%5E(1%2F2).jpeg "(1+i)^(1/2)")
Exploring the Square Root of (1 + i)
The expression (1 + i)^(1/2) represents the square root of the complex number (1 + i). Let's delve into finding this root and understanding its significance.
Complex Numbers and Roots
Complex numbers extend the real number system by introducing the imaginary unit 'i', where i² = -1. A complex number is expressed as a + bi, where 'a' and 'b' are real numbers.
Finding the square root of a complex number is akin to finding a number that, when multiplied by itself, results in the original complex number.
Finding the Square Root of (1 + i)
We can approach this problem in two main ways:
1. Algebraic Approach:
- Assume the square root is of the form (x + yi), where x and y are real numbers.
- Square this assumed root: (x + yi)² = (x² - y²) + (2xy)i
- Equate the real and imaginary parts to the original number:
- x² - y² = 1
- 2xy = 1
- Solve the system of equations: This yields two possible solutions:
- x = √2 / 2, y = √2 / 2
- x = -√2 / 2, y = -√2 / 2
Therefore, the square roots of (1 + i) are (√2 / 2 + √2 / 2 i) and (-√2 / 2 - √2 / 2 i).
2. Polar Form Approach:
- Convert (1 + i) to polar form:
- r = |1 + i| = √2
- θ = arctan(1/1) = π/4
- (1 + i) = √2 (cos(π/4) + i sin(π/4))
- Apply De Moivre's Theorem: (r(cos θ + i sin θ))^(1/2) = √r (cos(θ/2 + kπ) + i sin(θ/2 + kπ)), where k = 0, 1.
- Calculate the roots:
- For k = 0: √2 (cos(π/8) + i sin(π/8))
- For k = 1: √2 (cos(5π/8) + i sin(5π/8))
These solutions are equivalent to the ones found algebraically.
Conclusion
Finding the square root of (1 + i) demonstrates the elegance and power of complex number operations. The existence of two distinct square roots is a consequence of the fundamental nature of complex numbers and their representation in the complex plane. Understanding the square root of (1 + i) provides insight into the intricate world of complex numbers and their applications in various fields, including mathematics, physics, and engineering.