Problem 18 - Algebra 1 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

The imaginary part of \(i^{i}\) is (a) 0 (b) 1 (c) 2 (d) \(-1\)

Step-by-Step Solution

Answer

The imaginary part of \(i^i\) is 0.

Step 1: Express Imaginary i in Exponential Form

The first step is to express the imaginary unit, i, in its exponential form using Euler's formula. The complex number i can be written as \( e^{i\frac{\pi}{2}} \) because it represents a 90-degree rotation in the complex plane.

Step 2: Apply Power to Exponential Form

To find \( i^i \), we apply the power to the exponential form: \( (e^{i\frac{\pi}{2}})^i = e^{i^2 \frac{\pi}{2}} = e^{-\frac{\pi}{2}} \). Here, we've used the fact that \( i^2 = -1 \).

Step 3: Simplify Expression to Identify Real and Imaginary Parts

The expression \( e^{-\frac{\pi}{2}} \) is a real number since the exponent is purely real. Therefore, the imaginary part of \( i^i = e^{-\frac{\pi}{2}} \) is \( 0 \).

Key Concepts

The Imaginary Unit

The imaginary unit, commonly denoted as \(i\), is a fundamental concept in complex numbers. It is defined as the square root of -1. This may seem strange at first because there is no real number that, when squared, gives a negative result. This property is unique to \(i\), making it useful for extending our understanding of numbers beyond the real number line.

The defining property of the imaginary unit is essential in complex number arithmetic. In essence, \(i^2 = -1\). This property opens up a whole new dimension of numbers, literally, as it allows us to represent and compute numbers that are not expressible in the standard number system.

With \(i\), we can perform operations that respect the normal rules of algebra. For example:

• Adding: \(a + bi\)
• Multiplying: \((a + bi)(c + di)\)
• Conjugating: \(a - bi\)

These capabilities reflect how the imaginary unit is combined with real numbers in a new system called complex numbers, forming numbers of the format \(a + bi\).

Understanding Euler's Formula

Euler's formula is a brilliant equation that connects complex exponentials with trigonometric functions. It states that:\[e^{ix} = \cos(x) + i\sin(x)\]where \(x\) is a real number.

Euler's formula is particularly useful because it bridges the gap between exponential expressions and rotations in the complex plane. It tells us how complex exponentials behave in terms of circles and angles. This formula is used to express complex numbers in their polar form, a representation that emphasizes the magnitude and angle.

By utilizing Euler's formula, complex numbers like \(i\) can be expressed elegantly in terms of an exponential function. For \(i\), it can be written as:\[i = e^{i\frac{\pi}{2}}\]This representation portrays \(i\) as a 90-degree rotation, which can be very insightful when visualizing and working with complex numbers in the plane.

Exponential Form and its Use

Exponential form provides a powerful way to express complex numbers, making calculations involving powers and roots much more straightforward. The exponential form of a complex number relies on Euler's formula. Instead of working with rectangular coordinates, we switch to polar coordinates.

For a complex number \(z = re^{i\theta}\), \(r\) is the magnitude (or modulus) and \(\theta\) is the argument (or angle) in the complex plane. This format is especially advantageous when multiplying or dividing complex numbers, as multiplication and division are much easier to handle:

• Multiplication: \(z_1 \cdot z_2 = r_1e^{i\theta_1} \cdot r_2e^{i\theta_2} = r_1r_2e^{i(\theta_1 + \theta_2)}\)
• Division: \(\frac{z_1}{z_2} = \frac{r_1}{r_2}e^{i(\theta_1 - \theta_2)}\)

These operations show the elegance of the exponential form when compared to the component-wise approach in rectangular representation.

Visualizing the Complex Plane

The complex plane is an extension of the real number line into two dimensions, allowing us to visualize complex numbers more naturally.

In the complex plane, the horizontal axis is the real axis, and the vertical axis is the imaginary axis. Each complex number \(a + bi\) is represented as a point, or vector, in this plane.

Understanding the complex plane is crucial because it provides insight into how complex numbers behave, particularly regarding addition, multiplication, and how they can represent rotations and scaling:

• Adding complex numbers corresponds to vector addition.
• Multiplying complex numbers relates to rotating and scaling vectors.
• Conjugation reflects a complex number across the real axis.

The complex plane also helps to conceptualize the exponential form of complex numbers; polar coordinates in the plane give a natural feel to the magnitude and direction, enriching our ability to interpret and solve complex number problems.