- A non-real complex number takes the form \(a + bi\) where \(b
eq 0\).
- Examining the provided expressions shows that \(\frac{9 + 3\sqrt{5}}{2}\), \(5\sqrt{\frac{1}{3}} - \frac{9}{\sqrt{7}}\), and \(2 - \frac{1}{\sqrt{11}}\) result in real numbers.
- The expression \(\frac{8}{3} + \sqrt{-\frac{7}{3}}\) contains the square root of a negative number, resulting in an imaginary component \(\sqrt{\frac{7}{3}}i\).
- Thus, the non-real complex number is \(\boxed{\frac{8}{3} + \sqrt{-\frac{7}{3}}}\).
Explanation
- Defining Non-Real Complex Numbers
A non-real complex number is typically expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers and \(b
eq 0\). In essence, a non-real complex number contains an imaginary part that involves the square root of a negative real number. To determine which of the provided options is a non-real complex number, we need to evaluate each one to see if it includes an expression of the form \(\sqrt{r}\) where \(r < 0\).
- Evaluating the Options
Let's analyze the given options one by one:
- \(\frac{8}{3} + \sqrt{-\frac{7}{3}}\): This expression involves the square root of a negative number, \(\sqrt{-\frac{7}{3}}\). Since \(\sqrt{-x} = \sqrt{x} \cdot i\) for \(x > 0\), this can be written as \(\frac{8}{3} + \sqrt{\frac{7}{3}}i\). Because the imaginary part \(\sqrt{\frac{7}{3}}\) is not zero, this is a non-real complex number.
- \(\frac{9 + 3\sqrt{5}}{2}\): This expression contains only real numbers \(9, 3, 5,\) and \(2\), and the square root of a positive number (\(\sqrt{5}\)). Therefore, this represents a real number.
- \(5\sqrt{\frac{1}{3}} - \frac{9}{\sqrt{7}}\): This expression consists solely of operations on positive real numbers. Since all components are real, the result is a real number.
- \(2 - \frac{1}{\sqrt{11}}\): This expression involves only real numbers, and the square root of a positive number (\(\sqrt{11}\)). Thus, this is also a real number.
- Conclusion
Based on the evaluation, only the first option, \(\frac{8}{3} + \sqrt{-\frac{7}{3}}\), contains an imaginary component. All other options result in real numbers. Therefore, the first option is the only non-real complex number. The final answer is \(\boxed{\frac{8}{3} + \sqrt{-\frac{7}{3}}}\).
Examples
In electrical engineering, alternating current (AC) circuits are analyzed using complex numbers to represent impedance. A non-real complex number like \(Z = R + jX\) helps engineers describe how a component (like a resistor and an inductor) affects the voltage and current, with the real part representing resistance and the imaginary part (represented by \(j = \sqrt{-1}\)) representing reactance. Identifying whether a value is real or non-real is crucial for calculating power factors and ensuring the stability of electrical systems.