(2i)/(1+i)

Jasonbradley

2 min read

·

Jun 16, 2024

10

Share

Simplifying Complex Fractions: (2i) / (1+i)

This article will walk you through the process of simplifying the complex fraction (2i) / (1+i).

Understanding Complex Numbers

Before diving into the simplification, let's briefly recap 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.

Simplifying the Fraction

To simplify a complex fraction, we need to get rid of the complex number in the denominator. This is achieved by multiplying both the numerator and denominator by the complex conjugate of the denominator. The complex conjugate of a number of the form a + bi is a - bi.

Here's how to simplify (2i) / (1+i):

1. Find the complex conjugate of the denominator: The complex conjugate of (1 + i) is (1 - i).

2. Multiply both numerator and denominator by the complex conjugate:

   (2i) / (1 + i) * (1 - i) / (1 - i)

3. Simplify the expression:

   • Numerator: (2i) * (1 - i) = 2i - 2i² = 2i + 2 (remembering that i² = -1)
   • Denominator: (1 + i) * (1 - i) = 1 - i² = 1 + 1 = 2

4. Combine the simplified numerator and denominator:

   (2i + 2) / 2

5. Simplify further by dividing each term by 2:

   i + 1

Final Result

Therefore, the simplified form of (2i) / (1 + i) is i + 1. This is now a complex number expressed in the standard form a + bi, where a = 1 and b = 1.