- A complex number \(a+bi\) is non-real if the imaginary part \(b
eq 0\).
- Simplifying each expression, we find that \(5-\sqrt{\frac{9}{4}}\) and \(\frac{2-\sqrt{12}}{5}\) are purely real numbers.
- The expressions \(1+\sqrt{-3}\), \(4-3\sqrt{-16}\), \(\frac{3+2\sqrt{-9}}{7}\), and \(9\sqrt{-\frac{7}{5}}\) contain imaginary units with non-zero coefficients.
- The set of non-real complex numbers is \(\boxed{1+\sqrt{-3}, 4-3\sqrt{-16}, \frac{3+2\sqrt{-9}}{7}, 9\sqrt{-\frac{7}{5}}}\).
Explanation
- Understanding Complex Numbers and the Classification Criteria
A complex number is written in the form \(a + bi\), where \(a\) and \(b\) are real numbers and \(i = \sqrt{-1}\). A complex number is considered 'non-real' if the imaginary part \(b\) is not equal to zero (\(b
eq 0\)). We will evaluate each expression in the provided table to determine if it can be written as \(a + bi\) with \(b
eq 0\).
- Evaluating the Expressions
We analyze each expression systematically:
- \(5-\sqrt{\frac{9}{4}} = 5 - \frac{3}{2} = 3.5\). Since there is no imaginary part, this is a real number.
- \(1+\sqrt{-3} = 1 + \sqrt{3}i\). Here, the imaginary part is \(\sqrt{3}
eq 0\), so this is a non-real complex number.
- \(4-3\sqrt{-16} = 4 - 3(4i) = 4 - 12i\). Here, the imaginary part is \(-12
eq 0\), so this is a non-real complex number.
- \(\frac{2-\sqrt{12}}{5} = \frac{2-2\sqrt{3}}{5} = \frac{2}{5} - \frac{2\sqrt{3}}{5}\). Since this expression consists only of real numbers, it is a real number.
- \(\frac{3+2\sqrt{-9}}{7} = \frac{3+2(3i)}{7} = \frac{3+6i}{7} = \frac{3}{7} + \frac{6}{7}i\). Here, the imaginary part is \(\frac{6}{7}
eq 0\), so this is a non-real complex number.
- \(9\sqrt{-\frac{7}{5}} = 9i\sqrt{\frac{7}{5}}\). Here, the imaginary part is \(9\sqrt{\frac{7}{5}}
eq 0\), so this is a non-real complex number.
- Final Classification of Non-Real Complex Numbers
Based on our evaluation, the expressions that result in non-real complex numbers are \(1+\sqrt{-3}\), \(4-3\sqrt{-16}\), \(\frac{3+2\sqrt{-9}}{7}\), and \(9\sqrt{-\frac{7}{5}}\). These expressions contain the imaginary unit \(i\) with a non-zero coefficient. The final answer is \(\boxed{1+\sqrt{-3}, 4-3\sqrt{-16}, \frac{3+2\sqrt{-9}}{7}, 9\sqrt{-\frac{7}{5}}}\).
Examples
Complex numbers are essential in engineering, especially in electrical circuit analysis. For instance, impedance in alternating current (AC) circuits is represented as a complex number \(Z = R + Xi\), where \(R\) is the resistance (real part) and \(X\) is the reactance (imaginary part). Recognizing whether a value is real or non-real helps engineers determine if a circuit will consume power (resistance) or store energy (reactance) in capacitors and inductors, playing a crucial role in building efficient electronic systems.