Problem 46 - College Algebra and Calculus: An Applied Approach

Question

Write the quotient in standard form. $$\frac{5}{4-2 i}$$

Step-by-Step Solution

Answer

The quotient in standard form is \( \frac{5}{3} + \frac{5}{6}i \).

Step 1: Identify the Complex Conjugate

Find the complex conjugate of the denominator. The complex conjugate of any complex number \(a + bi\) is \(a - bi\). Here, the complex conjugate of \(4 - 2i\) is \(4 + 2i\).

Step 2: Multiply by the Complex Conjugate

Multiply both the top and bottom of the fraction by the complex conjugate. You get \[\frac{5}{4-2i} \times \frac{4+2i}{4+2i} = \frac{20 +10i}{4^2 + (2i)^2} = \frac{20 +10i}{16 - 4},\]where \(i^2 = -1\) has been used in the denominator.

Step 3: Simplify the Resulting Expression

Simplify the expression obtained in step 2 to get the answer:\[\frac{20 +10i}{12} = \frac{20}{12} + \frac{10i}{12} = \frac{5}{3} + \frac{5}{6}i.\]

Key Concepts

Standard Form of a Complex Number

Understanding the standard form of a complex number is fundamental in working with complex numbers. A complex number is composed of a real part and an imaginary part. In the standard form, it is expressed as a + bi, where a represents the real part and b signifies the imaginary part, with i being the symbol for the imaginary unit.

The imaginary unit i is defined by its property that i^2 = -1. For instance, in the complex number 3 + 4i, 3 is the real part and 4i is the imaginary part. When working with complex numbers, it's crucial to express them in this standard form for simplicity in carrying out operations such as addition, subtraction, multiplication, and division.

Multiplying Complex Numbers

When it comes to multiplying complex numbers, the process involves a few more steps than multiplication with real numbers. To multiply two complex numbers, you use the distributive property, commonly known as the FOIL method (First, Outer, Inner, Last) used for binomials.

The product of (a + bi)(c + di) is ac + adi + bci + bdi^2. Since i^2 = -1, it simplifies to (ac - bd) + (ad + bc)i. It is essential to account for the i^2 term turning negative which is a cornerstone in the multiplication of complex numbers and distinguishes it from the multiplication of real numbers.

An understanding of this multiplication principle is crucial for further operations with complex numbers, such as division, where multiplying by the complex conjugate is a key step.

Simplifying Complex Fractions

The process of simplifying complex fractions involving complex numbers often includes the use of the complex conjugate. When faced with a fraction like \(\frac{z}{w}\), where both z and w are complex numbers, the goal is to eliminate the imaginary part from the denominator. To do this, multiply both the numerator and the denominator by the complex conjugate of the denominator.

This is exemplified in the exercise, where you multiply \(\frac{5}{4-2i}\) by \(\frac{4+2i}{4+2i}\) to get rid of the complex number in the denominator. The multiplication creates a real number in the denominator, allowing the fraction to be split into its real and imaginary parts and simplified further into standard form. It's an elegant method that brings complex fractions into a form that's far easier to understand and work with.