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Chapter 2: Problem 82

Perform the indicated operations and write the result in standard form. $$\frac{1+i}{1+2 i}+\frac{1-i}{1-2 i}$$

Short Answer

Expert verified
After following the above steps, we find that the solution to the given problem is \(1\).

Step by step solution

01

Complex Conjugate Pairing

First, to simplify the division, multiply both the numerator and the denominator by the complex conjugate of the denominator. Let's start with the first fraction, perform the multiplication and simplify the result. The denominator of the first fraction is \(1+2i\), so it's conjugate is \(1-2i\). Hence, we multiply both the numerator and denominator by \(1-2i\). This gives: \((1 + i)(1 - 2i)\) / \((1+2i)(1-2i)\). Calculate the resulting complex number.
02

First Complex Fraction Simplification

To simplify this fraction, perform the multiplication in the numerator and the denominator. The numerator will result in: \((1-i)\). The denominator is the result of a difference of squares: \((a - b)(a + b) = a^2 - b^2\), thus will result in \(1 - (2i)^2\). Now compute the resulting complex number.
03

Second Complex Fraction Simplification

Now perform the same steps with the second fraction: \((1 - i)\) / \((1-2i)(1+2i)\). Multiply both top and bottom by \(1+2i\), giving us: \((1 - i)(1 + 2i)\) / \((1+2i)(1-2i)\). Simplify the multiplication to claim the resulting complex number.
04

Addition of Simplified Fractions

We now have two simplified fractions. The next step is to add these two fractions. When we do this, simplify the complex number.

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