Problem 154 - A Complete Resource Book in Mathematics for JEE Main

Question

\(\sqrt{i}-\sqrt{-i}\) is equal to (A) \(i \sqrt{2}\) (B) \(\frac{1}{i \sqrt{2}}\) (C) 0 (D) \(-i \sqrt{2}\)

Step-by-Step Solution

Answer

The correct answer is (A) \(i \sqrt{2}\).

Step 1: Express Complex Numbers in Exponential Form

Start by expressing the complex number \(i\) and \(-i\) in polar/exponential form. We know that \(i = e^{i\frac{\pi}{2}}\) and \(-i = e^{-i\frac{\pi}{2}}\).

Step 2: Calculate Square Roots using Exponential Form

The square root of a complex number \(e^{i\theta}\) can be written as \(e^{i\theta/2}\). Therefore, \(\sqrt{i} = e^{i\frac{\pi}{4}}\) and \(\sqrt{-i} = e^{-i\frac{\pi}{4}}\).

Step 3: Simplify using Euler's Formula

We use Euler's formula, \(e^{i\theta} = \cos(\theta) + i\sin(\theta)\), to find \(\sqrt{i}\) and \(\sqrt{-i}\). Thus, \(\sqrt{i} = \frac{1}{\sqrt{2}} + i\frac{1}{\sqrt{2}}\) and \(\sqrt{-i} = \frac{1}{\sqrt{2}} - i\frac{1}{\sqrt{2}}\).

Step 4: Subtract the Values

Subtract \(\sqrt{-i}\) from \(\sqrt{i}\): \[ \left(\frac{1}{\sqrt{2}} + i\frac{1}{\sqrt{2}}\right) - \left(\frac{1}{\sqrt{2}} - i\frac{1}{\sqrt{2}}\right) = 0 + 2i\frac{1}{\sqrt{2}} = i\sqrt{2}\]

Step 5: Conclusion

The expression \(\sqrt{i} - \sqrt{-i}\) simplifies to \(i\sqrt{2}\), which matches option (A).

Key Concepts

Exponential Form

The exponential form is a powerful way to express complex numbers, which makes calculations, like taking square roots or powers, much easier. Each complex number has a real part and an imaginary part. In exponential form, it can be represented as \( re^{i\theta} \), where \( r \) is the magnitude, and \( \theta \) is the angle it makes with the positive x-axis, also known as the argument. The angle \( \theta \) is measured in radians.

• The magnitude \( r \) can also be seen as the distance of the point from the origin in the complex plane.
• The argument \( \theta \) represents the direction, making it a polar coordinate.

This expression allows the multiplication and division of complex numbers to transform into adding or subtracting their angles. For example, the number \( i \) can be expressed as \( e^{i\frac{\pi}{2}} \), and when squared it becomes \( e^{i\pi} = -1 \). Understanding this principle is crucial for grasping how complex numbers interact in mathematical operations.

Euler's Formula

Euler's Formula is a key mathematical formula connecting exponential functions and trigonometric functions. Given by \( e^{i\theta} = \cos(\theta) + i\sin(\theta) \), this equation reveals the deep relationship between trigonometry and the complex exponential function. It is particularly useful for converting complex numbers from their exponential form back into rectangular form.

• The \( \cos(\theta) \) term represents the real part of the complex number.
• The \( i\sin(\theta) \) term represents the imaginary part.

For example, with \( \sqrt{i} = e^{i\frac{\pi}{4}} \), Euler's formula helps us find that \( \sqrt{i} = \frac{1}{\sqrt{2}} + i\frac{1}{\sqrt{2}} \). Consequently, Euler's Formula is indispensable for converting complex numbers between different forms, which simplifies the computations considerably.

Polar Coordinates

Polar coordinates provide an alternative way to represent complex numbers, parallel to the customary rectangular (also known as Cartesian) coordinates. They are very useful in the context of complex numbers, because they align with the intuition of dealing with magnitudes and angles.

• A point in polar coordinates is given as \( (r, \theta) \), where \( r \) is the radial distance from the origin, and \( \theta \) is the polar angle.
• This system of coordinates elegantly describes the complex number \( z \) as \( z = r(\cos(\theta) + i\sin(\theta)) \), which is the crux of converting it to exponential form \( re^{i\theta} \).

By using the angle \( \theta \) and the radius \( r \), calculations such as finding square roots or rotations are straightforward. For instance, knowing \( i \) as \( (1, \frac{\pi}{2}) \) provides an easy transition to its exponential \( e^{i\frac{\pi}{2}} \) or its trigonometric representation. Polar coordinates simplify many mathematical problems into handling rotations and magnitudes.