1. Definition
A rational number is any number that can be written in the form \( \dfrac{p}{q} \), where:
- \(p\) and \(q\) are integers
- \(q \neq 0\)
2. Symbol and Representation
The set of all rational numbers is represented by the symbol \( \mathbb{Q} \).
This symbol comes from the word Quotient, because every rational number can be written as a quotient (fraction) of two integers.
3. Examples of Rational Numbers
Here are some examples of rational numbers explained using the definition \( \dfrac{p}{q} \), where \(p\) and \(q\) are integers and \(q \neq 0\):
- \( \dfrac{1}{2} \): This is a simple rational number where \(p = 1\) and \(q = 2\). Both are integers, and the denominator is not zero.
- \( -\dfrac{3}{4} \): This is a negative rational number. Here \(p = -3\) and \(q = 4\). The negative sign can be with the numerator or denominator, and it is still a rational number.
- Whole number \(5\): Even though 5 looks like a whole number, it can be written as \( \dfrac{5}{1} \). So \(p = 5\) and \(q = 1\), which makes it a rational number.
- Zero: Zero can be written as \( \dfrac{0}{1} \), where \(p = 0\) and \(q = 1\). Since the denominator is not zero, 0 is a rational number.
- Negative whole number \(-7\): This can be written as \( \dfrac{-7}{1} \). So it also satisfies the definition of a rational number.
- A rational number with a negative denominator, like \( \dfrac{5}{-8} \): This is still a rational number because only the value matters. It can be rewritten as \( -\dfrac{5}{8} \).
All these examples show that any number that can be expressed in the form \( \dfrac{p}{q} \) with integers \(p, q\) and \(q \neq 0\) is a rational number.
4. Decimal Representation
Every rational number can be written in decimal form. These decimals are of two types.
4.1. Terminating Decimals
A terminating decimal is a decimal number that ends after a certain number of digits. It does not continue indefinitely.
For example, \(0.75\) and \(0.125\) are terminating decimals.
Fractions whose denominators (after simplifying) contain only the prime factors 2 or 5 give terminating decimals.
4.2. Repeating Decimals
A repeating decimal is a decimal that continues forever, with a digit or a group of digits repeating in a pattern.
For example, \(0.333...\) and \(0.83\overline{3}\) are repeating decimals.
Fractions whose denominators (after simplifying) contain prime factors other than 2 or 5 produce repeating decimals.
4.3. Examples Table
| Fraction | Decimal | Type |
|---|---|---|
| \(\dfrac{1}{8}\) | 0.125 | Terminating |
| \(\dfrac{5}{6}\) | 0.83\(3\) | Repeating |
5. Properties of Rational Numbers
Rational numbers follow several important mathematical properties. These properties describe how rational numbers behave under different operations.
5.1. Positive, Negative, and Zero
Rational numbers can be positive, negative, or zero. This depends on the signs of the numerator and denominator.
5.1.1. Positive Rational Numbers
A rational number is positive when both the numerator and denominator have the same sign.
- Example: \( \dfrac{5}{8} \)
- Example: \( -\dfrac{7}{-9} \)
5.1.2. Negative Rational Numbers
A rational number is negative when numerator and denominator have opposite signs.
- Example: \( -\dfrac{3}{4} \)
- Example: \( \dfrac{6}{-11} \)
5.1.3. Zero as a Rational Number
Zero is rational because it can be written as \( \dfrac{0}{1} \). Any non-zero denominator works.
5.2. Closure Property
Rational numbers are closed under addition, subtraction, multiplication, and division (except by zero). This means performing these operations on rational numbers gives another rational number.
5.2.1. Closure Under Addition
Adding two rational numbers always gives a rational number.
Example:
- \( \dfrac{2}{3} + \dfrac{1}{6} = \dfrac{5}{6} \)
- \( -\dfrac{5}{4} + \dfrac{3}{2} = \dfrac{1}{4} \)
5.2.2. Closure Under Subtraction
Subtracting rational numbers always produces another rational number.
