Sum or Product of Two Rational Numbers is Rational

Introduction

Key Concepts

Defining Rational Numbers

A rational number is any number that can be expressed as the quotient or fraction $\frac{p}{q}$ of two integers, where $p$ (the numerator) and $q$ (the denominator) are integers, and $q \neq 0$. Examples include $\frac{1}{2}$, $-3$, and $0.75$.

Properties of Rational Numbers

Rational numbers possess several key properties, particularly when performing arithmetic operations such as addition and multiplication:

• Closure Property: The set of rational numbers is closed under addition and multiplication, meaning the sum or product of any two rational numbers is also a rational number.
• Commutative Property: For addition and multiplication, the order of the numbers does not affect the result. For example, $a + b = b + a$ and $a \times b = b \times a$.
• Associative Property: The grouping of numbers does not affect the sum or product. For instance, $(a + b) + c = a + (b + c)$ and $(a \times b) \times c = a \times (b \times c)$.
• Distributive Property: Multiplication distributes over addition: $a \times (b + c) = a \times b + a \times c$.

Sum of Two Rational Numbers

When adding two rational numbers, the result is always rational. Suppose we have two rational numbers $r_1 = \frac{a}{b}$ and $r_2 = \frac{c}{d}$, where $a, b, c,$ and $d$ are integers, and $b, d \neq 0$. The sum is:

Since $ad + bc$ and $bd$ are both integers (as the product and sum of integers are integers), the sum $\frac{ad + bc}{bd}$ is a rational number.

Product of Two Rational Numbers

Similarly, the product of two rational numbers is always rational. Using the same representations, the product is:

Here, $ac$ and $bd$ are integers, making the product $\frac{ac}{bd}$ a rational number.

Examples

• Example 1: Sum

  Let $r_1 = \frac{2}{3}$ and $r_2 = \frac{4}{5}$. Then:

  $$ r_1 + r_2 = \frac{2}{3} + \frac{4}{5} = \frac{10 + 12}{15} = \frac{22}{15} $$

  Since $\frac{22}{15}$ is a fraction of two integers, it is rational.

• Example 2: Product

  Let $r_1 = -\frac{3}{4}$ and $r_2 = \frac{5}{6}$. Then:

  $$ r_1 \times r_2 = -\frac{3}{4} \times \frac{5}{6} = -\frac{15}{24} = -\frac{5}{8} $$

  Since $-\frac{5}{8}$ is a fraction of two integers, it is rational.

Proof of Closure

To formally prove that the sum or product of two rational numbers is rational, consider two arbitrary rational numbers:

where $a, b, c, d \in \mathbb{Z}$ and $b, d \neq 0$.

Sum:

Both $ad + bc$ and $bd$ are integers, and $bd \neq 0$. Therefore, $\frac{ad + bc}{bd}$ is rational.

Product:

Both $ac$ and $bd$ are integers, and $bd \neq 0$. Therefore, $\frac{ac}{bd}$ is rational.

Conclusion on Key Concepts

The closure property of rational numbers under addition and multiplication ensures that performing these operations on any two rational numbers yields another rational number. This fundamental property is pivotal in various mathematical applications and problem-solving scenarios within the Cambridge IGCSE syllabus.

Advanced Concepts

In-depth Theoretical Explanations

While the closure property is straightforward, understanding its implications requires delving into more complex territory. Consider the structure of the rational numbers, $\mathbb{Q}$, as a field. In field theory, $\mathbb{Q}$ serves as the simplest example of a field, where both addition and multiplication are defined, and every non-zero element has a multiplicative inverse.

Moreover, exploring the density of rational numbers offers deeper insights. Between any two rational numbers, no matter how close, there exists another rational number. This property underscores the infinitude and intricate structure of $\mathbb{Q}$ within the real numbers, $\mathbb{R}$.

Mathematical Derivations and Proofs

Beyond proving the closure property, it's essential to understand the implications of these operations in the context of algebraic structures. For instance, in ring theory, the set $\mathbb{Q}$ forms a commutative ring with unity, where the sum and product operations satisfy specific axioms, including associativity, commutativity, distributivity, and the existence of additive and multiplicative identities.

One can also explore the implications of these properties in polynomial equations. For example, solving linear equations with rational coefficients ensures that the solutions (roots) are either rational or irrational, depending on the discriminant.

Complex Problem-Solving

Consider solving systems of equations involving rational numbers:

1. Problem: Solve for $x$ and $y$: $$ \begin{cases} \frac{1}{2}x + \frac{3}{4}y = \frac{5}{6} \\ \frac{2}{3}x - \frac{1}{4}y = \frac{1}{2} \end{cases} $$
2. Solution:
   • Multiply the first equation by 12 to eliminate denominators: $$ 12 \times \left( \frac{1}{2}x + \frac{3}{4}y \right) = 12 \times \frac{5}{6} $$ $$ 6x + 9y = 10 $$>
   • Multiply the second equation by 12: $$ 12 \times \left( \frac{2}{3}x - \frac{1}{4}y \right) = 12 \times \frac{1}{2} $$ $$ 8x - 3y = 6 $$
   • Solve the system: $$ \begin{cases} 6x + 9y = 10 \\ 8x - 3y = 6 \end{cases} $$> Multiply the first equation by 3 and the second by 9 to align coefficients: $$ \begin{cases} 18x + 27y = 30 \\ 72x - 27y = 54 \end{cases} $$> Add the equations: $$ 90x = 84 \Rightarrow x = \frac{84}{90} = \frac{14}{15} $$> Substitute $x = \frac{14}{15}$ into the second original equation: $$ 8 \times \frac{14}{15} - 3y = 6 $$ $$ \frac{112}{15} - 3y = 6 \Rightarrow -3y = 6 - \frac{112}{15} = \frac{90 - 112}{15} = -\frac{22}{15} $$ $$ y = \frac{22}{45} $$
   • Solution: $x = \frac{14}{15}$, $y = \frac{22}{45}$

Interdisciplinary Connections

The concept that the sum or product of two rational numbers is rational finds applications beyond pure mathematics:

• Computer Science: In algorithms that involve numerical computations, understanding rational operations ensures precision and accuracy, especially in programming languages that handle fractions and floating-point numbers.
• Physics: When dealing with ratios and proportions in formulas, rational number operations are fundamental in deriving accurate measurements and calculations.
• Engineering: Rational calculations are essential in circuit design, where resistances and other parameters are often represented as fractions.
• Economics: Financial models frequently use ratios to represent rates, prices, and other economic indicators, relying on the properties of rational numbers for stability and predictability.

Challenges and Common Pitfalls

Students often encounter challenges when dealing with rational numbers, particularly in simplifying fractions and performing operations involving negative numbers or mixed fractions. Misapplication of the closure property can lead to errors in problem-solving. Additionally, recognizing when a result is rational or irrational is crucial, especially in higher-level mathematics where irrational numbers play a significant role.

Comparison Table

Summary and Key Takeaways