Perform the indicated operations and write the result in standard form. (1 + i)/(1 + 2i) + (1 - i)/(1 - 2i)

검증된 단계별 안내

Identify the complex fractions: \( \frac{1+i}{1+2i} \) and \( \frac{1-i}{1-2i} \).

To simplify each fraction, multiply the numerator and the denominator by the conjugate of the denominator. For \( \frac{1+i}{1+2i} \), multiply by \( \frac{1-2i}{1-2i} \). For \( \frac{1-i}{1-2i} \), multiply by \( \frac{1+2i}{1+2i} \).

Simplify the numerators and denominators separately by using the distributive property (FOIL method) and remember that \( i^2 = -1 \).

Combine the simplified fractions by adding them together. Ensure that the denominators are the same before adding.

Write the final result in standard form \( a + bi \), where \( a \) and \( b \) are real numbers.

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주요 개념

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Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where a is the real part and b is the imaginary part. In this context, 'i' represents the imaginary unit, defined as the square root of -1. Understanding how to manipulate complex numbers is essential for performing operations like addition and division.

Standard Form of Complex Numbers

The standard form of a complex number is a + bi, where a and b are real numbers. When performing operations on complex numbers, it is important to express the final result in this form. This involves simplifying the expression and ensuring that the imaginary unit 'i' is clearly separated from the real part.

Rationalizing the Denominator

Rationalizing the denominator is a technique used to eliminate complex numbers from the denominator of a fraction. This is typically done by multiplying the numerator and denominator by the conjugate of the denominator. In this problem, applying this technique will help simplify the fractions before performing the addition.