%2F(3%2B4i).jpeg "(1+i)/(3+4i)")
Simplifying Complex Fractions: (1 + i) / (3 + 4i)
This article will guide you through the process of simplifying the complex fraction (1 + i) / (3 + 4i).
Understanding Complex Numbers
Before diving into the simplification, let's refresh our understanding of complex numbers. A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, defined as the square root of -1 (i² = -1).
Simplifying Complex Fractions
To simplify a complex fraction, we aim to eliminate the imaginary term from the denominator. We achieve this by multiplying both the numerator and denominator by the complex conjugate of the denominator.
1. Finding the Complex Conjugate:
The complex conjugate of 3 + 4i is 3 - 4i. It's obtained by simply changing the sign of the imaginary term.
2. Multiplying by the Conjugate:
(1 + i) / (3 + 4i) * (3 - 4i) / (3 - 4i)
3. Expanding and Simplifying:
- Numerator: (1 + i)(3 - 4i) = 3 - 4i + 3i - 4i² = 3 + i + 4 = 7 + i
- Denominator: (3 + 4i)(3 - 4i) = 9 - 12i + 12i - 16i² = 9 + 16 = 25
4. Final Result:
(7 + i) / 25
Therefore, the simplified form of (1 + i) / (3 + 4i) is (7 + i) / 25.