- \( \dfrac{7}{5} - \dfrac{2}{5} = 1 \)
- \( -\dfrac{4}{3} - \dfrac{2}{3} = -2 \)
5.2.3. Closure Under Multiplication
Multiplying rational numbers always results in a rational number.
- \( \dfrac{3}{4} \times \dfrac{2}{3} = \dfrac{1}{2} \)
- \( -\dfrac{5}{6} \times \dfrac{3}{2} = -\dfrac{5}{4} \)
5.2.4. Closure Under Division
Dividing rational numbers (except by zero) gives another rational number.
- \( \dfrac{4}{5} \div \dfrac{2}{3} = \dfrac{6}{5} \)
- Division by zero is not allowed.
5.3. Density Property
Between any two rational numbers, there is always another rational number. This means rational numbers are dense on the number line.
5.3.1. Finding a Rational Number Between Two Rational Numbers
You can find a rational number between any two rational numbers using:
Formula: \( \dfrac{a + b}{2} \)
5.3.2. Examples
- Between 1 and 2: \( \dfrac{1 + 2}{2} = \dfrac{3}{2} \)
- Between \( \dfrac{2}{5} \) and \( \dfrac{3}{5} \): \( \dfrac{1}{2} \)
5.4. Equality Property
Two rational numbers are equal if their simplified (lowest terms) form is the same. Multiplying or dividing numerator and denominator by the same non-zero number gives an equivalent rational number.
5.4.1. Examples of Equivalent Rational Numbers
- \( \dfrac{2}{4} = \dfrac{1}{2} \)
- \( \dfrac{3}{9} = \dfrac{1}{3} \)
- \( \dfrac{5}{10} = \dfrac{1}{2} \)
5.5. Reciprocal Property
The reciprocal of a rational number \( \dfrac{p}{q} \) is \( \dfrac{q}{p} \), as long as \( p \neq 0 \). A number multiplied by its reciprocal equals 1.
5.5.1. Examples of Reciprocals
| Number | Reciprocal | Product |
|---|---|---|
| \( \dfrac{3}{4} \) | \( \dfrac{4}{3} \) | 1 |
| \( -\dfrac{5}{7} \) | \( -\dfrac{7}{5} \) | 1 |
| 2 | \( \dfrac{1}{2} \) | 1 |
5.5.2. Note
Zero does not have a reciprocal because division by zero is undefined.
5.6. Commutative Property
Rational numbers follow the commutative property for addition and multiplication.
5.6.1. Addition
\( a + b = b + a \)
Example: \( \dfrac{1}{3} + \dfrac{2}{5} = \dfrac{2}{5} + \dfrac{1}{3} \)
5.6.2. Multiplication
\( a \times b = b \times a \)
Example: \( \dfrac{4}{7} \times 3 = 3 \times \dfrac{4}{7} \)
5.7. Associative Property
Rational numbers follow the associative property for addition and multiplication.
5.7.1. Addition
\( (a + b) + c = a + (b + c) \)
Example: \( (\dfrac{1}{2} + \dfrac{1}{3}) + \dfrac{1}{4} = \dfrac{1}{2} + (\dfrac{1}{3} + \dfrac{1}{4}) \)
5.7.2. Multiplication
\( (a \times b) \times c = a \times (b \times c) \)
Example: \( (2 \times \dfrac{3}{5}) \times \dfrac{4}{7} = 2 \times (\dfrac{3}{5} \times \dfrac{4}{7}) \)
5.8. Distributive Property
The distributive property connects multiplication with addition and subtraction.
5.8.1. Multiplication Over Addition
\( a(b + c) = ab + ac \)
Example: \( 2(\dfrac{3}{4} + \dfrac{1}{4}) = 2(1) = 2 \)
5.8.2. Multiplication Over Subtraction
\( a(b - c) = ab - ac \)
Example: \( \dfrac{1}{2}(5 - 3) = 1 \)