91Ó°ÊÓ

In complex fractions, the real part is the non-imaginary component of a complex number. It represents the value that does not involve the imaginary unit \(i\). For instance, in the expression \(9 - 3i\), the real part is 9.

Identifying the real part: When given a complex fraction, such as \(\frac{9-3i}{-6}\), we first identify and separate the real component. Here, 9 is the real part.
Simplifying: Next, we need to divide this real part by the given denominator, which in this case is -6. Mathematically, we do the following calculation:
\(\frac{9}{-6} = -\frac{3}{2}\). This gives us the simplified real component of the fraction.

The imaginary part of a complex fraction includes the imaginary unit \(i\). In our example, \(9 - 3i\), the imaginary part is \(-3i\). This signifies a component in the complex plane that is perpendicular to the real axis.

Separation: Just like with the real part, we separate out the imaginary component from the complex fraction \(\frac{9 - 3i}{-6}\). The imaginary part here is \(-3i\).
Simplifying: To simplify the imaginary portion, we divide by the denominator:
\(\frac{-3i}{-6} = \frac{1}{2}i\). Note that the negatives cancel each other out, simplifying to a positive \(\frac{1}{2}i\). This final form represents the imaginary part of the fraction in its reduced form.

Simplifying fractions involves making them as straightforward as possible. For complex numbers, this means reducing both the real and imaginary parts.

Initial Steps: You start with the complex fraction, e.g., \(\frac{9-3i}{-6}\). Next, separate into real and imaginary parts:
\(\frac{9}{-6} - \frac{3i}{-6}\).
Simplification of Real and Imaginary Parts: Each part is then simplified individually:
* Real Part: \(\frac{9}{-6} = -\frac{3}{2}\)
* Imaginary Part: \(\frac{-3i}{-6} = \frac{1}{2}i\)
Combining: After simplifying, combine both parts to form the final simplified version of the complex number:
\(-\frac{3}{2} + \frac{1}{2}i\). This makes it easier to work with and understand in mathematical contexts.