%2F(3-i).jpeg "(2+i)/(3-i)")
Simplifying Complex Fractions: (2 + i) / (3 - i)
This article will guide you through the process of simplifying the complex fraction (2 + i) / (3 - i).
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 (i.e., i² = -1).
Simplifying the Fraction
To simplify the fraction, we need to eliminate the imaginary term in the denominator. This is achieved by multiplying both the numerator and denominator by the complex conjugate of the denominator.
The complex conjugate of (3 - i) is (3 + i).
Step 1: Multiply both numerator and denominator by the complex conjugate
(2 + i) / (3 - i) * (3 + i) / (3 + i)
Step 2: Expand the numerator and denominator
Numerator: (2 + i)(3 + i) = 6 + 2i + 3i + i² = 6 + 5i - 1 = 5 + 5i Denominator: (3 - i)(3 + i) = 9 + 3i - 3i - i² = 9 + 1 = 10
Step 3: Simplify the fraction
(5 + 5i) / 10 = (5/10) + (5/10)i = 1/2 + 1/2i
Conclusion
The simplified form of the complex fraction (2 + i) / (3 - i) is 1/2 + 1/2i. This process of multiplying by the complex conjugate is a standard technique used to simplify complex fractions and express them in the standard form a + bi.