Problem 59 - Algebra A Combined Function

Question

Divide. Write your answers in the form \(a+b i\) $$ \frac{6 i}{1-2 i} $$

Step-by-Step Solution

Answer

The division results in \(-\frac{12}{5} + \frac{6}{5}i\).

Step 1: Identify the complex conjugate

To divide complex numbers, we can multiply the numerator and the denominator by the complex conjugate of the denominator. The complex conjugate of a complex number of the form \(a + bi\) is \(a - bi\). For the denominator \(1 - 2i\), the complex conjugate is \(1 + 2i\).

Step 2: Multiply the numerator and the denominator by the complex conjugate

We multiply both the numerator and the denominator by the complex conjugate \(1 + 2i\): \[\frac{6i}{1-2i} \cdot \frac{1+2i}{1+2i}\] This simplification will help us eliminate the imaginary part in the denominator.

Step 3: Simplify the denominator

Apply the difference of squares formula: \((a - b)(a + b) = a^2 - b^2\). Here, we have: \[ (1 - 2i)(1 + 2i) = 1^2 - (2i)^2 = 1 - 4i^2 \] Since \(i^2 = -1\), this becomes: \[ 1 - 4(-1) = 1 + 4 = 5 \] Thus, the denominator simplifies to 5.

Step 4: Simplify the numerator

Distribute \(6i\) across \(1 + 2i\): \[ 6i(1 + 2i) = 6i + 12i^2 \] Substitute \(i^2 = -1\): \[ 6i + 12(-1) = 6i - 12 \] The numerator simplifies to \(-12 + 6i\).

Step 5: Form the complex number

Combine the results from Steps 3 and 4 to form the complex number \(a + bi\): \[ \frac{-12 + 6i}{5} = -\frac{12}{5} + \frac{6}{5}i \] This final expression is in the form \(a + bi\).

Key Concepts

Division of Complex Numbers

Dividing complex numbers can initially seem a bit tricky, but when approached step by step, it becomes quite manageable. The basic idea involves removing the imaginary part from the denominator by multiplying both the numerator and the denominator by a special number known as the "complex conjugate." This changes the division into a simpler expression.
For the division \[\frac{6i}{1-2i}\], we see that directly dividing isn't straightforward, so we use the complex conjugate method, which leads into a structure where division becomes more like simple fraction arithmetic. This technique is crucial for tidy arithmetic, as it ensures there's no imaginary unit remaining in the denominator, simplifying further calculations.
Essentially, the step of finding the correct conjugate is the first step to solving the division practically.

Complex Conjugate

The term "complex conjugate" can sound intimidating, but it really isn't. Simply put, the complex conjugate of a complex number reverses the sign of the imaginary part. For example, if our complex number is \(1 - 2i\), its conjugate will be \(1 + 2i\).

• Complex conjugates are vital because they help in rationalizing denominators, which simplifies our expressions.
• By using the identity \((a - bi)(a + bi) = a^2 + b^2\), the product of a complex number and its conjugate is a real number, which is often very useful in algebraic manipulation.

In our problem, multiplying by the complex conjugate \(1 + 2i\) gets us from a hard-to-simplify fraction into a neater one with real numbers in the denominator. This shows that mastering complex conjugates is not just a cumbersome trick, but a powerful tool in the mathematics toolkit.

Simplifying Expressions

Once your fraction involves only real numbers in the denominator, simplifying becomes much more straightforward.

• The approach is to handle multiplications and simplify your terms according to standard algebraic rules (like distributing multiplication over addition).
• A key part of simplifying involves being comfortable with the property that \(i^2 = -1\). This converts any squared imaginary unit into a real number, which helps eliminate imaginary bases in final results.

For example, when you simplify \[6i(1 + 2i) = 6i + 12i^2\], noticing that \(12i^2 = 12(-1) = -12\) allows us to collect like terms effectively.
Ultimately, the result of division was simplified to reach the form \( -\frac{12}{5} + \frac{6}{5}i \), a clear complex number which demonstrates all the steps of multiplication and division of complex expressions performed correctly. This leads to a much more tidy and usable form.