Operations on Real Numbers

In this section, we explore the operations on real numbers with particular attention to the interactions between rational and irrational numbers.

1. Rational Numbers: It is established that rational numbers satisfy the commutative, associative, and distributive laws for addition and multiplication. Moreover, performing basic operations (addition, subtraction, multiplication, or division, except by zero) with rational numbers yields a result that remains within the set of rational numbers, affirming their closure properties.
2. Irrational Numbers: Similar to rational numbers, irrational numbers also adhere to the commutative, associative, and distributive laws. However, the results of operations involving irrational numbers can produce either rational or irrational outcomes, highlighting the complexity of their interactions.
3. Operations with Rational and Irrational Numbers:
4. The sum, difference, or product of a rational number and an irrational number is always an irrational number.
5. Conversely, the addition or subtraction of two irrational numbers may yield a rational number. This introduces the concept of results varying between rational and irrational based on the specific numbers involved.
6. Examples: Multiple examples and exercises illustrate how to addition, multiplication, and square roots of these numbers operate, further reinforcing the underpinning theories and laws governing real numbers.

Overall, this section lays the foundational understanding necessary for working with real numbers in mathematical operations, especially emphasizing the distinction and interaction between rational and irrational sets.

Example: Multiply \( 8 \sqrt{8} \) by \( 3 \sqrt{2} \).

Solution: \( 8 \sqrt{8} \times 3 \sqrt{2} = 8 \times 3 \times \sqrt{8} \times \sqrt{2} = 24 \times \sqrt{16} = 24 \times 4 = 96 \).

Example:

Divide \( 12\sqrt{20} \) by \( 3\sqrt{5} \).

Solution:

\[ 12\sqrt{20} \div 3\sqrt{5} = \frac{12\sqrt{20} \times \sqrt{5}}{3\sqrt{5}} = 4\sqrt{20} \text{ (after simplification)} \]

\[ = 4\sqrt{4 \times 5} = 4 \times 2\sqrt{5} = 8\sqrt{5} \]

Example : Rationalize the Denominator

Solution:

We want to write \( \frac{1}{\sqrt{3}} \) as an equivalent expression in which the denominator is a rational number. We know that \( \sqrt{3} \) is irrational. We also know that multiplying \( \frac{1}{\sqrt{3}} \) by \( \frac{\sqrt{3}}{\sqrt{3}} \) will give us an equivalent expression, since \( \sqrt{3} \times \sqrt{3} = 3 \). So, we put these two facts together to get:

\[ \frac{1}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{3} \]

In this form, it is easy to locate \( \frac{\sqrt{3}}{3} \) on the number line. It is halfway between 0 and \( 1 \).