Detailed Summary
In section 1.3, we reaffirm the characteristics of irrational numbers introduced in previous classes, defining an irrational number as one that cannot be expressed as a fraction of two integers (i.e., in the form
\( \frac{p}{q} \), where \( q \neq 0 \)). Examples given include \( \sqrt{2} \), \( \sqrt{3} \), and numbers like \( \pi \) and non-repeating decimals.
The section goes on to establish a proof that numbers like \( 2 \) and \( 3 \) are irrational through a method known as proof by contradiction. The proof utilizes the Fundamental Theorem of Arithmetic, which states that every composite number can uniquely be expressed as a product of prime factors. By assuming that \( 2 \) and \( 3 \) can be expressed as rational numbers and following the logical consequences of that assumption, we arrive at contradictions that validate their irrationality. The section also reiterates key principles regarding operations involving rational and irrational numbers, reinforcing that operations such as addition, subtraction, multiplication, and division yield outcomes that uphold the properties of irrational numbers. These discussions are crucial not just for understanding the mathematical landscape but also for solving problems involving irrational numbers.