Question

Find each of the following in rectangular \((a+b i)\) form if \(z=2-3 i ;\) if \(z=x+i y\) $$z / \bar{z}$$

Short answer

The expression in rectangular form is \(-\frac{5}{13} - \frac{12}{13}i\).

Step-by-step solution

Identify Conjugate of z

The conjugate of a complex number is found by changing the sign of the imaginary part. If the complex number is given by \(z = 2 - 3i\), then its conjugate is given by \(\bar{z} = 2 + 3i\).

Perform Division

Next, we need to divide \(z\) by \(\bar{z}\). This can be written as \(\frac{z}{\bar{z}} = \frac{2 - 3i}{2 + 3i}\). To simplify this, multiply the numerator and the denominator by the conjugate of the denominator. This gives: \(\frac{2 - 3i}{2 + 3i} \times \frac{2 - 3i}{2 - 3i}\).

Multiply Numerator and Denominator

Multiply the numerator and the denominator by the conjugate of the denominator. This results in: \[\frac{(2 - 3i)(2 - 3i)}{(2 + 3i)(2 - 3i)}\].

Apply Formula For Multiplying Conjugates

We use the formula for multiplying conjugates: \((a+bi)(a-bi) = a^2 + b^2\). For the denominator: \((2 + 3i)(2 - 3i) = 2^2 + 3^2 = 4 + 9 = 13\).

Expand Numerator

Expand the numerator: \((2 - 3i)^2 = 4 - 12i + 9i^2\). Remind that \(i^2 = -1\) gives: \((2 - 3i)^2 = 4 - 12i + 9(-1) = 4 - 12i - 9 = -5 - 12i\).

Simplify the Fraction

Plug the expanded numerator and calculated denominator back in to simplify the fraction: \[\frac{-5 - 12i}{13}\]. Thus \((x + yi)\) form of the expression is: \(-\frac{5}{13} - \frac{12}{13}i\).

Key concepts

complex number conjugate

The conjugate of a complex number changes the sign of the imaginary component. If you have a number like \(z = 2 - 3i\), the conjugate, represented as \(\bar{z}\), flips the sign of \i\. So in this case, \(\bar{z} = 2 + 3i\).
Conjugates are very useful in simplifying complex number operations, especially when dealing with division. By using conjugates, complex denominators can be turned into real numbers, which are much easier to handle.

rectangular form

Rectangular form is one way to express complex numbers, represented as \(a + bi\). Here, \a\ is the real part and \b\ is the imaginary part.
For example, in our problem, \(z = 2 - 3i\) means the real part is 2 and the imaginary part is -3. Converting complex numbers into rectangular form helps keep track of both real and imaginary components easily.

fraction simplification

Simplification often involves turning a complex fraction into a more manageable form. For our division problem, this means multiplying by the conjugate of the denominator.
Let's take \(\frac{2 - 3i}{2 + 3i}\). We multiply both the numerator and the denominator by the conjugate of the denominator (in this case, \(2 - 3i\)) to get: \(\frac{(2 - 3i)(2 - 3i)}{(2 + 3i)(2 - 3i)}\).
This method gets rid of the complex number in the denominator, making it easier to simplify.

complex number operations

Complex number operations might seem tricky, but breaking them down step-by-step helps. Let's focus on multiplication and division.
When multiplying complex numbers, use the distributive property: \((a + bi)(c + di) = ac + adi + bci + bdi^2 \). Don't forget that \(i^2 = -1\).
Division involves multiplying the numerator and the denominator by the conjugate of the denominator. For example, \( \frac{2 - 3i}{2 + 3i} \times \frac{2 - 3i}{2 - 3i} \). After multiplying, use the formula \((a+bi)(a-bi) = a^2 + b^2\) for the denominator. Finally, simplify the result, converting it back into rectangular form if necessary.