Problem 31 - Algebra and Trigonometry

Question

Write the complex number in polar form with argument \(\theta\) between 0 and 2\(\pi\) $$ \sqrt{2}-\sqrt{2} i $$

Step-by-Step Solution

Answer

The polar form is \( 2(\cos\frac{7\pi}{4} + i\sin\frac{7\pi}{4}) \).

Step 1: Identify the real and imaginary parts

The given complex number is \( \sqrt{2} - \sqrt{2}i \). Here, \( \sqrt{2} \) is the real part and \( -\sqrt{2} \) is the imaginary part.

Step 2: Calculate the modulus

The modulus of a complex number \( a + bi \) is given by \( r = \sqrt{a^2 + b^2} \). Here, the modulus is \( r = \sqrt{(\sqrt{2})^2 + (-\sqrt{2})^2} = \sqrt{2 + 2} = \sqrt{4} = 2 \).

Step 3: Calculate the argument

For a complex number in the form \( a + bi \), the argument \( \theta \) is calculated as \( \tan^{-1}(\frac{b}{a}) \). Here, \( \theta = \tan^{-1}(\frac{-\sqrt{2}}{\sqrt{2}}) = \tan^{-1}(-1) = \frac{7\pi}{4} \) since the complex number is in the fourth quadrant.

Step 4: Write the polar form

The polar form of a complex number is given by \( r(\cos\theta + i\sin\theta) \). Substitute the calculated values to get \( 2(\cos\frac{7\pi}{4} + i\sin\frac{7\pi}{4}) \).

Key Concepts

Complex Numbers

Complex numbers are numbers that have two parts: a real part and an imaginary part. They are usually written in the form \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part with \( i \) representing the imaginary unit. The imaginary unit \( i \) is defined by the property \( i^2 = -1 \). This allows complex numbers to extend the concept of one-dimensional number lines to two dimensions, providing a way to measure quantities that require both a magnitude and a direction.
One of the simplest applications of complex numbers is in electrical engineering, where they are used to model AC circuits. In mathematics, they simplify the calculation of roots of polynomials. A familiar example of a complex number is \( \sqrt{2} - \sqrt{2}i \), where \( \sqrt{2} \) is the real part and \( -\sqrt{2}i \) is the imaginary part. This number can be plotted on the complex plane, where the horizontal axis represents the real part and the vertical axis represents the imaginary part.

Modulus of a Complex Number

The modulus of a complex number, often referred to as its magnitude, measures its distance from the origin on the complex plane. For a complex number \( a + bi \), the modulus is calculated using the formula \( r = \sqrt{a^2 + b^2} \).
This formula is derived from the Pythagorean theorem, as it finds the hypotenuse of a right triangle formed by the real and imaginary parts. For our example, the complex number \( \sqrt{2} - \sqrt{2}i \) has a modulus \( r = \sqrt{(\sqrt{2})^2 + (-\sqrt{2})^2} = 2 \).
The modulus tells you how far the point representing the complex number is from the origin. This is particularly helpful in physics and engineering, where the modulus can represent physical quantities like electricity or force magnitude.

Argument of a Complex Number

The argument of a complex number is the angle formed with the positive real axis on the complex plane. It gives us the direction of the vector representing the complex number. The argument is usually denoted by \( \theta \), and it can be calculated using the formula \( \theta = \tan^{-1}\left(\frac{b}{a}\right) \), where \( b \) is the imaginary part and \( a \) is the real part of the complex number.
In our example, the complex number \( \sqrt{2} - \sqrt{2}i \) has an argument \( \theta = \tan^{-1}\left(\frac{-\sqrt{2}}{\sqrt{2}}\right) = \frac{7\pi}{4} \). This angle is obtained by assessing which quadrant the complex number falls into. Since both \( \sqrt{2} \) and \( -\sqrt{2} \) indicate the fourth quadrant, the argument is correctly \( \frac{7\pi}{4} \) since it represents a direction in this sector.
Knowing the argument helps in transforming rectangular coordinates (real and imaginary) into polar coordinates, where a complex number can be fully expressed by its modulus and argument. This is particularly useful in trigonometry and when dealing with waves or oscillations.