbers can be located on the number line.

It is impossible to locate all fractions because they are infinite. However, as we mentioned earlier, rational numbers do not complete the number line, as there are empty spaces that correspond to irrational numbers.

Operations with Rational Numbers

The basic and fundamental operations with rational numbers are addition, subtraction, multiplication, and division. We will define each of them.

Addition and Subtraction: *\dfrac{a}{b}±\dfrac{c}{d}=\dfrac{ad±bc}{bd}*

Multiplication: *\dfrac{a}{b}\cdot\dfrac{c}{d}=\dfrac{ac}{bd}*

Division: *\dfrac{a}{b}:\dfrac{c}{d}=\dfrac{ad}{bc}*

You can see each operation and examples in more detail in this article:

Properties of the Operations

We will briefly state the properties of operations with rational numbers.

1) Closure Property: when performing an operation between two rational numbers, the result is also a rational number. It is important to note that division by zero is not allowed.

2) Commutative Property: the order of the numbers does not affect the result. This holds for addition and multiplication.

*\dfrac{a}{b}+\dfrac{c}{d}=\dfrac{c}{d}+\dfrac{a}{b}*

*\dfrac{a}{b}\cdot \dfrac{c}{d}=\dfrac{c}{d}\cdot \dfrac{a}{b}*

3) Associative Property: the grouping of the numbers does not affect the result. This holds for addition and multiplication.

*\dfrac{a}{b}+\left(\dfrac{c}{d}+\dfrac{e}{f}\right)=\left(\dfrac{a}{b}+\dfrac{c}{d}\right)+\dfrac{e}{f}*

*\dfrac{a}{b}\cdot \left(\dfrac{c}{d}\cdot\dfrac{e}{f}\right)=\left(\dfrac{a}{b}\cdot \dfrac{c}{d}\right)\cdot \dfrac{e}{f}*

4) Distributive Property: multiplication distributes over addition.

*\dfrac{a}{b}\cdot \left(\dfrac{c}{d}+\dfrac{e}{f}\right)=\dfrac{a}{b}\cdot \dfrac{c}{d}+\dfrac{a}{b}\cdot \dfrac{e}{f}*

5) Identity Element: there is an identity element for addition and multiplication in the set of rational numbers.

• For addition, the identity element is *\dfrac{0}{1},* since *\dfrac{a}{b}+\dfrac{0}{1}=\dfrac{a}{b}.*
• For multiplication, the identity element is *\dfrac{1}{1},* since *\dfrac{a}{b}\cdot \dfrac{1}{1}=\dfrac{a}{b}.*

6) Inverse Elements: there are inverse elements for addition and multiplication.

• For addition, the inverse of a fraction *\dfrac{a}{b}* is *-\dfrac{a}{b},* because *\dfrac{a}{b}+\left(-\dfrac{a}{b}\right)=0*
• For multiplication, the inverse of a fraction *\dfrac{a}{b}* is *\dfrac{b}{a},* because *\dfrac{a}{b}\cdot \dfrac{b}{a}=1*

Limitations of Rational Numbers

The limitation of rational numbers is related to the fact that they are unable to fill all the spaces on the number line. Indeed, there are numbers that cannot be expressed as fractions; these are called irrational numbers. Some examples of this limitation are:

• The hypotenuse of a right triangle whose legs have a length of 1 cannot be expressed as a rational number. Applying the Pythagorean Theorem shows that the hypotenuse has a length of *\sqrt{2},* which is not a rational number.
• There is no rational number that solves the equation *x^2=3,* as one of the solutions is *\sqrt{3},* which is not a rational number.
• The ratio of a circle's circumference to its diameter is not a rational number. This number is *\pi,* a well-known irrational number.

To overcome these limitations, a broader set of numbers is introduced: the real numbers. This set includes both numbers that can be written as fractions (rational) and those that cannot (irrational).