Problem 89 - Precalculus

Question

Graphing calculators are able to convert complex numbers from rectangular to polar form using the \(\overline{\mathbf{a b s}}\) command to find the modulus and the angle command to find the angle. Find \(\operatorname{abs}(1+i) .\) Find angle \((1+i) .\) Write \(1+i\) in polar form.

Step-by-Step Solution

Answer

The polar form of \(1+i\) is \(\sqrt{2}(\cos \frac{\pi}{4} + i\sin \frac{\pi}{4})\) or \(\sqrt{2} e^{i\frac{\pi}{4}}\).

Step 1: Understand the Rectangular Form

The complex number given is \(1 + i\). In this form, the real part is \(1\) (along the x-axis) and the imaginary part is \(i\) (along the y-axis). This is known as the rectangular or Cartesian form of a complex number.

Step 2: Calculate the Modulus

The modulus (or absolute value) of a complex number \(a + bi\) is given by \(\sqrt{a^2 + b^2}\). For \(1 + i\), we have \(a = 1\) and \(b = 1\). Therefore, the modulus is \(\sqrt{1^2 + 1^2} = \sqrt{2}\).

Step 3: Calculate the Argument (Angle)

The angle (argument) \(\theta\) for a complex number \(a + bi\) is calculated using \(\tan^{-1}(\frac{b}{a})\). For \(1 + i\), this becomes \(\tan^{-1}(\frac{1}{1}) = \tan^{-1}(1)\), which equals \(\frac{\pi}{4}\) radians or 45 degrees.

Step 4: Write in Polar Form

A complex number in polar form is written as \(r(\cos \theta + i\sin \theta)\) or in exponential form as \(re^{i\theta}\). Substituting \(r = \sqrt{2}\) and \(\theta = \frac{\pi}{4}\), the polar form is \(\sqrt{2} \left( \cos \frac{\pi}{4} + i \sin \frac{\pi}{4} \right)\) or \(\sqrt{2} e^{i \frac{\pi}{4}}\).

Key Concepts

Polar Form

The polar form of a complex number is a way to express the number in terms of its magnitude and direction. Instead of using the traditional Cartesian format (real part + imaginary part), polar form uses

• modulus (the distance from the origin) and
• argument (the angle from the positive real axis).

In polar form, a complex number is represented by \[ r(\cos \theta + i \sin \theta) \] or in exponential form as \[ re^{i\theta}. \]Here, \( r \) is the modulus, or absolute value, of the complex number, indicating how far away the number is from the origin. Meanwhile, \( \theta \) is the argument, which tracks the rotation required to reach the point from the origin.
This notation becomes especially useful when multiplying or dividing complex numbers, simplifying the process to manipulate angles and moduli directly.

Modulus of a Complex Number

The modulus of a complex number, also known as its absolute value, measures the distance of the complex number from the origin in the complex plane. It is calculated using the formula \[ |a + bi| = \sqrt{a^2 + b^2}. \]For the complex number \(1+i\), both the real and imaginary parts are \(1\). Plugging these values into the formula gives us: \[ |1 + i| = \sqrt{1^2 + 1^2} = \sqrt{2}. \]This result tells us that the complex number \(1+i\) is \(\sqrt{2}\) units away from the origin.
Understanding the modulus is crucial because it represents the size or the magnitude of the complex number without considering its direction.

Argument of a Complex Number

The argument of a complex number describes its direction in the complex plane relative to the positive real axis. It is essentially the "angle" that the vector makes with the positive x-axis and is denoted by \(\theta\). We find this angle using the formula \[ \theta = \tan^{-1}(\frac{b}{a}). \]For \(1+i\), both \(a\) and \(b\) are \(1\), leading to \[ \theta = \tan^{-1}(1) = \frac{\pi}{4}. \]This angle, or argument, is \(45\) degrees or \(\frac{\pi}{4}\) radians. It indicates how the number \(1+i\) is oriented in the plane.
The argument is vital in distinguishing where the complex number "points" or is located directionally. It's worth noting that the argument can also be expressed in terms of complete circular rotations by adding \(2\pi\) multiples to \(\theta\).

Graphing Calculators

Graphing calculators are quite powerful when working with complex numbers and can greatly simplify the process of converting between different forms. By using functions such as

• \(\operatorname{abs}\) for finding the modulus, and
• a specific angle function (often named differently depending on the calculator model) to determine the argument,

students can quickly and accurately convert complex numbers from their rectangular form (like \(1+i\)) to their polar form.
To calculate the modulus, the command \(\operatorname{abs}(a + bi)\) will give the distance from the origin to the point representing the complex number. For the argument, the calculator can provide the angle made with the positive x-axis.
These capabilities make graphing calculators an invaluable tool in learning and confirming the manual steps involved with complex numbers, speeding up calculations and allowing for immediate checks of your work.