\( (3+\sqrt{5}) \) and \( (3-\sqrt{5}) \)

The expressions \( (3 + \sqrt{5}) \) and \( (3 - \sqrt{5}) \) are **conjugate pairs** involving the square root of 5. These numbers have several interesting mathematical properties and applications. Here's a detailed overview: ### 1. **Basic Properties** - **Sum and Difference:** \[ \begin{align*} \text{Sum} &= (3 + \sqrt{5}) + (3 - \sqrt{5}) = 6 \\ \text{Difference} &= (3 + \sqrt{5}) - (3 - \sqrt{5}) = 2\sqrt{5} \end{align*} \] - **Product:** \[ (3 + \sqrt{5})(3 - \sqrt{5}) = 3^2 - (\sqrt{5})^2 = 9 - 5 = 4 \] - **Quotient:** \[ \frac{3 + \sqrt{5}}{3 - \sqrt{5}} = \frac{(3 + \sqrt{5})^2}{(3)^2 - (\sqrt{5})^2} = \frac{9 + 6\sqrt{5} + 5}{4} = \frac{14 + 6\sqrt{5}}{4} = \frac{7 + 3\sqrt{5}}{2} \] ### 2. **Roots of a Quadratic Equation** These two numbers are the roots of the quadratic equation derived from their sum and product. - **Quadratic Equation:** \[ x^2 - (\text{Sum})x + (\text{Product}) = 0 \implies x^2 - 6x + 4 = 0 \] Solving \( x^2 - 6x + 4 = 0 \) using the quadratic formula: \[ x = \frac{6 \pm \sqrt{36 - 16}}{2} = \frac{6 \pm \sqrt{20}}{2} = \frac{6 \pm 2\sqrt{5}}{2} = 3 \pm \sqrt{5} \] ### 3. **Irrationality** Both \( 3 + \sqrt{5} \) and \( 3 - \sqrt{5} \) are **irrational numbers** since they involve the square root of a non-square integer (5). ### 4. **Recurrence Relations and Powers** These numbers can be used to express terms in certain recurrence relations, especially those resembling the Fibonacci sequence. - **Powers:** The powers of \( (3 + \sqrt{5}) \) and \( (3 - \sqrt{5}) \) can be expressed using binomial expansion. Additionally, since they are roots of the quadratic equation \( x^2 = 6x - 4 \), higher powers can be simplified using this relation. - **Example:** \[ (3 + \sqrt{5})^2 = 9 + 6\sqrt{5} + 5 = 14 + 6\sqrt{5} = 6(3 + \sqrt{5}) - 4 \] This simplifies calculations involving higher powers. ### 5. **Geometric Interpretation** In the context of the quadratic equation \( x^2 - 6x + 4 = 0 \), the numbers \( 3 + \sqrt{5} \) and \( 3 - \sqrt{5} \) represent the points where the parabola \( y = x^2 - 6x + 4 \) intersects the x-axis. ### 6. **Applications** - **Algebraic Identities:** They are useful in simplifying expressions involving radicals. - **Number Theory:** These conjugate pairs appear in the study of quadratic fields and Diophantine equations. - **Geometry:** They can represent lengths, distances, or other geometric quantities that involve irrational numbers. ### 7. **Graphical Representation** If you plot these numbers on the real number line: - \( 3 + \sqrt{5} \) is approximately \( 3 + 2.236 = 5.236 \) - \( 3 - \sqrt{5} \) is approximately \( 3 - 2.236 = 0.764 \) They are symmetric around the center point 3. ### 8. **Conjugate Properties in Complex Numbers** While \( (3 + \sqrt{5}) \) and \( (3 - \sqrt{5}) \) are real numbers, the concept of conjugates extends to complex numbers, where conjugate pairs are of the form \( a + bi \) and \( a - bi \). These pairs share similar properties, such as having the same real part and opposite imaginary parts. --- If you have a specific question or application in mind regarding \( (3 + \sqrt{5}) \) and \( (3 - \sqrt{5}) \), feel free to ask for a more targeted explanation!