Book cover for Precalculus: A Right Triangle Approach

Precalculus: A Right Triangle Approach

Judith A. Beecher, Marvin L. Bittinger, David J. Ellenbogen, Judith A. Penna

ISBN #9780321783967

5th Edition

5,905 Questions

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25,202 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

The complex-number system expands the real numbers by introducing the imaginary unit i, which satisfies i² = -1. This allows us to express square roots of negative numbers and solve equations that lack real solutions. Arithmetic operations with complex numbers follow standard algebraic rules with additional care given to the properties of i. Simplification techniques, including the cyclical powers of i and the use of conjugates in division, are fundamental skills necessary for working in various mathematical and engineering applications.

Learning Objectives

1

Define and explain the concept of complex numbers, including the importance of the imaginary unit i.

2

Perform arithmetic operations with complex numbers: addition, subtraction, multiplication, and division using conjugates.

3

Express square roots of negative numbers in terms of i and simplify powers of i using their cyclical properties.

4

Utilize complex numbers to solve equations that lack real-number solutions, such as quadratic equations with negative discriminants.

Key Concepts

CONCEPT

DEFINITION

Complex Number

A number of the form a + bi, where a and b are real numbers. Here, 'a' is the real part and 'b' is the imaginary part.

Imaginary Unit (i)

A number defined by the property i² = -1. It is used to express the square roots of negative numbers.

Real Part

In a complex number a + bi, the term 'a' is known as the real part.

Imaginary Part

In a complex number a + bi, the term 'b' represents the imaginary part.

Complex Conjugate

For a complex number a + bi, the complex conjugate is a - bi. Multiplying a complex number by its conjugate yields a real number.

Example Problems

Example 1

Express the number in terms of $i$. $$ \sqrt{-3} $$

Example 2

Express the number in terms of $i$. $$ \sqrt{-21} $$

Example 3

Express the number in terms of $i$. $$ \sqrt{-25} $$

Example 4

Express the number in terms of $i$. $$ \sqrt{-100} $$

Example 5

Express the number in terms of $i$. $$ -\sqrt{-33} $$
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Step-by-Step Explanations

QUESTION

Express √(-p) in terms of i, where p is a positive real number.

STEP-BY-STEP ANSWER:

Step 1: Recognize that √(-p) = √(p) * √(-1).
Step 2: Replace √(-1) with i.
Step 3: Write the expression as i√(p).
Final Answer: √(-p) = i√(p).

Expressing Square Roots of Negative Numbers

QUESTION

Multiply (a + bi) by (c + di) and simplify the result.

STEP-BY-STEP ANSWER:

Step 1: Use the distributive property (FOIL method): (a + bi)(c + di) = ac + adi + bci + bdi².
Step 2: Replace i² with -1 in the term bdi², so bdi² = -bd.
Step 3: Combine the real parts (ac - bd) and the imaginary parts (ad + bc)i.
Final Answer: (a + bi)(c + di) = (ac - bd) + (ad + bc)i.

Multiplying Complex Numbers

QUESTION

Divide (2 - 5i) by (1 - 6i) and simplify the expression.

STEP-BY-STEP ANSWER:

Step 1: Write the division as (2 - 5i) / (1 - 6i).
Step 2: Multiply numerator and denominator by the conjugate of the denominator: (1 + 6i).
Step 3: The expression becomes ((2 - 5i)(1 + 6i)) / ((1 - 6i)(1 + 6i)).
Step 4: Compute the denominator: (1 - 6i)(1 + 6i) = 1 + 6i - 6i - 36i² = 1 + 36 = 37.
Step 5: Expand the numerator: (2 - 5i)(1 + 6i) = 2 + 12i - 5i - 30i² = 2 + 7i + 30 (since -30i² = 30).
Step 6: Combine terms: numerator becomes 32 + 7i.
Final Answer: (2 - 5i) / (1 - 6i) = (32 + 7i)/37, or expressed as 32/37 + (7/37)i.

Dividing Complex Numbers Using Conjugates

QUESTION

Simplify i^n by expressing it in terms of i, -1, or 1.

STEP-BY-STEP ANSWER:

Step 1: Recall that the powers of i cycle every four terms: i, -1, -i, 1, ...
Step 2: Divide the exponent n by 4 and find the remainder r.
Step 3: Use the remainder to determine: if r = 0 then i^n = 1; if r = 1 then i^n = i; if r = 2 then i^n = -1; if r = 3 then i^n = -i.
Final Answer: i^n simplifies based on the remainder when n is divided by 4.

Simplifying Powers of i

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Common Mistakes

  • Failing to replace i² with -1 during multiplication or simplification.
  • Incorrectly adding or subtracting real and imaginary parts; they must be combined separately.
  • Overlooking the cyclical nature of powers of i, leading to errors when simplifying i^n.
  • Not using the complex conjugate properly when dividing complex numbers, which can result in a non-standard form answer.