%2F(3%2B3i).jpeg "(5-2i)/(3+3i)")
Simplifying Complex Fractions: A Step-by-Step Guide
In mathematics, complex numbers often arise in various calculations. One common operation is dividing complex numbers. This guide will demonstrate how to simplify the complex fraction (5 - 2i) / (3 + 3i).
Understanding Complex Numbers
Complex numbers are numbers 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. The real part is represented by a, and the imaginary part is represented by b.
Simplifying the Fraction
To simplify the complex fraction (5 - 2i) / (3 + 3i), we need to eliminate the imaginary unit from the denominator. This is achieved by multiplying both the numerator and denominator by the complex conjugate of the denominator.
1. Finding the Complex Conjugate
The complex conjugate of a complex number is found by changing the sign of the imaginary part. The complex conjugate of (3 + 3i) is (3 - 3i).
2. Multiplying by the Complex Conjugate
Multiply both the numerator and denominator by (3 - 3i):
(5 - 2i) / (3 + 3i) * (3 - 3i) / (3 - 3i)
3. Expanding the Multiplication
Expanding the multiplication in the numerator and denominator, we get:
(15 - 15i - 6i + 6i^2) / (9 - 9i + 9i - 9i^2)
4. Simplifying Using i^2 = -1
Remember that i^2 = -1. Substitute this into the expression:
(15 - 15i - 6i - 6) / (9 + 9)
5. Combining Real and Imaginary Terms
Combine the real and imaginary terms:
(9 - 21i) / 18
6. Final Simplification
Simplify the expression by dividing each term by 9:
(1 - 7/6i) / 2
Therefore, the simplified form of the complex fraction (5 - 2i) / (3 + 3i) is (1 - 7/6i) / 2.
Conclusion
Simplifying complex fractions involves multiplying both the numerator and denominator by the complex conjugate of the denominator. This process eliminates the imaginary unit from the denominator, resulting in a simplified expression in the form a + bi.