Problem 57 - Precalculus with Limits

Question

Multiplying or Dividing Complex Numbers In Exercises \(57-62,(\) a) write the trigonometric forms of the complex numbers, (b) perform the indicated operation using the trigonometric forms, and (c) perform the indicated operation using the standard forms, and check your result with that of part (b). $$(2+2 i)(1-i)$$

Step-by-Step Solution

Answer

The result of multiplying the complex numbers \(2 + 2i\) and \(1 - i\) is \(4\). This is confirmed by achieving the same result while using both the trigonometric and standard forms.

Step 1: Convert to trigonometric forms

To convert a complex number to trigonometric form, use the formulas \( r = \sqrt{a^{2} + b^{2}} \) for the magnitude and \( \theta = arctan(\frac{b}{a})\) for the argument. For \(2 + 2i\), the magnitude \( r_{1} \) is \( \sqrt{2^{2} + 2^{2}} = 2\sqrt{2} \) and the argument \( \theta_{1} \) is \( arctan(\frac{2}{2}) = \frac{\pi}{4} \). Likewise, for \(1 - i\), the magnitude \( r_{2} \) is \( \sqrt{1^{2} + (-1)^{2}} = \sqrt{2} \) and the argument \( \theta_{2} \) is \( arctan(\frac{-1}{1}) = -\frac{\pi}{4} \). So, their trigonometric forms are \(2 \sqrt{2} e^{i \frac{\pi}{4}}\) and \( \sqrt{2} e^{-i \frac{\pi}{4}}\) respectively.

Step 2: Perform multiplication using trigonometric forms

To perform multiplication in trigonometric form, multiply the magnitudes and add the angles. Multiplying, we get \( r = r_{1} * r_{2} = 2\sqrt{2} * \sqrt{2} = 4 \). Similarly, adding the arguments gives \( \theta = \theta_{1} + \theta_{2} = \frac{\pi}{4} -\frac{\pi}{4} = 0 \). The product thus in the trigonometric form is \(4 e^{i 0} = 4(1+0i) = 4 \).

Step 3: Perform multiplication using standard forms

Multiplying \(2 + 2i\) and \(1 - i\) directly using standard forms as per the FOIL method gives \(2*1 + 2*-i + 2i*1 + 2i*-i = 2 - 2i + 2i - 2 = 4\). This checks with the result obtained in the previous step.

Key Concepts

Trigonometric Form of Complex Numbers

Complex numbers, which are the sum of a real part and an imaginary part, can be expressed in a unique format known as the trigonometric form. This form takes advantage of the polar coordinate system, where any point on a plane can be identified by its distance from the origin, and the angle it makes with the positive x-axis.

The trigonometric form of a complex number is written as: \(r (\cos \theta + i\sin \theta)\), or more succinctly using Euler's formula as \(r e^{i\theta}\), where \(r\) represents the magnitude (or modulus) of the complex number, and \(\theta\), known as the argument, is the angle made with the positive real axis.

To convert a complex number from its standard \(a + bi\) form to its trigonometric form, the formulas \(r = \sqrt{a^{2} + b^{2}}\) for finding the magnitude and \(\theta = \arctan(\frac{b}{a})\) for finding the argument are used. The beauty of the trigonometric form lies in its simplicity when it comes to multiplication and division, as these operations become a matter of manipulating the magnitudes and summing or subtracting the angles, respectively.

Operations with Complex Numbers

Complex numbers can be added, subtracted, multiplied, and divided using various methods depending on their form. When in standard form (\(a + bi\)), addition and subtraction are straightforward, involving combining like terms. Multiplication, however, often employs the FOIL (First, Outer, Inner, Last) method, which can become cumbersome especially with larger expressions.

Alternatively, when complex numbers are in trigonometric form, multiplication and division are more elegant—simply multiply the magnitudes for multiplication, and add the angles. Likewise, divide the magnitudes for division, and subtract the angles. This results in a cleaner and often faster computation, which can be particularly advantageous in settings involving numerous complex number operations.

FOIL Method in Algebra

The FOIL method is a technique used in algebra to simplify the multiplication of two binomials. The acronym FOIL stands for First, Outer, Inner, Last, referring to the terms in each binomial that are multiplied together in sequence.

First:

Multiply the first terms in each binomial.

Outer:

Multiply the outermost terms in the product.

Inner:

Multiply the inner terms.

Last:

Multiply the last terms in the binomials.

Once each of these pairs is multiplied, the results are added together to give the final product. This method is especially useful when dealing with complex numbers in standard form, as it correctly accounts for the real and imaginary parts of the product. However, for larger expressions or products involving complex conjugates, the trigonometric form might offer a simpler alternative.