Problem 76 - College Algebra

Question

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

Step-by-Step Solution

Answer

The result of the given expression in standard form is \(0 + 0i\), or simply \(0\).

Step 1: Identification and Conjugation

Identify the complex numbers from the given expression and their conjugates. A conjugate of a complex number \(a+bi\) is \(a-bi\). So, the conjugate of \(1+2i\) is \(1-2i\) and the conjugate of \(1-2i\) is \(1+2i\). Multiply each fraction by the conjugate of its respective denominator.

Step 2: Simplification

After multiplying, simplify the multiplication in the numerator and denominator separately by using the formula \((a+bi)(a-bi) = a^2 + b^2\).

Step 3: Final Result

After simplification, the fraction terms are finally added to obtain the answer in standard form.

Key Concepts

Conjugate

In the world of complex numbers, the concept of a conjugate is quite essential. When we talk about a complex number in the form of \(a + bi\), its conjugate is expressed as \(a - bi\). This simply means you switch the sign of the imaginary part while keeping the real part unchanged.
Understanding conjugates is vital because they are used to simplify expressions, especially when dividing complex numbers. This process helps eliminate the imaginary component from the denominator.

• For example, if you have a complex number like \(1 + 2i\), its conjugate would be \(1 - 2i\).
• Similarly, the conjugate of \(1 - 2i\) would be \(1 + 2i\).

By multiplying a complex number by its conjugate, you can simplify expressions as the product of a complex number and its conjugate is always a real number: \(a^2 + b^2\). This principle was pivotal in the exercise to simplify the fractions.

Standard Form

Writing complex numbers in standard form means expressing them as \(a + bi\) where \(a\) and \(b\) are real numbers. This form makes it straightforward to interpret and perform arithmetic operations on complex numbers.
The real part, \(a\), represents the number without the imaginary unit, marked by \(b\), which is multiplied by \(i\), the imaginary unit. When writing results from complex operations, presenting them in standard form ensures clarity and consistency.

• The standard form serves as a norm to assess whether expressions are simplified to their simplest form.
• In the case of our exercise, after simplifying the fractions using their conjugates, the results were further added and rewritten in standard form before finalizing the answer.

This is often the last step in solving equations involving complex numbers as it provides a neat and efficient summary of the solution.

Operations with Complex Numbers

Working with complex numbers often involves a variety of operations such as addition, subtraction, multiplication, and division. Each operation works similarly to arithmetic with real numbers but requires special attention to the imaginary unit, \(i\).

• For addition and subtraction, you combine the real parts and the imaginary parts separately.
• When multiplying, you distribute each term, using the fact that \(i^2 = -1\). This is crucial for simplifying terms involving \(i\).
• Division can be more intricate as it involves the conjugate. You multiply the numerator and the denominator by the conjugate of the denominator. This step uses the property \((a+bi)(a-bi) = a^2 + b^2\) to convert the denominator into a real number, which simplifies the division.

The exercise shared required the usage of these operations, especially focusing on division, where conjugates were utilized to bring the complex numbers into a solvable form, leading to a neat solution in standard form.