Complex Number Division Explained

A complex number is a number that can be expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit, satisfying \(i^2 = -1\). The number \(a\) is called the real part, and \(b\) is called the imaginary part.

When we need to divide one complex number by another, say \(\frac{Z_1}{Z_2}\), where \(Z_1 = a + bi\) and \(Z_2 = c + di\), we use a specific technique. This involves multiplying both the numerator and the denominator by the complex conjugate of the denominator.

The complex conjugate of a complex number \(c + di\) is \(c - di\). Multiplying a complex number by its conjugate results in a real number, which simplifies the division process.

Simplifying the Given Complex Number Z

We are given the complex number \({\rm{Z}}\) in the form of a fraction:

\({\rm{Z}} = \frac{{2 - 3{\rm{i}}}}{{ - 5 + {\rm{i}}}}\)

To express \({\rm{Z}}\) in the standard form \(x + yi\), we need to eliminate the complex number from the denominator. We do this by multiplying the numerator and the denominator by the complex conjugate of the denominator.

The denominator is \(-5 + {\rm{i}}\). Its complex conjugate is \(-5 - {\rm{i}}\).

Let's perform the multiplication:

\({\rm{Z}} = \frac{{(2 - 3{\rm{i}})}}{{( - 5 + {\rm{i}})}} \times \frac{{( - 5 - {\rm{i}})}}{{( - 5 - {\rm{i}})}}\)

Numerator Calculation

First, let's calculate the numerator:

\((2 - 3{\rm{i}})(-5 - {\rm{i}})\)

We distribute the terms:

\(= 2(-5) + 2(-{\rm{i}}) - 3{\rm{i}}(-5) - 3{\rm{i}}(-{\rm{i}})\)

\(= -10 - 2{\rm{i}} + 15{\rm{i}} + 3{\rm{i}}^2\)

Since we know that \({\rm{i}}^2 = -1\), substitute this value:

\(= -10 - 2{\rm{i}} + 15{\rm{i}} + 3(-1)\)

\(= -10 - 2{\rm{i}} + 15{\rm{i}} - 3\)

Combine the real parts and the imaginary parts:

\(= (-10 - 3) + (-2 + 15){\rm{i}}\)

\(= -13 + 13{\rm{i}}\)

Denominator Calculation

Next, let's calculate the denominator:

\(( - 5 + {\rm{i}})(-5 - {\rm{i}})\)

This is in the form \((a+b)(a-b)\), which simplifies to \(a^2 - b^2\). Here, \(a = -5\) and \(b = {\rm{i}}\).

\(= (-5)^2 - ({\rm{i}})^2\)

\(= 25 - {\rm{i}}^2\)

Again, substitute \({\rm{i}}^2 = -1\):

\(= 25 - (-1)\)

\(= 25 + 1\)

\(= 26\)

Final Expression of Z

Now, substitute the calculated numerator and denominator back into the expression for \({\rm{Z}}\):

\({\rm{Z}} = \frac{{ - 13 + 13{\rm{i}}}}{{26}}\)

To express this in the form \(x + yi\), we separate the real and imaginary parts:

\({\rm{Z}} = \frac{{ - 13}}{{26}} + \frac{{13{\rm{i}}}}{{26}}\)

Simplify the fractions:

\({\rm{Z}} = -\frac{1}{2} + \frac{1}{2}{\rm{i}}\)

Converting to decimal form:

\({\rm{Z}} = -0.5 + 0.5{\rm{i}}\)

Comparing with Options

Let's compare our result with the given options:

• Option 1: − 0.5 − 0.5i
• Option 2: − 0.5 + 0.5i
• Option 3: 0.5 − 0.5i
• Option 4: 0.5 + 0.5i

Our calculated value for \({\rm{Z}}\) is \(-0.5 + 0.5{\rm{i}}\), which matches Option 2.