Problem 5 - Advanced Engineering Mathematics

Question

Write the given complex number in polar form. \(1+i\)

Step-by-Step Solution

Answer

The polar form is \(\sqrt{2}(\cos \frac{\pi}{4} + i\sin \frac{\pi}{4})\).

Step 1: Identify Real and Imaginary Parts

In the complex number \(1+i\), the real part is \(1\) and the imaginary part is also \(1\). This set the stage for converting to polar form.

Step 2: Calculate the Magnitude

The magnitude (also called the modulus) is calculated using the formula \(|z| = \sqrt{a^2 + b^2}\), where \(a\) is the real part and \(b\) is the imaginary part. For \(1+i\), this gives us \(|z| = \sqrt{1^2 + 1^2} = \sqrt{2}\).

Step 3: Determine the Argument

The argument \(\theta\) of the complex number is calculated using \(\theta = \tan^{-1}\left(\frac{b}{a}\right)\). For the complex number \(1+i\), \(\theta = \tan^{-1}\left(\frac{1}{1}\right) = \frac{\pi}{4}\).

Step 4: Construct the Polar Form

The polar form of a complex number is given by \(z = r(\cos \theta + i\sin \theta)\), where \(r\) is the magnitude. Substituting the values we obtained, the polar form of \(1+i\) is \(\sqrt{2}(\cos \frac{\pi}{4} + i\sin \frac{\pi}{4})\).

Key Concepts

Polar Form

Complex numbers can be represented in different ways, and one of the most insightful representations is the polar form. The polar form provides a geometric understanding, emphasizing the number’s magnitude and direction in the complex plane. To express a complex number in polar form, you need two key components:

• The magnitude, denoted as \( r \), represents the distance from the origin to the point in the complex plane.
• The argument, denoted as \( \theta \), defines the angle formed with the positive real axis.

Given a complex number \( z = a + bi \), its polar form is written as \( z = r(\cos \theta + i\sin \theta) \). An alternative expression is \( z = re^{i\theta} \), which employs Euler’s formula, linking exponential functions to trigonometric ones. This expression elegantly combines both magnitude and argument into a single formula, highlighting the strong connection between trigonometry and complex analysis.

Magnitude

The magnitude of a complex number, sometimes called the modulus, quantifies its size without considering direction. For any complex number \( z = a + bi \), the magnitude can be found using the formula:

• \(|z| = \sqrt{a^2 + b^2}\)

This equation follows directly from the Pythagorean theorem. Engagingly, it treats the real part \( a \) and the imaginary part \( b \) as the two legs of a right triangle, with the magnitude serving as the hypotenuse. In our example, with \( 1+i \), the magnitude is computed as:

• \(|z| = \sqrt{1^2 + 1^2} = \sqrt{2}\)

The magnitude is crucial for understanding how large a complex number is, irrespective of its direction, and is an essential part of the polar form representation.

Argument

Understanding the argument of a complex number involves determining the angle it makes with the positive real axis in the complex plane. For a complex number \( z = a + bi \), the argument \( \theta \) is calculated using:

• \(\theta = \tan^{-1}\left(\frac{b}{a}\right)\)

The argument helps describe the direction of the complex number relative to the origin. Another useful perspective is considering it as the phase angle, which tells you how far the point on the complex plane is rotated from the positive real axis. In the case of \( 1+i \), the calculation is straightforward:

• \(\theta = \tan^{-1}\left(\frac{1}{1}\right) = \frac{\pi}{4}\)

Recognizing the argument provides deep insight into the orientation and rotational aspects of complex numbers, thus complementing the magnitude to fully define a complex number's polar form.