Problem 29 - Precalculus : Building Concepts and Connections

Question

Use De Moivre's Theorem to find each expression. $$(1+i)^{4}$$

Step-by-Step Solution

Answer

The result of \((1 + i)^4\) is -4.

Step 1: Convert Complex Number to Polar Form

The complex number is given as \(1 + i\). First, calculate the modulus \(r\), \(r = \sqrt{a^2 + b^2} = \sqrt{1^2 + 1^2} = \sqrt{2}\). Now calculate the argument \(\theta\), using the formula \(\tan^{-1}(b/a)\), which gives \(\tan^{-1}(1/1) = \pi/4\). Therefore, the complex number can be expressed in polar form as \(\sqrt{2}(\cos(\pi/4) + i \sin(\pi/4))\)

Step 2: Apply De Moivre's Theorem

According to De Moivre's theorem, raising a complex number to a power is the same as raising its modulus to the power and multiplying its argument by the power. Therefore, \((1 + i)^4 = \sqrt{2}^4(\cos(4 \cdot \pi/4) + i \sin(4· \pi/4)) = 4(\cos(\pi) + i \sin(\pi))\)

Step 3: Convert back to Rectangular Form

Now this must be converted back to rectangular form x + yi. Use the definitions \(\cos(\pi) = -1\) and \(\sin(\pi) = 0\) to get \(4(\cos(\pi) + i \sin(\pi)) = 4(-1 + 0i) = -4\)

Key Concepts

Complex Numbers

Complex numbers may sound complicated, but they're simply a combination of a real number and an imaginary number. They're usually written in the form \(a + bi\), where:

• \(a\) is the real part.
• \(b\) is the imaginary part.
• \(i\) is the imaginary unit, with the property \(i^2 = -1\).

For example, the number \(1 + i\) has 1 as its real part and 1 as its coefficient of the imaginary part. Complex numbers are essential in mathematics because they allow us to work with equations that have no real solutions. They also enable rotations and translations in the complex plane, facilitating a new dimension of analysis.
When working with problems involving exponential powers of complex numbers, it's often advantageous to express these numbers in a different form known as polar form.

Polar Coordinates

Polar coordinates offer a unique way to think about complex numbers. Instead of expressing a complex number as \(a + bi\), a polar representation uses a radius (or modulus) and an angle (or argument). Here's how it works:

• The modulus \(r\) of the complex number \(a + bi\) is calculated as \(\sqrt{a^2 + b^2}\). This represents the distance from the origin to the point in the complex plane.
• The argument \(\theta\) is the angle formed with the positive real axis, and it can be calculated using \(\tan^{-1}(\frac{b}{a})\).

In polar form, the complex number \(1 + i\) becomes \(\sqrt{2}(\cos(\pi/4) + i \sin(\pi/4))\). This representation is particularly useful when working with exponential powers due to its compactness and its relationship with trigonometric identities.

Modulus and Argument

The terms 'modulus' and 'argument' are crucial for understanding the polar form of a complex number.

• The modulus \(r\) measures the magnitude or length of the vector in the complex plane. For \(1 + i\), the modulus is computed as \(\sqrt{1^2 + 1^2} = \sqrt{2}\).
• The argument \(\theta\) indicates the direction of the vector, given by the angle it forms with the positive real axis. For \(1 + i\), this is \(\pi/4\).

These two components are used in combination to express a complex number in its polar form. Using De Moivre's Theorem, complex numbers raised to a power can be simplified significantly by raising the modulus to the power and multiplying the argument by the power. This reduced form is then converted back to the rectangular form \(x + yi\) to finalize the calculation, providing solutions that are directly usable in various applications. In the example of \((1 + i)^4\), we used both modulus and argument to find that \( (1 + i)^4 = -4\).