Problem 23 - Algebra and Trigonometry

Question

Divide and express the result in standard form. $$ \frac{2 i}{1+i} $$

Step-by-Step Solution

Answer

The result in standard form is \(1 + i\)

Step 1: Identify the Complex Conjugate

The complex conjugate of a complex number \(a + bi\) is \(a - bi\). In this case, the complex conjugate of the denominator \(1 + i\) is \(1 - i\).

Step 2: Multiply by the Complex Conjugate

Multiply the numerator and the denominator by the complex conjugate of the denominator. So, \(\frac{2i}{1 + i}\) will become \(\frac{2i(1 - i)}{(1 + i)(1 - i)}\).

Step 3: Simplify

Simplify both the numerator and the denominator. In the numerator, distribute \(2i\) across \((1 - i)\) to get \(2i - 2i^2\). In the denominator, use difference of squares which states that \(x^2 - y^2 = (x - y)(x + y)\). Here the denominator becomes \(1 - i^2\).

Step 4: Replace \(i^2\) with -1

In complex numbers, \(i^2\) is equal to -1. Replace each instance of \(i^2\) with -1. The numerator becomes \(2i - 2(-1)\) which is \(2i + 2\), and the denominator becomes \(1 -(-1)\) which is \(1 + 1\) or \(2\).

Step 5: Express in Standard Form

The resulting complex number should be in the form \(a + bi\). Here \(a\) is the real part and \(b\) is the imaginary part. So, the final result is \(\frac{2i + 2}{2}\) which simplifies to \(i + 1\) or \(1 + i\), which is in the standard form.