%2F(1-5i).jpeg "(2+3i)/(1-5i)")
Simplifying Complex Fractions: (2 + 3i) / (1 - 5i)
This article will guide you through the process of simplifying the complex fraction (2 + 3i) / (1 - 5i).
Understanding Complex Numbers
Before we begin, 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 (i.e., i² = -1).
Simplifying the Fraction
To simplify the fraction (2 + 3i) / (1 - 5i), we need to eliminate the imaginary term from the denominator. We achieve this by multiplying both the numerator and the denominator by the complex conjugate of the denominator.
The complex conjugate of (1 - 5i) is (1 + 5i).
Step 1: Multiply numerator and denominator by the conjugate
(2 + 3i) / (1 - 5i) * (1 + 5i) / (1 + 5i)
Step 2: Expand the numerator and denominator
[(2 + 3i) * (1 + 5i)] / [(1 - 5i) * (1 + 5i)]
Step 3: Simplify using distributive property and i² = -1
[(2 + 10i + 3i + 15i²)] / [(1 + 5i - 5i - 25i²)]
[(2 + 13i - 15)] / [(1 + 25)]
Step 4: Combine real and imaginary terms
(-13 + 13i) / 26
Step 5: Express in standard form (a + bi)
(-13/26) + (13/26)i
Step 6: Simplify
-1/2 + 1/2i
Therefore, the simplified form of (2 + 3i) / (1 - 5i) is -1/2 + 1/2i.