Question

Find each of the following in rectangular \((a+b i)\) form if \(z=2-3 i ;\) if \(z=x+i y\) $$\frac{1}{z+1}$$

Short answer

\(\frac{1}{6} + \frac{1}{6}i\)

Step-by-step solution

- Write the given complex number

The given complex number is \(z = 2 - 3i\). We need to find \(\frac{1}{z + 1}\).

- Add 1 to the complex number

Add 1 to the complex number: \(z + 1 = (2 - 3i) + 1 = 3 - 3i\).

- Express the result as a single fraction

We need to find \(\frac{1}{3 - 3i}\).

- Multiply numerator and denominator by the conjugate

Multiply both the numerator and the denominator by the conjugate of the denominator: \[\frac{1}{3 - 3i} \cdot \frac{3 + 3i}{3 + 3i} = \frac{3 + 3i}{(3 - 3i)(3 + 3i)}\].

- Simplify the denominator

Simplify the denominator using the difference of squares formula: \[(3 - 3i)(3 + 3i) = 3^2 - (3i)^2 = 9 - (-9) = 18\].

- Simplify the fraction

Simplify the fraction: \[\frac{3 + 3i}{18} = \frac{1}{6} + \frac{1}{6}i\].

- Write the final solution

Thus, \(\frac{1}{z + 1}\) in rectangular form is \(\frac{1}{6} + \frac{1}{6}i\).

Key concepts

rectangular form

Rectangular form is a way of expressing complex numbers. It involves writing the complex number as a sum of its real part and its imaginary part. For example, if we have a complex number written as \(z = x + i y\), then \(x\) is the real part, and \(y\) is the imaginary part, multiplied by \(i\) (which is the imaginary unit). The rectangular form is very useful for adding, subtracting, and visualizing complex numbers on the complex plane. In our solved example, we start with \(z = 2 - 3i\) and add 1 to it to obtain a new complex number in rectangular form: \(3 - 3i\).

complex conjugate

The complex conjugate of a complex number is obtained by changing the sign of its imaginary part. For instance, if we have the complex number \(z = 3 - 3i\), its conjugate is \(3 + 3i\). The complex conjugate is very useful in simplifying expressions involving complex numbers, especially in division operations. When we multiplied both the numerator and the denominator by the conjugate \(3 + 3i\) in our example, it helped remove the imaginary part from the denominator, making it easier to simplify.

fraction simplification

Fraction simplification when dealing with complex numbers often involves multiplying by the conjugate to get rid of imaginary components in the denominator. In our example, to simplify \(\frac{1}{3 - 3i}\), we multiplied the numerator and the denominator by the conjugate \(3 + 3i\). This transforms the denominator into a real number, which simplifies our calculation. After performing the multiplication, we simplify the fraction further to get the final result \(\frac{1}{6} + \frac{1}{6}i\).

difference of squares

The difference of squares formula states that \(a^2 - b^2 = (a - b)(a + b)\). This formula is especially helpful when dealing with products of complex numbers and their conjugates. In our problem, we used the difference of squares to simplify the denominator \((3 - 3i)(3 + 3i)\), which results in \(3^2 - (-3i)^2 = 9 - (-9) = 18\). Using this formula makes the simplification much more straightforward.

complex number addition

Addition of complex numbers involves adding their real parts together and their imaginary parts together. When we added 1 to the complex number \((2 - 3i)\), we added the real number 1 to the real part 2, resulting in \(3 - 3i\). This process is straightforward and can be done component-wise, providing an easy-to-understand way to combine complex numbers. This technique is foundational when solving more complicated problems involving complex numbers.