Express $\sqrt{-16}$ as a complex number in standard form: $a+b i$
$\sqrt{-16}=$
To express \(\sqrt{-16}\) as a complex number in the form \(a+bi\), we first identify the properties of the imaginary unit \(i\). The imaginary unit \(i\) is defined as the principal square root of \(-1\), meaning \(i = \sqrt{-1}\) or \(i^2 = -1\). For any positive real number \(x\), the square root of its negative, \(\sqrt{-x}\), can be expressed as:
\[\sqrt{-x} = \sqrt{x \cdot (-1)} = \sqrt{x} \cdot \sqrt{-1} = i\sqrt{x}.\]
This property allows us to work with negative radicands by pulling out the factor of \(i\).
We apply the property \(\sqrt{-x} = i\sqrt{x}\) to the given expression \(\sqrt{-16}\). Here, \(x = 16\). Therefore:
\[\sqrt{-16} = i\sqrt{16}.\]
Next, we calculate the square root of \(16\). Since \(4 \times 4 = 16\), we have \(\sqrt{16} = 4\). Substituting this back into our expression, we get:
\[\sqrt{-16} = i \cdot 4 = 4i.\]
A complex number is written in the standard form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. In our result \(4i\), the real part \(a\) is \(0\), and the imaginary part \(b\) is \(4\). Therefore, we can write \(4i\) in the standard form as:
\[0 + 4i.\]
Thus, the expression \(\sqrt{-16}\) in standard form is \(\boxed{0+4i}\).
Complex numbers are essential in electrical engineering for analyzing alternating current (AC) circuits. When engineers calculate the impedance of an inductor or a capacitor, they encounter values involving square roots of negative numbers. For example, if an analysis leads to an impedance term proportional to \(\sqrt{-16}\) ohms, they use the complex notation \(4i\) (or sometimes \(4j\) in engineering) to perform calculations that model the phase shift and resistance in the circuit, allowing them to design reliable power grids and communication devices.