Question

The value of \((-i)^{(-1)}=\) (a) \(-(\pi / 2)\) (b) \((\pi / 2)\) (c) \(\mathrm{e}^{-(\pi / 2)}\) (b) \(\mathrm{e}^{(\pi / 2)}\)

Short answer

\( \mathrm{e}^{-(\pi / 2)} \)

Step-by-step solution

Express -i in exponential form

The exponential form of a complex number is given by \(re^{i\theta}\), where r is the magnitude of the complex number and \(\theta\) is the argument (angle) of the complex number. Here, we have -i, which can be written as \(0 - i\). The magnitude of -i is: \[r = \sqrt{0^2 + (-1)^2} = 1\] Now we need to find the angle \(\theta\). Since -i is along the negative imaginary axis, the angle is \(\frac{3\pi}{2}\). So, the exponential form of -i is: \[-i = 1e^{i\frac{3\pi}{2}}\]

Raise the exponential form of -i to -1

Now we need to raise this to the power of -1: \[(-i)^{-1} = (1e^{i\frac{3\pi}{2}})^{-1}\] We know that \((a^b)^c = a^{bc}\), so we can multiply exponents: \[(1e^{i\frac{3\pi}{2}})^{-1} = 1e^{-i\frac{3\pi}{2}}\] Comparing this result to the given options, we can see that the answer is (c), which is \(\mathrm{e}^{-(\pi / 2)}\)

Key concepts

Exponential Form

The exponential form of complex numbers is a powerful tool that simplifies many operations, including multiplication and division. The formula is given by \( re^{i\theta} \), where:

• \( r \) is the magnitude of the complex number.
• \( \theta \) is the argument or angle from the positive x-axis in the complex plane.

To convert a complex number like \( -i \) to exponential form, we start by identifying its magnitude and angle. For \( -i \), which lies on the negative imaginary axis, the magnitude \( r \) is 1 since \( \sqrt{0^2 + (-1)^2} = 1 \).
The angle \( \theta \) is \( \frac{3\pi}{2} \), because \( -i \) forms an angle of \( \frac{3\pi}{2} \) in the complex plane from the positive x-axis. Therefore, in the exponential form, it is expressed as \( 1e^{i(\frac{3\pi}{2})} \).
Using the exponential form streamlines the process of manipulating complex numbers, making it easier to deal with operations like exponentiation.

Imaginary Unit

The imaginary unit is denoted by \( i \), which is defined by the property \( i^2 = -1 \). This might seem puzzling at first, but it is a core concept in complex numbers, allowing us to extend the real number system. When we talk about complex numbers, we often write them in the format \( a + bi \), where \( a \) and \( b \) are real numbers.
The imaginary unit enables us to express quantities that can’t be represented on the real number line alone. Without \( i \), special numbers like square roots of negative numbers can't be represented in our numerical system.
In the context of this exercise with \(-i\), the imaginary unit plays a crucial role. Here, \(-i\) signifies that its real component is zero, and it lies on the imaginary axis itself, specifically in the negative direction.

Complex Conjugates

Complex conjugates are another fundamental concept related to complex numbers. If you have a complex number \( a + bi \), its conjugate is \( a - bi \). This changes the sign of the imaginary component while keeping the real part the same. Complex conjugates are especially useful for division and simplifying expressions in complex number arithmetic.
When you multiply a complex number by its conjugate, you get a real number. Specifically, \((a + bi)(a - bi) = a^2 - (bi)^2 = a^2 + b^2\). This result is significant because it simplifies the denominator when dividing complex numbers.

• The conjugate of \(-i \) is \( 0 + i \).
• Multiplying \(-i \) by \( i \) gives us \(-1\), ano matter how imaginary the numbers may seem.

Understanding conjugates helps not just in computation, but also in comprehending the geometrical representation and transformation of complex numbers.