Problem 29 - Algebra and Trigonometry

Question

Write the complex number in polar form with argument \(\theta\) between 0 and 2\(\pi\) $$ 1+i $$

Step-by-Step Solution

Answer

Polar form is \( \sqrt{2} e^{i\frac{\pi}{4}} \).

Step 1: Identify Real and Imaginary Parts

The complex number given is \(1 + i\). Here, the real part \(a = 1\) and the imaginary part \(b = 1\).

Step 2: Calculate the Magnitude

The magnitude \(|z|\) of a complex number \(z = a + bi\) is given by the formula: \[ |z| = \sqrt{a^2 + b^2} \] For \(1 + i\), we have: \[ |z| = \sqrt{1^2 + 1^2} = \sqrt{2} \]

Step 3: Determine the Argument

The argument \(\theta\) is calculated using the tangent function: \[ \tan(\theta) = \frac{b}{a} \] For \(1 + i\), this becomes: \[ \tan(\theta) = \frac{1}{1} = 1 \] Using the fact that \(\tan(\frac{\pi}{4}) = 1\), we determine \(\theta = \frac{\pi}{4}\).

Step 4: Write in Polar Form

The polar form of a complex number is given by:\[ z = r (\cos \theta + i \sin \theta) \] or \[ z = r e^{i\theta} \] For \(1 + i\), using \(|z| = \sqrt{2}\) and \(\theta = \frac{\pi}{4}\), the polar form is:\[ \sqrt{2} (\cos \frac{\pi}{4} + i \sin \frac{\pi}{4}) \] or equivalently,\[ \sqrt{2} e^{i\frac{\pi}{4}} \]

Key Concepts

Complex Numbers

Complex numbers are numbers that have two components: a real part and an imaginary part. They are expressed in the form \(a + bi\), where \(a\) represents the real part and \(b\) represents the imaginary part. The imaginary component is multiplied by \(i\), where \(i\) is the imaginary unit. It is essential to understand that \(i\) is the square root of \(-1\), a fundamental concept that makes complex numbers distinct from real numbers.
- Real Parts: Represented by the real number \(a\). - Imaginary Parts: Represented by \(b\) times \(i\).
Complex numbers are plotted on a complex plane, where the x-axis represents the real part, and the y-axis represents the imaginary part. This visualization helps in understanding their behavior similarly to vectors. With this two-dimensional approach, complex numbers become powerful tools in various fields such as engineering, physics, and mathematics.

Magnitude of Complex Numbers

The magnitude, often referred to as the absolute value or modulus, of a complex number provides a measure of its size, irrespective of its direction in the complex plane. For a given complex number \(z = a + bi\), the magnitude is symbolized by \(|z|\) and is calculated using the formula: \[ |z| = \sqrt{a^2 + b^2} \] This formula stems from the Pythagorean theorem, representing the length of the vector from the origin to the point \((a, b)\) in the complex plane.
- For \(1 + i\), \(|z| = \sqrt{1^2 + 1^2} = \sqrt{2}\).
Understanding the magnitude is crucial as it allows us to express complex numbers in polar form and gives an indication of how far the point is from the origin, creating a bridge between the algebraic form \(a + bi\) and the geometric interpretation on the complex plane.

Argument of a Complex Number

The argument of a complex number is the angle \(\theta\) formed between the positive real axis and the line representing the complex number in the complex plane. This angle is always measured positively from the real axis, moving counterclockwise. Calculating the argument involves the tangent function: \[ \tan(\theta) = \frac{b}{a} \]where \(a\) and \(b\) are the real and imaginary parts, respectively. For \(1 + i\), we have:- \(\tan(\theta) = \frac{1}{1} = 1\), which gives the angle \(\theta = \frac{\pi}{4}\).
It is important to note that the argument can lie within a specific range, commonly between \(0\) and \(2\pi\). In polar representation, knowing the argument along with the magnitude allows you to accurately express any complex number using trigonometric or exponential forms: \(z = r (\cos \theta + i \sin \theta)\) or equivalently, \(z = r e^{i\theta}\).
This dual representation eases complex number computations, especially for multiplication and division operations, where adding or subtracting the arguments respectively replaces the usual rules of algebra.