The real part of a complex number is the component that is not multiplied by the imaginary unit, \(i\). It behaves like a regular, real number that we are familiar with over our mathematical journey. When working with complex numbers, we split them into two parts:
- The real part
- The imaginary part
In our exercise, we start with two complex numbers: \(-\left(\frac{3}{2}+\frac{5}{2} i\right)\) and \(+\left(\frac{5}{3}+\frac{11}{3}i\right)\). Here, the real parts are \(-\frac{3}{2}\) and\(\frac{5}{3}\).
To find the resulting real part, we perform the operation:\(-\frac{3}{2} + \frac{5}{3}\).
This involves a simple subtraction/addition operation, where common fractions need to have a common denominator:
- Convert \(-\frac{3}{2}\) to \(-\frac{9}{6}\)
- Convert \(\frac{5}{3}\) to \(\frac{10}{6}\)
- Add/Subtract these values: \(-\frac{9}{6} + \frac{10}{6} = \frac{1}{6}\)
Providing the real component of the result.