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Chapter 2: Problem 23

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

Short Answer

Expert verified
The result of the division \(\frac{2 i}{1+i}\) in standard form is \(1 + i\).

Step by step solution

01

Identify the Complex Conjugate

A complex conjugate of a number \(a + ib\) is given by \(a - ib\). So, the complex conjugate of the denominator \(1 + i\) is \(1 - i\).
02

Multiply by the Complex Conjugate

We multiply both the numerator and the denominator by the complex conjugate \(1 - i\). Thus, the expression becomes \(\frac{2i(1 - i)}{(1 + i)(1 - i)}\).
03

Simplify

Now, we simplify both the numerator and the denominator separately. The numerator becomes \(2i - 2i^2\) and the denominator becomes \(1 - i^2\). Since \(i^2\) is \(-1\), the expression hence simplifies to \(\frac{2i + 2}{2}\). The final answer is thus \(\frac{2i + 2}{2} = 1 + i\).

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